Rings, Modules, and Closure Operations (Springer Monographs in Mathematics) 3030244008, 9783030244002

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Springer Monographs in Mathematics

Jesse Elliott

Rings, Modules, and Closure Operations

Springer Monographs in Mathematics Editors-in-Chief Isabelle Gallagher, Paris, France Minhyong Kim, Oxford, UK Series Editors Sheldon Axler, San Francisco, USA Mark Braverman, Princeton, USA Maria Chudnovsky, Princeton, USA Tadahisa Funaki, Tokyo, Japan Sinan C. Güntürk, New York, USA Claude Le Bris, Marne la Vallée, France Pascal Massart, Orsay, France Alberto A. Pinto, Porto, Portugal Gabriella Pinzari, Padova, Italy Ken Ribet, Berkeley, USA René Schilling, Dresden, Germany Panagiotis Souganidis, Chicago, USA Endre Süli, Oxford, UK Shmuel Weinberger, Chicago, USA Boris Zilber, Oxford, UK

This series publishes advanced monographs giving well-written presentations of the “state-of-the-art” in fields of mathematical research that have acquired the maturity needed for such a treatment. They are sufficiently self-contained to be accessible to more than just the intimate specialists of the subject, and sufficiently comprehensive to remain valuable references for many years. Besides the current state of knowledge in its field, an SMM volume should ideally describe its relevance to and interaction with neighbouring fields of mathematics, and give pointers to future directions of research.

More information about this series at http://www.springer.com/series/3733

Jesse Elliott

Rings, Modules, and Closure Operations

123

Jesse Elliott Department of Mathematics California State University, Channel Islands Camarillo, CA, USA

ISSN 1439-7382 ISSN 2196-9922 (electronic) Springer Monographs in Mathematics ISBN 978-3-030-24400-2 ISBN 978-3-030-24401-9 (eBook) https://doi.org/10.1007/978-3-030-24401-9 © Springer Nature Switzerland AG 2019, corrected publication 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Contents

Preface . . . . . . . . . . . . . . . Overview . . . . . . . . . . . Conventions. . . . . . . . . List of Symbols . . . . . . Outline . . . . . . . . . . . . Dependence Chart . . . . Acknowledgements . . .

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0 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.1 Algebraic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.2 Ordered Algebraic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Semistar Operations on Commutative Rings . . . . . . . . . . 2.1 Total Quotient Rings and Kaplansky Fractional Ideals 2.2 Dedekind Rings, Prüfer Rings, and Invertibility . . . . . 2.3 Integral Closure and Complete Integral Closure . . . . . 2.4 Semistar Operations . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 H-Dedekind Rings and H-Prüfer Rings . . . . . . . . . . . 2.6 TV, TW, H-Noetherian, and Related Conditions . . . . 2.7 H-Marot Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 H-PIRs and H-Bézout Rings . . . . . . . . . . . . . . . . . . . 2.9 r-UFRs, Factorial Rings, and UFDs . . . . . . . . . . . . .

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63 63 73 82 93 104 114 123 128 137

1 Introductory Survey of Multiplicative Ideal Theory 1.1 Integral Domains . . . . . . . . . . . . . . . . . . . . . . 1.2 Some Examples and Counterexamples . . . . . . . 1.3 A Preview of Rings with Zerodivisors . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.10 Weak Counterparts of H-Dedekind Rings and H-Prüfer Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 2.11 Further Characterizations of H-Dedekind and H-Prüfer Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 3 Semistar Operations on Commutative Rings: Local Methods . 3.1 Valuation Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 H-Prime Ideals and Applications to H-Prüfer Domains . . . . 3.3 Associated Primes, Strong Krull Primes, and t-Primes . . . . 3.4 Krull Domains and Dedekind Domains . . . . . . . . . . . . . . . 3.5 Large Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Valuations and Valuation Pairs . . . . . . . . . . . . . . . . . . . . . 3.7 Krull Rings, Dedekind Rings, and r-UFRs . . . . . . . . . . . . 3.8 PVMRs, Prüfer Rings, and H-Prüfer Rings . . . . . . . . . . . . 3.9 Integrally Closed Rings, H-Prüfer Rings, and Content Ideals 3.10 Valuation Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Spectral Semistar Operations and H-Prüfer-Like Conditions Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Semiprime, Star, and Semistar Operations on Commutative Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Star Operations and Fractional and Nonunital Star Operations 4.2 Nonunital Semistar Operations . . . . . . . . . . . . . . . . . . . . . . 4.3 Semiprime Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Integral Closure of Ideals and Submodules . . . . . . . . . . . . . 4.5 Complete Integral Closure and Related Operations on Ideals 4.6 Tight Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Noncommutative Rings and Closure Operations on Submodules 5.1 Chapter Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Closure Operations on Ideals and Submodules . . . . . . . . . . 5.3 Stable and Finite Type Closure Operations . . . . . . . . . . . . . 5.4 Divisorial and Codivisorial Semiprime Operations . . . . . . . . 5.5 Preradicals, Pretorsion Theories, and Systems of Closure Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Torsion Theories and Left Exact Preradicals . . . . . . . . . . . . 5.7 Semistar Operations on Algebras . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Closure Operations and Nuclei . . 6.1 Ordered Magmas and Nuclei 6.2 The Poset of All Nuclei . . . . 6.3 Divisorial Nuclei . . . . . . . . .

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6.4 Finitary Nuclei . . . . . . . . 6.5 Associated Finitary Nuclei 6.6 Stable Nuclei . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . .

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References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

Preface

Overview This book is a self-contained and systematic exposition of the various applications of closure operations to both commutative and noncommutative algebra. Its primary goal is to apply closure operations to the study of rings and modules, with an emphasis on commutative rings and ideals. The text can be read as a monograph or used as a textbook for a graduate course or special topics course, primarily by students of or researchers in commutative algebra, ring theory, or multiplicative ideal theory. Historically, closure operations in commutative algebra have fallen into two main categories. One includes operations like divisorial closure and t-closure of fractional ideals that are typically applied to the study of integral domains, and the other includes operations like integral closure and tight closure of ideals and submodules that are typically applied to the study of Noetherian rings. Although much work has been done to unify these two approaches, this is the first text to take on such a project. The text consists of six chapters, along with a preliminary chapter that provides a comprehensive list of definitions and results from abstract algebra that are assumed throughout the text. Chapter 1 is an introductory survey of multiplicative ideal theory. Chapters 2 and 3 adapt an extension of Krull’s technique of star operations to commutative rings and use this to generalize the theories of Dedekind domains, unique factorization domains, Krull domains, and Prüfer domains to much larger classes of rings. No prior knowledge of these classes of domains, or of star operations, is assumed. While closure operations occupy a central role in this book, several other important tools are developed in Chapters 2 and 3, including integral closure, complete integral closure, weak Bourbaki associated primes, strong Krull primes, large localization, and Manis valuations and valuation pairs. Chapter 4 develops the theory of semiprime operations on commutative rings, with particular attention to integral closure, complete integral closure, and tight closure of ideals. Chapter 5 then generalizes the theories of semiprime and semistar operations to

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modules over noncommutative rings, relating the resulting theory to the theories of radicals, torsion theories, and noncommutative localization. Finally, Chapter 6 develops the general theory of nuclei on ordered magmas. A more detailed outline of the contents is provided in the Outline section of this Preface. The first four chapters focus on various generalizations of ideals in commutative rings. Although modules are far more general than ideals, ideals possess a natural operation of multiplication that has no precise analogue for modules, the closest analogue being the tensor product. Ideal theory and its generalizations are thus commonly called multiplicative ideal theory to highlight this difference. There have been several influential books in multiplicative ideal theory, most notably (and in chronological order) Krull’s Idealtheorie [124], Jaffard’s Les Systèms d’Idéaux [111], Gilmer’s Multiplicative Ideal Theory [78, 79], Larsens’s and McCarthy’s Multiplicative Theory of Ideals [130], Huckaba’s Commutative Rings with Zero Divisors [107], Halter-Koch’s Ideal Systems: An Introduction to Multiplicative Ideal Theory [95], Fontana’s, Huckaba’s, and Papick’s Prüfer domains [63], Knebusch’s and Zhang’s Manis Valuations and Prüfer Extensions I: A New Chapter in Commutative Algebra [121], and Knebusch’s and Kaiser’s Manis Valuations and Prüfer Extensions II [122]. Many of the results proved in this text are drawn from or are inspired by the extensive literature on multiplicative ideal theory, which includes not only the nine books listed above but also the extensive journal literature. I have made an effort whenever possible to cite the original sources; however, undoubtedly I have failed to do so in many cases. Most of the results in Chapters 2 and 3 are known already for integral domains. The main challenge in writing those two chapters was to generalize all such results to commutative rings from first principles, without relying on any external sources or assuming unnecessarily restrictive hypotheses. Some of less trivial results in Chapters 2 and 3 that are new include Theorems 2.9.23, 2.9.25, 2.10.14, 2.11.4, 3.2.3, 3.3.23, 3.5.14, 3.6.22, 3.7.9, 3.8.4, 3.8.8, 3.9.5, 3.9.8, 3.11.3, 3.11.5, and 3.11.10, and Propositions 2.4.23, 2.4.30, 2.4.34, 2.4.40, 2.7.18, 2.8.7, 3.3.2, 3.5.8, and 3.5.10. The majority of the results in Chapters 4–6 are new. Chapter 1 is a primer on PIDs, Dedekind domains, UFDs, Krull domains, Bézout domains, Prüfer domains, GCD domains, PVMDs, completely integrally closed domains, v-domains, and integrally closed domains, among many other classes of domains, along with various generalizations to rings with zerodivisors. The chapter is a concise but in-depth survey of multiplicative ideal theory and the existing literature. Its aim is to inform the reader of some of the major results of multiplicative ideal theory. To achieve this goal most efficiently, proofs of all of the results are postponed until Chapters 2 and 3. Section 1.2 provides many counterexamples, with ample supply coming from rings lying between D[X] and K[X], where D is an integral domain with quotient field K. The section provides a systematic way of organizing counterexamples, making extensive use of tables and implication diagrams. Section 1.3 provides a brief preview of the various generalizations to rings with zerodivisors that are discussed in Chapters 2 and 3. Chapter 1 aims to provide the reader with some necessary background for reading research articles in multiplicative ideal theory. The reader who assimilates Chapters 2 and 3,

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on the other hand, should be fully prepared to conduct research in multiplicative ideal theory and may enjoy trying to tackle any of the several open problems provided. Since its primary intended audience is researchers in the fields of commutative algebra, noncommutative algebra, and order theory, this book is first and foremost a research monograph. However, the text is equipped with exercises (organized within each section by approximate level of difficulty) and is pedagogically friendly, and thus it has also been designed to be used as a graduate-level textbook. For example, in Fall 2016 and in Spring 2019, I based a Master’s-level algebra course at Cal State Channel Islands on Chapters 1–3 and 5–7 of Atiyah’s and MacDonald’s Introduction to Commutative Algebra [17] and on drafts of Chapters 2 and 3. A concerted effort has been made to arrange the results in this text in such a way that many of them possess a trivial verification from prior results and thus may appear to be self-evident. In those cases, the proofs are consigned to the exercises. Decisions have been made that occasionally lie in the interest of pedagogy at the expense of conciseness and expedience. For example, definitions and results are often stated for integral domains, then later generalized to commutative rings, and then later generalized, whenever possible, to noncommutative rings. Much effort in this book has also been made to avoid restrictive hypotheses as typified by the domain, Noetherian, and Marot hypotheses. A commutative ring R is said to be Marot if every ideal of R containing a non-zerodivisor is generated by non-zerodivisors. All integral domains and Noetherian rings are Marot, and many important results on integral domains and Noetherian rings generalize more readily to Marot rings than to commutative rings in general. Because of this, it is very tempting in multiplicative ideal theory to assume the Marot hypothesis, as is done in [10, 11, 107, 118, 144], for example. Results in this book that require the domain, Noetherian, Marot, or related hypotheses are almost always subsidiary to more general results that do not. The foundations of commutative algebra and multiplicative ideal theory trace back to the work of E. Noether and W. Krull. In 1921, Noether provided the first axiomatization of general commutative rings and ideals [157], and, six years later, initiated the study of Dedekind domains [158], which form the basis of modern treatments of algebraic number theory. The class of Dedekind domains is the intersection of four important and well-studied classes of (commutative) rings: the integral domains, the Noetherian rings, the rings of dimension at most one, and the integrally closed rings. Most of the rings that arise in algebraic geometry are Noetherian, and most of those arising in algebraic number theory are of dimension at most one. Integrally closed rings are also ubiquitous, if only because the integral closure of any ring is integrally closed. Chapters 1 through 3 of this book undertake a study of various classes of integrally closed rings that behave in many ways like the Dedekind domains, including domains that are more general, even, than the Krull domains and the Prüfer domains, such as the Prüfer v-multiplication domains, or PVMDs. Nevertheless, it is important to emphasize that all of the theory developed in Chapters 1–3 is inspired by the theory of Dedekind domains. The three chapters are therefore appropriate material for a graduate-level course in commutative algebra,

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particularly those with a view toward algebraic number theory or multiplicative ideal theory. In the 1920s, Krull also developed the tools of localization and completion, which are now essential to modern commutative algebra, algebraic number theory, and algebraic geometry. A less widely known tool utilized in this text is also due to Krull. In his influential 1935 book, Idealtheorie, Krull defined a ′-operation to be an operation I
! I 0 , on the set of all nonzero fractional ideals of an integral domain, satisfying six axioms, to which he added two more in a paper the following year [124, p. 118; 125]. Following Gilmer’s 1968 book, Multiplicative Ideal Theory [78], ′-operations came to be known as star operations, defined as follows. A fractional ideal of an integral domain D with quotient field K is a D-submodule I of K such that aI  D for some nonzero a 2 D. The set Freg ðDÞ of all nonzero fractional ideals of D is a partially ordered commutative monoid under the operation of multiplication, where the product IJ is the D-submodule of K generated by the set fab : a 2 I; b 2 Jg, and where the partial ordering is the subset relation. (“reg” stands for “regular,” which for a domain is equivalent to “nonzero”) A star operation on D is a self-map  : Freg ðDÞ
! Freg ðDÞ,  : I
! I  , of the ordered monoid Freg ðDÞ that is (1) (2) (3) (4) (5)

order-preserving: I  J implies I   J  , expansive: I  I  , idempotent: ðI  Þ ¼ I  , sub-multiplicative: I  J   ðIJÞ , and unital: D ¼ D,

where each of these axioms holds for all I; J 2 Freg ðDÞ. The first three star operation axioms come from the well-known axioms for a closure operation on a partially ordered set. The fourth axiom relates the operation to multiplication in the ordered monoid Freg ðDÞ, and that axiom combined with the first three closure axioms state that the operation  is a nucleus on the ordered monoid Freg ðDÞ. The theory of star operations is a substantial branch of multiplicative ideal theory, and for good reason: star operations are useful for generalizing the standard results in commutative algebra on Dedekind domains, for example, to rings that are still closely connected to number theory but may be non-Dedekind and even non-Noetherian, such as the unique factorization domains (e.g., Z½X1 ; X2 ; . . .), the pffiffiffiffiffiffiffi Krull domains (e.g., Z½
5½X1 ; X2 ; . . .), the Bézout domains (e.g., the ring of all algebraic integers), and the Prüfer domains (e.g., the ring IntðZÞ of all polynomial functions from Z to Z with rational coefficients). This is typically accomplished for a given class of domains by requiring a familiar ideal-theoretic characterization of that class to hold only up to -closure. In essence, this amounts to generalizing the theory of divisors on an integral domain [23, Chapter VII] to the setting of such closure operations. A striking example is provided by the operation of t-closure, or the t-operation, which is a quintessential example of a star operation and was first introduced by Jaffard in 1960 [111]. The operation of t-closure on a domain D is uniquely

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characterized as the largest star operation on D that is continuous when the poset Freg ðDÞ is endowed with the Scott topology. It can be used, for example, to generalize the definition of a principal ideal domain (PID) to an equivalent characterization of the unique factorization domains (UFDs). Indeed, an integral domain D is a UFD if and only if it is a t-PID, that is, if and only if every t-closed ideal of D is principal. This can be deduced immediately from the facts that a UFD is equivalently a Krull domain with trivial divisor class group and that a Krull domain is equivalently an integral domain in which every nonzero ideal is invertible up to t-closure. In Chapter 2, where star operations and the t-operation are generalized to any commutative ring, I provide a proof that generalizes to rings with or without zerodivisors: a ring R (commutative with identity) is factorial in the sense of [10, 11, 118] if the t-closure of any ideal of R generated by non-zerodivisors is principal, and, furthermore, every regular non-unit of a ring R is a product of primes (unique up to rearrangement and associates) if and only if the t-closure of any ideal of R containing a non-zerodivisor is principal. I show in Section 2.9 that the latter class of rings is equivalently the class of Krull rings with trivial divisor class group and therefore in my view provides the most natural notion of a “unique factorization ring with zerodivisors.” We call such rings r-unique factorization rings, r-UFRs, or t-PIRs. We also show in Chapter 3 that one can use the t-operation to uniquely characterize the (Manis) valuation rings: a ring R is a valuation ring if and only if R is a PVMR that has at most one ideal that is maximal among the proper t-closed ideals of R, if and only if R is a PVMR for which the set of all regular t-closed ideals of R is totally ordered: see Corollary 3.8.6 and Proposition 3.10.7. Another nontrivial result in Chapter 3, which was proved for integral domains by Houston and Zafrullah [104], is that any TV ring is of finite t-character. This result is subsequently used to streamline the proofs of various well-known characterizations of Krull domains, as well as to generalize these characterizations to characterizations of Krull rings with zerodivisors. The main goal of Chapters 2 and 3 is to generalize large portions of the theories of star and semistar operations on integral domains to commutative rings with zerodivisors. To this end, we define a semistar operation on a commutative ring R to be a nucleus on the ordered monoid of K(R) all R-submodules of the total quotient ring of R, that is, it is a closure operation H on the poset K(R) such that I H J H  ðIJÞH for all I; J 2 KðRÞ. The total quotient ring TðRÞ of a commutative ring R and the regular ideals of R both figure very prominently in the theories of semistar and star operations. One may obtain natural alternatives to these theories by replacing T(R) with larger quotient rings, for example, the ring of finite fractions Q0 ðRÞ  TðRÞ of R, where the semiregular R-submodules of Q0 ðRÞ play the role of the regular R-submodules of T(R), or the complete ring of quotients QðRÞ  Q0 ðRÞ of R, where the dense R-submodules of Q(R) play the role of the regular R-submodules of T(R). Nevertheless, the corresponding theories one obtains in this manner are in fact less general, rather than more general, than the theory of semistar operations. For an example of this phenomenon, note that a ring R is said to be

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Dedekind (resp., Q0 -Dedekind, Q-Dedekind) if every regular ideal (resp., every semiregular ideal, every dense ideal) of R is invertible (resp., Q0 -invertible, Q-invertible), and one has the irreversible implications Q-Dedekind ) Q0 -Dedekind ) Dedekind. One studies such stronger conditions in order to wield greater control over the zerodivisors of a commutative ring, but this lies beyond the scope of this book. It could be said more specifically that the main goal of Chapters 2 and 3 is to study the eight classes of integrally closed rings in the implication lattice below, along with their generalizations to arbitrary semistar operations. r-PIR r-UFR

Dedekind

r-Bézout

Krull

r-GCD

Prüfer

PVMR

Each of the classes of rings above can be characterized by the t-closure semistar operation in several nontrivial and surprising ways. Moreover, using the t-operation, their respective theories to a large extent can be developed along the lines of that of the Dedekind domains. The theory of star and semistar operations is quite extensive but is constrained in a few ways that prevent its wider embrace. First, the greater part of the existing literature does not treat rings with zerodivisors, while many of the closure operations in commutative algebra, such as integral closure and tight closure, are on ideals of commutative rings rather than on fractional ideals of integral domains. Second, in the commutative algebra, number theory, and algebraic geometry communities, non-Noetherian rings are often unfairly assumed to be irrelevant or pathological (with many notable exceptions, such as perfectoid algebras and rings of Witt vectors). Third, there are few up-to-date treatments of the fundamentals of the theory and many of the more central results and their proofs remain scattered throughout the literature, which makes it difficult to supply complete proofs of many facts that are taken for granted by the multiplicative ideal theory community but that are not obvious to the nonexpert. This book aims to overcome these three obstacles by generalizing the theories of semiprime, star, and semistar operations to commutative rings, with a view toward rings that may be non-Noetherian but still behave much like Dedekind domains, and providing a self-contained exposition of these theories that assumes only a rudimentary knowledge of ring theory, namely: rings, ideals, prime and maximal ideals, fields, integral domains, localization, modules, exact sequences, flatness, chain conditions, and Krull dimension. A more than sufficient background for Chapters 1–4 can be obtained from Chapters 1–3 and 6–7 of Atiyah’s and MacDonald’s Introduction to Commutative Algebra [17], Chapters 1–4 of Zariski’s

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and Samuel’s Commutative Algebra [191], Chapters 1–3 of Matsumura’s Commutative Ring Theory [145], or Chapters 1–3 of Eisenbud’s Commutative Algebra: with a View Toward Algebraic Geometry [43]. We also assume familiarity with the fundamentals of order theory, especially in Chapter 6. Chapter 0 provides a near exhaustive list of definitions and results from algebra and order theory that are assumed throughout the book. Only in Sections 5.5–5.7 do we assume some further results on noncommutative rings and modules, all of which (including basic facts about preradicals, torsion theories, injective envelopes, Gabriel filters, and localization at Gabriel filters) can be found in [177, Chapters VI and IX]. Besides the star and semistar operations, another type of closure operation in commutative algebra is known as a semiprime operation. A semiprime operation on a commutative ring R is a nucleus on the ordered monoid IðRÞ of all ideals of R, that is, it is a closure operation s on the poset IðRÞ such that I s J s  ðIJÞs for all ideals I and J of R [163]. Semiprime operations abound in commutative algebra. Among the most important and well studied are radical, integral closure, tight closure, Frobenius closure, plus closure, solid closure, and I-saturation for any finitely generated ideal I [55]. Traditionally, semiprime operations have been employed mainly in the setting of Noetherian rings, while star and semistar operations have been employed mainly in the setting of integral domains. An important bridge between the two was found when, in [54], N. Epstein proved an equivalence between the poset of all standard finite type semiprime operations on a commutative ring R and the poset of all finite type semistar operations on R. This serves to situate the theory of finite type semistar operations under the rubric of semiprime operations, and it implies that many important semiprime operations, including integral closure, tight closure, and plus closure, are induced by appropriate semistar operations. All of this is discussed in Chapter 4. Chapter 5 generalizes the theories of semiprime and semistar operations to closure operations on submodules of modules over noncommutative rings. A primary tool we introduce in Section 5.2 is a natural Galois connection between the poset of all closure operations on ModR ðMÞ and the poset of all closure operations on ModR ðNÞ for any R-modules M and N, where R is any ring and ModR ðPÞ for any R-module P denotes the poset of all R-submodules of P. Taking M to be the left R-module R yields a method for lifting any closure operation on the left ideals of R to a closure operation on ModR ðNÞ for any R-module N. This provides a novel way of lifting semiprime operations, including radical, tight closure, and integral closure, to operations on submodules, and, as a corollary, leads to a characterization of the semiprime operations on any commutative ring R that are induced by some semistar operation on R. Sections 5.5–5.7 draw heavily from B. Stenström’s Rings of Quotients: An Introduction to Methods of Ring Theory [177]. An important technique used here is a noncommutative version of localization (with respect to a Gabriel filter). Related and equally important tools are radicals and torsion theories, first introduced, respectively, by Maranda in 1962 [142] and Dickson and Lambek in 1966 [39, 129]. A new result in Section 5.5 is that any “cohereditary” radical is completely

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determined by the semiprime operation it induces on left ideals. This allows us to prove the equivalence of the following categories, for any ring R: (1) the poset of all semiprime operations on the left ideals of R; (2) the category of all cohereditary radicals on the category of left R-modules; and (3) the category of all closure operations on submodules of left R-modules that are minimal in a certain sense. We prove in Section 5.6 the equivalence of the following categories for any ring R: (1) the poset of all stable semiprime operations on the left ideals of R; (2) the category of all left exact radicals on the category of left R-modules; (3) the category of all hereditary torsion theories on the category of left R-modules; and (4) the poset of all Gabriel filters (or Gabriel topologies) on R. (It is known that, over a commutative Noetherian ring, left exact radicals are equivalently abstract local cohomology functors [185] and therefore generalize local cohomology functors in the sense of [182, Section 4.6].) Thus, the theory of semiprime operations generalizes a sizable portion of the theories of radicals and torsion theories. In 2011, my understanding of closure operations evolved dramatically after I stumbled upon a book by K. Rosenthal [173], where I learned of the notion of nucleus on an ordered monoid. The most basic object on which a nucleus is defined is an ordered magma, which is a partially ordered set M together with a binary operation on M that (when written multiplicatively) satisfies x  x0 ; y  y0 ) x y  x0 y0 for all x; y; x0 ; y0 2 M. A nucleus on an ordered magma M is simply a closure operation H on the partially ordered set M such that xH yH  ðxyÞH for all x; y 2 M. The notion of a nucleus allows for a unified approach to various types of closure operations in commutative algebra. For example, a semiprime operation on a commutative ring R is a nucleus on the multiplicative lattice (or commutative unital quantale) of ideals of R; a star operation on R is a nucleus  on the ordered monoid of all regular fractional ideals of R, such that R ¼ R; and a semistar operation on R is a nucleus on the multiplicative lattice of all R-submodules of the total quotient ring of R. Each of these types of operations associated to a ring has a number of uses in ring theory, and it is worthwhile to generalize results on them to the setting of nuclei. Chapter 6 of this text generalizes sizable portions of the theories of semiprime operations, star operations, semistar operations, and Halter-Koch’s ideal systems and module systems to nuclei on a coherent residuated lattice-ordered semiquantale. In my view, it demystifies many of the standard facts about star, semistar, and semiprime operations, and more importantly, it leads to several new results about them. The theory of nuclei is saved for this final chapter so that the abstractions discussed in the chapter are fully motivated by the chapters preceding it. My hope is that the chapter will serve to widen the audience on multiplicative ideal theory to those interested more broadly in closure operations, order theory, and the theories of ordered semigroups, quantales, multiplicative lattices, locales, and frames, as well as to inform algebraists, both commutative and noncommutative, both associative and nonassociative, about such ubiquitous structures. Much work remains to generalize multiplicative ideal theory to arbitrary rings. It is my hope that a sequel of this text might be written, potentially with many

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contributors, to address the many goals of multiplicative ideal theory that have yet to be accomplished. Possible topics include the following. (1) Solve any of the many open problems included in the text or in the extensive literature on multiplicative ideal theory, for example, as listed in [29]. (2) Generalize to rings with zerodivisors known results on the Nagata ring, Kronecker function ring, and e.a.b. semistar operation associated to a semistar operation on an integral domain. (3) Study how various ring-theoretic properties carry over to ring extensions, including polynomial rings and various overrings, such as Nagata rings and Kronecker function rings, as well as the integral closure of a ring in a finite or integral extension of its total quotient ring. (4) Generalize the theory of Prüfer extensions [121, 122] to the setting of closure operations (e.g., generalize Chapters 2 and 3 of this book to arbitrary extensions of commutative rings). (5) Prove analogues for the complete ring of quotients Q(R) and the ring of finite fractions Q0 ðRÞ of any results in this book that use the total quotient ring T(R) of a commutative ring R. (6) Generalize the theories of Dedekind rings, Prüfer rings, Krull rings, and PVMRs to noncommutative rings. (7) Further generalize the theories of semistar operations and w-envelopes to noncommutative rings. (8) Develop an algebro-geometric interpretation of the results in this book and of multiplicative ideal theory at large. Since Chevalley, Zariski, and Grothendieck, to name only a few of the pioneers of modern algebraic geometry, the elegance and beauty of commutative algebra has been somewhat overshadowed by its power, when equipped with homological methods, to form the bedrock of algebraic geometry. Yet, commutative algebra, in the tradition of Kummer, Dedekind, Noether, Krull, and Prüfer, among many others, is not just a tool to support algebraic geometry any more than mathematics is just a tool to support the natural sciences. The roots of commutative algebra lie in algebraic number theory, to which, among all of its many branches, multiplicative ideal theory is closest in spirit. It is my hope that this book, in addition to further advancing multiplicative ideal theory, will make more widely known the many and various uses of closure operations in the theory of rings and modules.

Conventions All rings and algebras are assumed unital and all ring homomorphisms are assumed to preserve unity. In Chapters 1–4, all rings and algebras are assumed commutative. If R is a ring, then we denote by R the group of units of R and by R the monoid R under multiplication. All R-modules are assumed to be left R-modules unless

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otherwise stated. Generally, the symbol “R” is reserved for a ring and “K,” if R is commutative, for its total quotient ring T(R), and the symbol “D” is reserved for an integral domain. The integral closure of a ring R in its total quotient ring is denoted R. Local rings are not assumed Noetherian, and fields are considered to be discrete valuation rings, Dedekind domains, unique factorization domains, etc. By the dimension of a ring, we always mean its Krull dimension. In a quotient R/I of a ring R by an ideal I, we occasionally blur the distinction between an element of R and an element of R/I. In this book, Z denotes the ring of integers, and Q, R, and C denote the fields of rational numbers, real numbers, and complex numbers, respectively. For any prime power q, we denote by Fq the finite field of cardinality q. We avail ourselves of standard abbreviations for various types of rings, such as the following. (1) “DVR” stands for “discrete valuation ring,” which means “discrete rank at most one valuation domain.” (2) “PID” stands for “principal ideal domain.” (3) “UFD” stands for “unique factorization domain.” (4) “PVMD” stands for “Prüfer v-multiplication domain.” We also use the following nonstandard abbreviations, especially within tables and implication diagrams. (5) “CIC” stands for “completely integrally closed.” (6) “IC” stands for “integrally closed.” (7) “f.g.” stands for “finitely generated.” This book contains many implication diagrams, such as the following. PID

UFD

Dedekind

Bézout

Krull

GCD

Prüfer

PVMD

The implication diagram above is to be interpreted as saying that any PID is a UFD, a Dedekind domain, and a Bézout domain; any UFD is a Krull domain and a GCD domain; and so on. It turns out that, in the implication diagram above, the intersection of any of the classes is the largest class in the diagram lying above them; for example, Krull & Prüfer , Dedekind, and UFD & Prüfer , PID. We summarize this by saying that the implication diagram is a full lattice of implications, or full implication lattice. We also say that the implication diagram is complete in the

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sense that none of the arrows is reversible. Almost all of the implication diagrams presented in this book are full implication lattices. In Chapters 2 and 3, all semistar operations are assumed to be unital. This convention is not followed in Chapters 4–6, where semistar operations are allowed to be nonunital. The reason for this convention in Chapters 2 and 3 is so that we may avoid incessant repetition of the word “unital.” S The union ‚2K X‚ of an indexed collection fX‚ : ‚ 2 Kg of sets X‚ is also S denoted by fX‚ : ‚ 2 Kg. More generally, if X is a set of sets, then the union S S X of the sets in X is denoted by X . This convention is particularly useful X2X when the particular symbol for the given indexing set K is too large to typeset S below the symbol. Similar conventions apply to intersections, direct products, etc. All definitions that are required in later sections are set aside as a numbered definition (e.g., Definition 2.1.1). Unnumbered definitions declared within the main dialogue or within a remark are either not required, or else are defined again, in later sections. A term that is being defined for the first time is written in boldface, while a term that hasn’t been defined yet or is being defined via a given cited reference is written in italics. For example, we might say, “Dedekind domains [17, Chapter 9; 145, Section 11] are a well-known and important class of integral domains,” versus, “a Dedekind domain is an integral domain that is Noetherian, integrally closed, and of dimension at most one.” A term that is imprecise or is as yet undefined may be written in quotation marks, or “scare quotes.” By convention, a word or phrase appearing in parentheses in some definition or theorem is meant to signify that the given definition or theorem holds whether or not that word or phrase is omitted. For example, if we write “a Dedekind ring is a ring in which every regular (fractional) ideal is invertible,” then this means that the word “fractional” may be included or not and the definition holds either way. Like many authors, I reserve the word “theorem” for a result that on its own is neither obvious nor possesses a trivial, easy, routine, or straightforward verification and is likely to be used in the proofs of other results. I use the word “proposition” for a result that is either obvious or else, given prior results, possesses a trivial, easy, routine, or straightforward verification, and often I will omit the proof or leave it as an exercise. Of course, what counts as trivial, easy, straightforward, clear, obvious, and the like, varies from person to person, so the labels are more indicative of the author’s attitude toward a given result than anyone else’s. Any numbered remark (e.g., Remark 2.2.23) can be overlooked or forgotten without any loss of continuity, but it may introduce terms or ideas that are necessary to complete some of the exercises. Open problems are included in the body of the text (e.g., Open Problem 3.8.20) and as exercises preceded by the symbol “ðÞ.” These are included to reveal areas where the theory as developed in the text is incomplete. Further conventions made within the text are labeled as such; specifically, these are Conventions 1.0.30, 2.1.27, 2.2.22, 2.4.6, 3.6.2, 4.0.24, 4.1.3, and 4.2.37.

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List of Symbols Symbol TðRÞ ModR ðTÞ; KðRÞ; IðRÞ ðI :T JÞ; I
1T ; I
1 FðRÞ; Freg ðRÞ; Kreg ðRÞ; Ireg ðRÞ R; R]  SemistarðRÞ d t; w Ht ; H; Hs ; Hw ðJÞ ðSÞ 0 0 H ’ H ;H H PrinðRÞ; InvðRÞ; ClðRÞ InvH ðRÞ; ClH ðRÞ InvH ðRÞ; ClH ðRÞ Rð þ ÞM gcdðSÞ; lcmðSÞ 0 H ’rg H ClHwk ðRÞ AssðMÞ; wAssðMÞ; sKðMÞ R½p ; ½IR½p T ; p ; 
1 ð1Þ T ; tT p ðIÞ pðnÞ cT ðf Þ; cðf Þ H½C ; HC SemistarðRÞ; USemistarðRÞ; RSemistarðRÞ sprimH ; H ; star H _ H st ; s; sw þ1 pclp pp bTR ; b; I redJ ðIÞ; J I Rreg ; Ro Il ðRÞ; Ir ðRÞ; IðRÞ R-Mod; Mod-R; ModR ðMÞ Ix; ðN :R XÞ; ðI nM NÞ 0 HdNe; H bMc dHe; bHc s Ht

vðJ Þ annl ðXÞ; annr ðXÞ Tr ; Fr Dr ; Cr Dr ; Cr rM ; rM Ht H

Definition 2.1.6 2.1.13 2.1.16 2.1.25 2.3.9 2.3.20 2.4.2 2.4.4 2.4.21 2.4.21, 4.2.7 2.4.26 2.4.29 2.4.35, 4.2.4 2.5.18 2.5.20 2.5.21 2.7.11 2.8.6 2.10.1 2.10.14 3.3.7 3.5.1 3.6.1 3.6.19 3.7.16 3.7.19 3.9.1 3.11.1, 4.2.28 4.1.2 4.1.5 4.2.3 4.3.11 4.3.18 4.3.29 4.3.31 4.4.1 4.4.4 4.5.1 5.2.2 5.2.2 5.2.3 5.2.26 5.2.31 5.3.26 5.3.33 5.4.5 5.4.7 5.5.2 5.5.23 5.5.24 5.5.27 6.5.3 6.6.5

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Outline The book is divided into six chapters. The preliminary chapter, Chapter 0, provides a comprehensive list of definitions and results from abstract algebra and order theory that are assumed throughout the text. It includes the definitions of groups, rings, fields, ideals, modules, localization, tensor products, and so on, most of which are contained in any graduate-level algebra text. It also contains definitions of various ordered structures and ordered algebraic structures (which are less widely known), such as posets, complete lattices, ordered monoids, and nuclei. Chapter 1 is a survey, without proofs, of the theory of star operations and its applications to the theory of integral domains. The entire chapter can be skipped without loss of continuity but is included to provide a concise overview of multiplicative ideal theory. (Instructors or readers who are pressed for time may elect to omit the chapter.) Unavoidably, the chapter introduces a lot of terminology, all of which are introduced again in Chapters 2 and 3 in the context of commutative rings. Section 1.2 provides, with references but without proofs, several examples and counterexamples among rings lying between D[X] and K[X], where D is an integral domain with quotient field K. Section 1.3 provides a brief preview of the various generalizations to rings with zerodivisors that are discussed in Chapters 2 and 3. Chapters 2 and 3 provide self-contained proofs of many of the results mentioned in the survey chapter, all generalized to rings with zerodivisors. Particular rings of interest include the Dedekind rings, factorial rings, r-UFRs, Krull rings, Prüfer rings, and PVMRs. The tools used are primarily ideal-theoretic and make heavy use of semistar operations. Chapter 2 restricts its attention to results whose proofs, with only a few exceptions, do not require local methods, e.g., the use of prime ideals, localization, completion, valuation domains, and DVRs. In this chapter, sizable portions of the theories of Dedekind domains, UFDs, and Krull domains are developed, and generalized to rings with zerodivisors, using just the basics of semistar operations, which supports the philosophy presented in the book that Krull’s technique of star operations is a powerful addition to the more standard techniques of commutative algebra. While Chapter 2 avoids local methods, Chapter 3 makes them its primary concern, focusing on results that require the tools of prime ideals, localization, large localization, valuation domains, (Manis) valuation rings, weak Bourbaki associated primes, and strong Krull primes (also known as Northcott attached primes). Besides prime ideals, localization, and modules, all of the tools used are defined and developed in this chapter.

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Chapter 4 studies various relationships between semistar operations and semiprime operations. One if its main goals is to show that semiprime operations such as integral closure and tight closure can be extended to unique semistar operations. An active area of research is to generalize ideal-theoretic closure operations to module-theoretic settings. This project is developed in Chapter 5, over noncommutative rings. Most notably, the chapter provides an equivalence between “cohereditary” radicals and semiprime operations and an equivalence between left exact radicals, hereditary torsion theories, stable semiprime operations, Gabriel filters, and Gabriel topologies. This serves in particular to situate semiprime operations in noncommutative ring theory under the general rubric of radicals. Chapter 6 generalizes various closure operations defined in the text to nuclei on ordered magmas satisfying various algebro-order-theoretic hypotheses. Although it is written to be independent of the other chapters, Chapter 6 contains results that demystify some of the results in Chapters 1–5, and occasionally a proof of a result in those chapters will require the reader to supply a proof or else to consult a particular result in Chapter 6.

Dependence Chart On the next page is a dependence chart for the various sections of the book. As can be seen from the diagram, Sections 1.2, 1.3, 2.11, 3.4, 3.9, 4.2, 4.6, 5.6, 5.7, and 6.6 are not used in any subsequent sections and therefore may be skipped without loss of continuity.

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1.3

xxiii

1.1

2.1

5.1

6.1

1.2

2.2

5.2

6.2

2.3

5.3

6.3

2.4

5.4

5.5

6.5

5.7

5.6

6.6

2.5

3.1

4.1

2.6

3.2

4.3

2.7

3.3

.

4.4

2.8

3.4

3.5

4.5

2.9

3.6

4.6

2.10

3.7

2.11

3.8

3.10

3.9

3.11

6.4

4.2

Acknowledgements I am indebted to my former Ph.D. advisor, Hendrik W. Lenstra Jr., for instilling in me a deeper respect and appreciation for generality and abstraction. I would also like to thank Paul-Jean Cahen, Jean-Luc Chabert, Marco Fontana, Evan Houston, Sarah Glaz, Salah Kabbaj, Tom Lucas, Bruce Olberding, and Muhammad Zafrullah for being once unwitting mentors and now superb colleagues. I have on occasion emailed them questions that have arisen in my work, and they were always patient with me and very generous with their replies. I would also like to thank Hwankoo Kim, who during a sabbatical at Cal State Channel Islands exposed me to the theory

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of radicals and provided very helpful dialogue and feedback on early drafts of Chapter 5, and Neil Epstein, who provided detailed comments on a draft of Chapters 4 and 5. I would also give a special thanks to Muhammad Zafrullah, whom I have met in person only once, very early in my career at a commutative algebra conference in Cortona in 2003, for his comments on early drafts of this book and for his generous sharing of information, particularly relating to counterexamples, and more broadly for his generous communications with me at various times throughout the last sixteen years or so on the various topics discussed here. It was at that conference in 2003 that I first heard of star operations and posed the question, to him and to myself, “Why do people care about star operations?” This book is my answer, sixteen years later. Finally, I would like to thank in advance the readers of this text who might be so generous as to provide feedback, corrections, or solutions to any of the unsolved problems posed herein. All such communications will be cited in any subsequent editions or in a possible sequel of this text. Camarillo, CA, USA

Jesse Elliott

The original version of the book was revised: the belated corrections have been incorporated throughout the book. The correction to this book is available at https:// doi.org/10.1007/978-3-030-24401-9_7.

Chapter 0

Preliminaries

In this brief chapter, we summarize some basic terminology and results that are assumed throughout the book.

0.1 Algebraic Structures In this section, we very briefly summarize the definitions from abstract algebra with which we assume familiarity. A more than sufficient background can be obtained from Chapters 1–7 of Atiyah’s and MacDonald’s Introduction to Commutative Algebra [17], Chapters 1–4 of Zariski’s and Samuel’s Commutative Algebra [191], Chapters 1–3 of Matsumura’s Commutative Ring Theory [145], or Chapters 1–4 of Eisenbud’s Commutative Algebra: with a View Toward Algebraic Geometry [43]. First, we provide definitions of some of the most important types of algebraic structures in mathematics. Definition 0.1.1 (Algebraic structures). (1) A binary operation on a set S is a function from S × S to S. If ∗ is a binary operation on a set S, then we write a ∗ b for ∗(a, b), for all a, b ∈ S. (2) Let ∗ be a binary operation on a set S. (a) ∗ is associative if (a ∗ b) ∗ c = a ∗ (b ∗ c) for all a, b, c ∈ S. (b) ∗ is commutative if a ∗ b = b ∗ a for all a, b ∈ S. (c) ∗ is unital if there exists a 1 ∈ S such that 1 ∗ a = a = a ∗ 1 for all a ∈ S. (3) A magma is a set M together with a binary ∗ operation on M (written multiplicatively, so that ab = a ∗ b for all a, b ∈ M, unless specified otherwise). (4) A magma M is associative (resp., commutative, unital) if the operation on M is associative (resp., commutative, unital). (5) A semigroup is an associative magma. © Springer Nature Switzerland AG 2019, corrected publication 2020 J. Elliott, Rings, Modules, and Closure Operations, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-24401-9_0

1

2

0 Preliminaries

(6) A monoid is a unital semigroup. (7) Let M be a monoid. (a) a ∈ M is invertible if there exists a b ∈ M such that ab = 1 = ba. (b) A group is a monoid in which every element is invertible. (c) The set M × of all invertible elements of M is a group. (d) An abelian group is a commutative group. (8) A field is a set K together with two binary operations + and · on K such that (1) K , + is an abelian group, (2) K − {0}, · is an abelian group, where 0 is the identity element of the abelian group K , +, and (3) a · (b + c) = (a · b) + (a · c) for all a, b, c ∈ K . (9) A (unital) ring is a set R together with two binary operations + and · on R such that (1) R, + is an abelian group, (2) R, · is a monoid, and (3) a · (b + c) = (a · b) + (a · c) and (b + c) · a = (b · a) + (c · a) for all a, b, c ∈ R. (10) Let R be a ring. (a) R is commutative if the operation · on R is commutative. (b) R is trivial if R = {0}, or equivalently if 0 = 1 in R. (c) R is nontrivial if R is not trivial. (d) a ∈ R is a unit of R if a is invertible in the monoid R, ·, that is, if ab = 1 = ba for some b ∈ R. (e) a ∈ R is a zerodivisor of R if ab = 0 or ba = 0 for some nonzero b ∈ R. (f) a ∈ R is a non-zerodivisor of R, or a regular element of R, if a is not a zerodivisor of R. If R is commutative, then the set R reg of all regular elements of R is a monoid under multiplication. Definition 0.1.2 (Substructures). (1) A submagma (resp., subsemigroup) of a magma (resp., semigroup) M is a nonempty subset of M that is closed under ·, that is, a nonempty subset N of M such that ab ∈ N for all a, b ∈ N . (2) A unital submagma of a unital magma M is a submagma of the magma M containing the identity element of M. (3) A submonoid of a monoid M is a unital submagma of M. (4) A subgroup of a group G is a submonoid of G that is closed under inverses, that is, a submonoid H of G such that a −1 ∈ H for all a ∈ H (or, equivalently, such that H × = H ). (5) A (unital) subring S of ring R is a subset S of R that is a subgroup of R, + and a submonoid of R, ·; equivalently, it is a subset of R that contains the identity elements 0 and 1 of R and is closed under +, −, and ·. (6) If X is a subset of a magma (resp., unital magma, semigroup, monoid, group) M, then there is a smallest submagma (resp., unital submagma, subsemigroup, submonoid, subgroup) of M containing X , called the submagma (resp., unital submagma, subsemigroup, submonoid, subgroup) of M generated by X . (7) If X is a subset of a ring R, then there is a smallest subring of R containing X , called the subring of R generated by X .

0.1 Algebraic Structures

3

In mathematics, we care not only about mathematical structures and their substructures, but also about maps between structures that “preserve” or “respect” their structure, which are called “morphisms.” Definition 0.1.3 (Morphisms). (1) A magma homomorphism is a function f : M −→ N , where M and N are magmas, such that f (ab) = f (a) f (b) for all a, b ∈ M. If M and N are unital, then f is unital if f (1) = 1. (2) A semigroup homomorphism is a magma homomorphism f : M → N , where M and N are semigroups. (3) A group homomorphism (resp., monoid homomorphism) is unital magma homomorphism f : M −→ N , where M and N are groups (resp., monoids). (4) A ring homomorphism is a function f : R −→ S, where R and S are rings, such that f is a group homomorphism from R, + to S, + and f is a monoid homomorphism from R, · to S, ·. (5) A magma isomorphism (resp., group isomorphism, semigroup isomorphism monoid isomorphism, ring isomorphism) is a magma homomorphism (resp., group homomorphism, semigroup isomorphism, monoid homomorphism, ring homomorphism) that is bijective (in which case its inverse function is also a morphism of the appropriate type). Definition 0.1.4 (Abelian groups). Let G be an abelian group, written additively. (1) Let N be a subgroup of G. Then G/N denotes the set {g + N : g ∈ G} of all cosets g = g + N = {g + n : n ∈ N } of N in G. The set G/N is an abelian group under the (well-defined) operation + defined by g + h = g + h for all g, h ∈ G. Equivalently, the set G/N has the unique structure of a group so that the map G −→ G/N given by g −→ g is a group homomorphism. The group G/N is called the quotient of G by N . (2) Let f : G −→ H be a homomorphism of abelian groups. The kernel of f is the subgroup ker f = {g ∈ G : f (g) = 0} of G. The image of f is the subgroup im f = { f (g) : g ∈ G} of H . (3) The first isomorphism theorem for abelian groups states that if f : G −→ H is any homomorphism of abelian groups, then there is a group isomorphism f : G/ ker f −→ im f such that f (g + ker f ) = f (g) for all g ∈ G. Definition 0.1.5 (Integral domains). (1) An (integral) domain is a ring that is a subring of some field. Equivalently, an integral domain is a nontrivial commutative ring that has no nonzero zerodivisors. (2) A field is equivalently a nontrivial commutative ring in which every nonzero element is a unit. (3) The quotient field, or field of fractions, of an integral domain D is a field K containing D as a subring that is minimal in the sense that, if L is any field containing D, then there exists a unique ring homomorphism K −→ L that is the identity on D. The quotient field of D is unique up to a unique isomorphism that is the identity on D.

4

0 Preliminaries

(4) Two elements a and b of an integral domain D are associate if a = ub for some unit u of D. A nonzero non-unit p of D is irreducible if p = ab implies that either a or b is a unit of D. (5) A ring D is a unique factorization domain, or UFD, if (1) D is an integral domain, (2) every nonzero non-unit of D can be written as a product p1 p2 · · · pk of irreducible elements pi of D, and (3) such factorizations are unique in the sense that, if p1 p2 · · · pk = q1 q2 · · · ql , where the pi and q j are irreducible, then k = l and, after an appropriate reordering of the q j , each pi is associate to qi , for all i. Ideals, including prime ideals and maximal ideals, are an extremely useful tool for studying commutative rings. Definition 0.1.6 (Ideals). Let R be a commutative ring. (1) An ideal I of R is a subgroup I of R, + such that ra ∈ I for all a ∈ I and all r ∈ R. (2) Let I be an ideal of R. Then R/I denotes the set {r + I : r ∈ R} of all cosets r = r + I = {r + a : a ∈ I } of I in R. The set R/I is a ring under the (welldefined) operations + and · defined by r + s = r + s and r · s = r · s for all r, s ∈ R. Equivalently, the set R/I has the unique structure of a ring so that the map R −→ R/I given by r −→ r is a ring homomorphism. The ring R/I is called the quotient of R by I . (3) Let f : R −→ S be a homomorphism of commutative rings. The kernel of f is the ideal ker f = {a ∈ R : f (a) = 0} of R. The image of f is the subring im f = { f (a) : a ∈ R} of S. (4) The first isomorphism theorem for commutative rings states that if f : R −→ S is any homomorphism of commutative rings, then there is a unique ring isomorphism f : R/ ker f −→ im f such that f (a + ker f ) = f (a) for all a ∈ R. (5) If X is a subset of R, then there is a smallest ideal of R containing X , called the nideal of R generated by X and denoted R X = (X ). One has ai xi : n ∈ Z>0 and ai ∈ R and xi ∈ X for all i  n}. R X = { i=1 (6) An ideal I of R is finitely generated if I = R X for some finite subset X = {a1 , . . . , an } of R. The ideal R X is also denoted (a1 , . . . , an ) and is equal to {r1 a1 + · · · + rn an : r1 , . . . , rn ∈ R}. (7) An ideal I of R is principal if I = R{a} for some a ∈ R. The principal ideal R{a} is also denoted (a), Ra, and a R and is equal to {ra : r ∈ R}. (8) R is Noetherian if every ideal of R is finitely generated, or equivalently if R satisfies the ascending chain condition on ideals, that is, if there are no infinite ascending chains I1  I2  I3  · · · of ideals of R. (9) R is a principal ideal domain, or PID, if R is an integral domain and every ideal of R is principal. (10) An ideal I of R is trivial if I = {0}. An ideal I of R is nontrivial if it is not trivial. (11) An ideal I of R is proper if I = R, or equivalently if 1 ∈ / I.

0.1 Algebraic Structures

5

(12) An ideal I of R is maximal if R/I is a field, or equivalently if I is maximal among the proper ideals of R. By Zorn’s lemma, every nontrivial ring has a maximal ideal. (13) An ideal I of R is prime if R/I is an integral domain, or equivalently if R − I is a submonoid of R, ·, or equivalently if I is proper and ab ∈ I implies a ∈ I or b ∈ I for all a, b ∈ R, or, equivalently still, if I is proper and H J ⊆ I implies H ⊆ I or J ⊆ I for all ideals H and J of R. (14) For any prime ideal p of R, the residue field of R at p is the quotient field of the integral domain R/p. (If p is maximal, then the residue field of R at p is just the field R/p.) (15) Spec R denotes the set of all prime ideals of R. (It is a topological space under the Zariski topology.) (16) Max R denotes the set of all maximal ideals of R. (17) R is local if it has exactly one maximal ideal, and R is semilocal if R has finitely many maximal ideals. (18) The height of a prime ideal p of R is the supremum of all nonnegative integers n such that there is an ascending chain p0  p1  p2  · · ·  pn = p of prime ideals of R terminating at p and of length n. (19) The (Krull) dimension of R is the supremum of all nonnegative integers n such that there is an ascending chain p0  p1  p2  · · ·  pn of prime ideals of R; equivalently, it is the supremum of the heights of the prime ideals of R. (20) A prime ideal of R is minimal if it has height zero, that is, if it is minimal among the prime ideals of R. Every prime ideal of a ring contains a minimal prime, and minimal primes consist only of zerodivisors. (21) Min R denotes the set of all minimal prime ideals of R. (22) Let I be an ideal of R. A prime ideal of R is minimal over I if it is minimal among the prime ideals of R containing I . (23) Every proper ideal I of R is contained in some maximal ideal of R as well as some prime ideal of R that is minimal over I . (24) The prime avoidance lemma states that if I is an ideal contained in a finite union p1 ∪ p2 ∪ · · · ∪√pn of prime ideals pi , then I is contained in pi for some i. n (25) The radical rad I = I of an ideal I of R is the ideal {a ∈ R : a ∈ I for some n ∈ Z>0 }. One has rad I = {p ∈√Spec R : p ⊇ I } for any ideal I . (26) An ideal I of R is radical if I = I , that is, if a ∈ I for all a ∈ R such that a n ∈ I for some positive integer n. The radical of any ideal I of R is the smallest radical ideal of R containing I . (27) a ∈ R is nilpotent if a n = 0 for some positive √ integer n. (0) = {a ∈ R : a is nilpotent} = (28) The nilradical nilrad(R) of R is the ideal   Spec R = Min R of R consisting of all nilpotent elements of R. (29) R is reduced if nilrad(R) = (0), that is, if 0 is the only nilpotent element of R. (30) The reduction of a ring R is the ring Rred = R/ nilrad(R), which is the “largest” quotient ring of R that is reduced. (31) a ∈ R is idempotent if a n = a for all positive integers n, or equivalently if a 2 = a. (32) R is connected if 0 and 1 are the only idempotent elements of R.

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Another useful tool for studying commutative rings is that of localization. Definition 0.1.7 (Localization). Let R be a commutative ring. (1) A multiplicative subset of R is a submonoid of R, ·. (2) Let U be multiplicative subset of R. There exists a ring U −1 R, unique up to isomorphism and equipped with a ring homomorphism f : R −→ U −1 R, such that (i) f (U ) is contained in the group (U −1 R)× of all invertible elements of U −1 R, and (ii) if ϕ : R −→ S is any ring homomorphism such that f (U ) is : contained in the group S × , then there exists a unique ring homomorphism ϕ  ◦ f = ϕ. The ring U −1 R is called the localization of U −1 R −→ S such that ϕ R at U . (3) Elements of U −1 R are equivalence classes a/u of pairs (a, u) with a ∈ R and u ∈ U , where (a, u) ∼ (a , u ) if v(u a − ua ) = 0 for some v ∈ U . Addition in U −1 R is given by (a/u) + (a /u ) = (u a + ua )/(uu ) and multiplication by (a/u)(a /u ) = aa /uu . (4) The ring homomorphism R −→ U −1 R acts by r −→ r/1 and has kernel {a ∈ R : ua = 0 for some u ∈ U }. (5) If p is a prime ideal of R, then Rp denotes the localization U −1 R, where U is the submonoid R − p of R, ·. The ring Rp is called the localization of R at p. It is a local ring with unique maximal ideal pRp . Conversely, if R is local with unique maximal ideal p, then the map R −→ Rp is an isomorphism. (6) The total quotient ring of R, denoted T (R), is the localization of R at the multiplicative subset R reg of all regular elements of R. There are several inequivalent notions of an “algebra” over a ring. For us the most relevant is the following. Definition 0.1.8 (Algebras and characteristic of a ring). (1) Let R be a ring. An R-algebra is a ring S equipped with a ring homomorphism R −→ S. (2) Let R be a ring, and let S and T be R-algebras. An R-algebra homomorphism from S to T is a ring homomorphism S −→ T such that the ring homomorphism R −→ T is the composition of the ring homomorphisms R −→ S −→ T . (3) For any ring R, there is a unique ring homomorphism Z −→ R. Thus a ring is equivalently a Z-algebra. (4) The characteristic of a ring R is the unique nonnegative integer n that generates the kernel of the unique ring homomorphism Z −→ R. A ring of characteristic n is equivalently a Z/nZ-algebra. Modules are yet another important tool for studying rings. Definition 0.1.9 (Modules). Let R be a ring. (1) If A is an abelian group, then the set End(A) of all group homomorphisms from A to A is a ring under the operations + and ◦, where + is pointwise addition and ◦ is composition of functions.

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(2) A (left) R-module, or (left) module over R, is an abelian group M together with a ring homomorphism f : R −→ End(M). We write r m = f (r )(m) for all r ∈ R and all m ∈ M, which defines a map R × M −→ M. A map R × M −→ M denoted (r, m) −→ r m induces the structure of an R-module on an abelian group M if and only if the following properties hold for all r, s ∈ R and all m, n ∈ M: (a) r (m + n) = r m + r n. (b) (r + s)m = r m + sm. (c) (r s)m = r (sm). (d) 1m = m. (3) If K is a field, then a K -vector space, or vector space over K , is a K -module. (4) A Z-module is equivalently an abelian group. (5) An R-submodule of an R-module M is a subgroup N of M under + such that r n ∈ N for all r ∈ R and all n ∈ N . (6) A map f : M −→ N of R-modules M and N is an R-module homomorphism, or R-linear, if f is a homomorphism of abelian groups such that f (r m) = r f (m) for all r ∈ R and all m ∈ M. (7) Let f : M −→ N be an R-module homomorphism. The kernel ker f is the R-submodule {m ∈ M : f (m) = 0} of M. The image im f is the R-submodule { f (m) : m ∈ M} of N . (8) Let N be an R-submodule of an R-module M. Then M/N denotes the set {m + N : m ∈ M} of all cosets m = m + N = {m + n : n ∈ N } of N in M. The set M/N is an R-module under the (well-defined) operations + and · defined by m + n = m + n and r · m = r · m for all m, n ∈ M and all r ∈ R. Equivalently, the set M/N has the unique structure of an R-module so that the map M −→ M/N given by m −→ m is an R-module homomorphism. The R-module M/N is called the quotient of M by N . (9) The first isomorphism theorem for modules states that, if f : M −→ N is any homomorphism of R-modules, then there is a unique R-module isomorphism f : M/ ker f −→ im f such that f (m + ker f ) = f (m) for all m ∈ M. (10) Let M be an R-module. An R-submodule N of M is a subgroup N of M, + such that r n ∈ N for all r ∈ R and all n ∈ N . (11) If X is a subset of an R-module M, then there is a smallest R-submodule of M containing X , called the R-submodule of M generated by X and n ai xi : n ∈ Z>0 and ai ∈ R and xi ∈ denoted R X = (X ). One has R X = { i=1 X for all i  n}. (12) An R-module M is finitely generated if M = R X for some finite subset X of M. (13) An R-module M is Noetherian if every R-submodule of M is finitely generated, or equivalently if M satisfies the ascending chain condition on submodules, that is, if there are no infinite ascending chains N1  N2  N3  · · · of R-submodules of M. A module over a Noetherian ring is Noetherian if and only if it is finitely generated. (14) A left ideal of R is an R-submodule of the left ideal R; equivalently, it is a subgroup I of R, + such that ra ∈ R for all a ∈ I and all r ∈ R. (15) Right modules and right ideals are defined in a similar manner. (16) An ideal of R is a subset of R that is both a left ideal and a right ideal of R.

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Definition 0.1.10 (Direct products, direct sums, and free modules). Let R be a ring and {Mλ }λ∈ and indexed collection of R-modules.  (1) The direct product λ∈ Mλ of the indexed collection {Mλ }λ∈ of R-modules is the unique R-module whose underlying set is the cartesian product of the given collection and where addition is coordinatewise and multiplication by any element of R is also  coordinatewise. (2) The direct sum λ∈ Mλ of the indexed  collection {Mλ }λ∈ of R-modules is the R-submodule of the direct product λ∈ Mλ consisting of all elements of  λ∈ Mλ whose coordinates are 0 for all but finitely many λ ∈ . (3) A subset X of an R-module M is a spanning set of M if M = R X . (4) A  subset X of an R-module M is linearly independent (in M over R) if x∈S a x x = 0 implies a x = 0 for all x ∈ S for every finite subset S of X and for any subset {ax : x ∈ S} of R indexed by S. (5) A subset X of an R-module M is a basis for M if X is a linearly independent spanning set of M. (6) A R-module M is free if it has a basis. (7) If  X is a subset of M, then the map x∈X R −→ M acting by (ax )x∈X −→ x∈X a x x is a well-defined R-module homomorphism. It is surjective if and only if X is a spanning set of M, it is injective if and only if X is linearly independent in M, and therefore it is an isomorphism if and only if X is a basis for M. Consequently, an R-module is free if and only it is isomorphic to x∈X R for some indexing set X . Definition 0.1.11 (Torsion-free, torsion, and faithful modules). Let R be a commutative ring and M an R-module. (1) The torsion submodule of M is the R-submodule Tors M = {m ∈ M : am = 0 for some a ∈ R reg }. (2) M is R-torsion-free if Tors M = 0, that is, if am = 0 implies m = 0 for all m ∈ M and all regular elements a of R. (3) M is R-torsion if Tors M = M, that is, if for all m ∈ M one has am = 0 for some regular element a of R. (4) If N is an R-submodule of M, then the annihilator of N is the ideal ann R (N ) = {a ∈ R : ax = 0 for all x ∈ N } of R. (5) M is faithful if ann R (M) = {0}. Definition 0.1.12 (Tensor products). Let R be a commutative ring, and let M, N , P be R-modules. (1) A map f : M × N −→ P is bilinear if the maps f (−, n) : M −→ P and f (m, −) : N −→ P are linear for all n ∈ N and all m ∈ M. (2) The tensor product M ⊗ R N of M and N is an R-module, unique up to isomorphism, equipped with a bilinear map f : M × N −→ M ⊗ R N such that, for any bilinear map ϕ : M × N −→ Q, there exists a unique linear map ϕ : M ⊗ R N −→ Q such that ϕ ◦ f = ϕ.

0.1 Algebraic Structures

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(3) Tensor products are functorial in the sense that any pair of R-module homomorphisms M −→ M and N −→ N induces a natural R-module homomorphism M ⊗ R N −→ M ⊗ R N in such a way that respects composition of homomorphisms. Definition 0.1.13 (Exactness and flatness). Let R be a commutative ring. f

g

(1) A sequence of R-module homomorphisms M −→ N −→ P is said to be exact at N if im f = ker g. (2) An R-module M is flat if for any sequence N −→N −→N

of R-module homomorphisms that is exact at N , the induced sequence M ⊗ R N −→ M ⊗ R N −→ M ⊗ R N

is exact at M ⊗ R N . (3) For an R-module M to be flat, it is necessary and sufficient that the R-module homomorphism I ⊗ R M −→ R ⊗ R M induced by the inclusion I −→ R is injective for every finitely generated ideal I of R. (4) A sequence · · · −→ Mi−1 −→ Mi −→ Mi+1 −→ · · · (finite or infinite in either direction) of R-module homomorphisms is said to be exact if it is exact at Mi for all i. Definition 0.1.14 (Localization of modules). Let U be multiplicative subset of a commutative ring R and M an R-module. (1) The localization of M at U is the U −1 R-module U −1 M whose elements are equivalence classes m/u of pairs (m, u) with m ∈ M and u ∈ U , where (m, u) ∼ (m , u ) if v(u m − um ) = 0 for some v ∈ U . Addition in U −1 M is given by (m/u) + (m /u ) = (u m + um )/(uu ) and scalar multiplication by (a/v)(m/u) = am/vu. (2) The map M −→ U −1 M given by m −→ m/1 is an R-module homomorphism with kernel {m ∈ M : um = 0 for some u ∈ U }. (3) The natural map (U −1 R) ⊗ R M −→ U −1 M is an isomorphism of R-modules. (4) If p is a prime ideal of R, then Mp denotes the localization U −1 M, where U = R − p. The Rp -module Mp is called the localization of M at p.

0.2 Ordered Algebraic Structures This book is concerned not only with algebraic structures, but also ordered structures, such as partially ordered sets, and ordered algebraic structures, such as ordered semigroups and nuclei. Definition 0.2.1 (Relations). Let S be a set. (1) A relation on S is a subset of S × S. For any relation R ⊆ S × S on S we write aRb as shorthand for (a, b) ∈ R. (2) A relation R on S is reflexive if aRa for all a ∈ S.

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(3) A relation R on S is transitive if aRb and bRc together imply aRc, for all a, b, c ∈ S. (4) A relation R on S is symmetric if aRb implies bRa for all a, b ∈ S. (5) A relation R on S is antisymmetric if aRb and bRa together imply a = b, for all a, b ∈ S. (6) A preorder on S is a reflexive and transitive relation on S. (7) An equivalence relation on S is a reflexive, transitive, and symmetric relation on S. (8) A partial ordering on S is a reflexive, transitive, and antisymmetric relation on S. (9) A partially ordered set, or poset, is a set S together with a partial ordering on S. (10) A relation R on S is total if aRb or bRa for all a, b ∈ R. (11) A total ordering on S is a partial ordering on S that is total. (12) A totally ordered set, or chain, is a poset whose partial ordering is total. Definition 0.2.2 (Suprema and infima). Let S be a poset with partial ordering , and let X be a subset of S. (1) An upper bound of X (in S) is an element u of S such that x  u for all x ∈ X . (2) A least upper bound, or supremum, of X (in S) is an upper bound u of X such that u  u for all upper bounds u of X . If it exists, it is unique and is denoted sup X . (3) A lower bound of X (in S) is an element l of S such that l  x for all x ∈ X . (4) A greatest lower bound, or infimum, of X (in S) is a lower bound l of X such that l  l for all lower bounds l of X . If it exists, it is unique and is denoted inf X . Definition 0.2.3 (Maximal and minimal elements, Zorn’s lemma, and the axiom of choice). (1) An element x of a poset S is maximal if x  y implies x = y, for all y ∈ S. (2) An element x of a poset S is minimal if y  x implies x = y, for all y ∈ S. (3) Zorn’s lemma states that, if S is any poset such that every totally ordered subset of S has an upper bound in S, then S contains at least one maximal element. (4) The axiom of choice states that the Cartesian product of any set of nonempty sets is nonempty. It is known that Zorn’s lemma is equivalent to the axiom of choice, given the Zermelo–Fraenkel axioms of set theory. Definition 0.2.4 (Posets). Let S be a poset with partial ordering . (1) S is a lattice, or lattice-ordered, if sup X and inf X exist for every nonempty finite subset X of S. (2) S is a join semilattice if sup X exists for every nonempty finite subset X of S. (3) S is a meet semilattice if inf X exists for every nonempty finite subset X of S. (4) S is a complete lattice, or complete, if sup X exists (or equivalently if inf X exists) for every subset X of S.

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Definition 0.2.5 (Self-maps). (1) Let X be a set. A self-map of X , or operation on X , is a unary operation on X , that is, a function from X to X . For any self-map  of a set X , we write x  = (x) for all x ∈ X and Y  = (Y ) for all Y ⊆ X . If x  = x, then x ∈ X is said to be -closed. (2) A self-map  of a set X is idempotent if (x  ) = x  for all x ∈ X . (3) If  is a self-map of a magma M, then the binary operation (x, y) −→ x  y = (x y) on M is called -multiplication on M. We then consider the set M  as a magma under -multiplication restricted to M  . Definition 0.2.6 (Morphisms and operations). (1) A map f : S −→ T of posets is order-preserving, or a homomorphism of posets, if x  y implies f (x)  f (y) for all x, y ∈ S. (2) An isomorphism of posets is a bijective poset homomorphism whose inverse is also a poset homomorphism. (3) An endomorphism of a poset S is a poset homomorphism from S to itself. (4) An operation  on a poset S is expansive, or inflationary, if x  x  for all x ∈ S. (5) An operation  on a poset S is contractive, or deflationary, if x   x for all x ∈ S. (6) A preclosure operation, or preclosure, on a poset S is an expansive endomorphism of S. (7) A closure operation on a poset S is an idempotent preclosure operation on S, that is, it is an order-preserving, expansive, and idempotent self-map of S. (8) A preinterior operation on a poset S is a contractive endomorphism of S. (9) An interior operation on a poset S is an idempotent preinterior operation on S, that is, it is an order-preserving, contractive, and idempotent self-map of S. (10) A map f : S −→ T of posets is reduced, or grounded, if inf S does not exist or else f (inf S) = inf T . Definition 0.2.7 (Ordered algebraic structures). (1) A (partially) ordered magma is a magma M equipped with a partial ordering  on M such that x  x and y  y implies x y  x y for all x, x , y, y ∈ M. (2) A (partially) ordered semigroup is an ordered magma that is a semigroup. (3) A (partially) ordered monoid is an ordered magma that is a monoid. (4) A (partially) ordered group is an ordered magma that is a group. (5) A map f : N −→ M from a magma N to an ordered magma M is submultiplicative if f (x) f (y)  f (x y) for all x, y ∈ N . Definition 0.2.8 (Nuclei). A nucleus on an ordered magma M is a sub-multiplicative closure operation on M, that is, it is a self-map  of M such that  is: (1) order-preserving: x  y implies x   y  , (2) expansive: x  x  ,

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(3) idempotent: (x  ) = x  , (4) sub-multiplicative: x  y   (x y) , where each of these axioms holds for all x, y ∈ M. The set Nucl(M) of all nuclei on

an ordered magma M is a partially ordered set, where one defines    if x   x  for all x ∈ M, or equivalently if every -closed element of M is  -closed. Definition 0.2.9 (Left and right nuclei). Let M be an ordered magma. A left nucleus on M is a closure operation  on M such that x y   (x y) for all x, y ∈ M. A right nucleus on M is a closure operation  on M such that x  y  (x y) for all x, y ∈ M. Definition 0.2.10 (Residuated ordered magmas). (1) Let x and a be elements of an ordered magma M. We let x/a, when it exists, denote the largest element z of M such that za  x. In other words, x/a, when it exists, is characterized as the unique element of M such that for all z ∈ M one has z  x/a ⇔ za  x. (2) Let x and a be elements of an ordered magma M. We let a\x, when it exists, denote the largest element z of M such that az  x. In other words, a\x when it exists, is characterized as the unique element of M such that for all z ∈ M one has z  a\x ⇔ az  x. (3) Some texts denote x/a and a\x by x ← a and a → x, respectively. (4) An ordered magma M is right residuated (resp., left residuated) if x/a exists (resp., a\x exists) for all x, a ∈ M. (5) An ordered magma is residuated if it is both left residuated and right residuated. Definition 0.2.11 (Scott topology and finitary maps). Let S be a poset. (1) A nonempty subset  of S is directed if every finite subset of  has an upper bound in . (2) A subset X of S is downward closed (resp., upward closed) if y ∈ X whenever y  x (resp., y  x) for some x ∈ X . (3) A subset X of S is Scott closed if X is a downward closed subset of S and for any directed subset  of X one has sup  ∈ X if sup  exists. (4) A subset X of S is Scott open if its complement is Scott closed, or equivalently if X is an upward closed subset of S and X ∩  = ∅ for any directed subset  of S with sup  ∈ X . (5) The Scott open subsets of a poset S form a topology on S called the Scott topology on S. (6) A map f : S −→ T of posets is Scott continuous, or finitary, if f is continuous when S and T are endowed with their respective Scott topologies. Equivalently, f is Scott continuous if and only if f (sup ) = sup f () for every directed subset  of S for which sup  exists. A Scott continuous map of posets is necessarily operation-preserving.

Chapter 1

Introductory Survey of Multiplicative Ideal Theory

This chapter provides an overview, without proofs, of multiplicative ideal theory. All of the information presented in this chapter is restated and proved in Chapters 2 and 3 in greater generality. The first section is a survey of the theory of integral domains, introducing some of the theory’s most central concepts and results. The reader may at any time wish to consult the various figures and tables at the end of Section 1.1, which summarize the characterizations of and interrelationships among the various classes of integral domains discussed here. The second section provides many examples and counterexamples. The third section provides a brief preview of the generalizations of Sections 1.1 and 1.2 to rings with zerodivisors that are discussed in Chapters 2 and 3.

1.1 Integral Domains An integral domain, typically defined as a nontrivial commutative ring with no nonzero zerodivisors, is most simply defined as a ring that is a subring of some field, that is, a subset of a field that itself contains 0 and 1 and is closed under addition, subtraction, and multiplication. The quintessential example is the ring Z of integers. As a branch of commutative algebra, the theory of integral domains has direct ties to field theory, algebraic geometry, and algebraic number theory. In algebraic geometry, integral domains correspond to reduced irreducible affine schemes, and in algebraic number theory, various important classes of integral domains arise in connection with algebraic number fields, including the Dedekind domains, Euclidean domains, principal ideal domains, and unique factorization domains. A more basic connection lies in the fact that the quotient R/I of a commutative ring R by an ideal I is an integral domain if and only if the ideal I is prime. Prime ideals have central importance in commutative algebra, algebraic geometry, and algebraic number theory and give rise to the extremely useful techniques of localization and completion, which © Springer Nature Switzerland AG 2019, corrected publication 2020 J. Elliott, Rings, Modules, and Closure Operations, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-24401-9_1

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were developed by Krull in the 1920s and applied soon after, with great fruition, to algebraic geometry. A less widely known tool known as a star operation is also due to Krull and is one of the main topics of this book. The notion of a star operation is founded on a generalization of the notion of an ideal called a fractional ideal. Fractional ideals and star operations are most easily defined and studied in the context of integral domains. However, as we show in Chapters 2 and 3, all definitions and results on integral domains described in this chapter generalize, albeit not uniquely, to commutative rings with (or without) (nonzero) zerodivisors. Let D be an integral domain with quotient field K . A fractional ideal of D is a D-submodule I of K such that a I ⊆ D for some nonzero a ∈ D, or equivalently, such that the D-submodule I −1 = (D : K I ) of K is nonzero (in which case I −1 is also a fractional ideal of D). The partially ordered set (or poset) Freg (D) of all nonzero fractional ideals of D ordered by the subset relation is a (partially) ordered commutative monoid under the operation of multiplication, where I J is defined to be the D-submodule of K generated by the set {ab : a ∈ I, b ∈ J }, and where the partial ordering is the subset relation. A star operation on D is a self-map ∗ : I −→ I ∗ of the set Freg (D) such that ∗ is: (1) (2) (3) (4) (5)

order-preserving: I ⊆ J implies I ∗ ⊆ J ∗ , expansive: I ⊆ I ∗ , idempotent: (I ∗ )∗ = I ∗ , sub-multiplicative: I ∗ J ∗ ⊆ (I J )∗ , and unital: D ∗ = D

where each of these axioms holds for all I, J ∈ Freg (D). Star operations were first introduced by Krull as  -operations in his influential 1935 book Idealtheorie [124, p. 118]. The axioms for star operations are both natural and intuitive and lead to a vast and important theory called the theory of star operations. As discussed in Chapter 0 and in greater detail in Chapter 6, the first three star operation axioms are more generally those for a closure operation on a partially ordered set. Moreover, the first four axioms are those for a nucleus, or sub-multiplicative closure operation, on an ordered semigroup. More precisely, a star operation on D is equivalently a nucleus ∗ on the ordered monoid Freg (D) such that D ∗ = D [51]. Consequently, axioms (1)–(4) can be replaced with the single axiom (1–4) (I ∗ : K J ) = (I ∗ : K J ∗ ) (or equivalently H J ⊆ I ∗ iff H J ∗ ⊆ I ∗ ) for all H, I, J ∈ Freg (D) [51]. One of the many reasons that star operations are so useful in commutative algebra is that many ideal-theoretic characterizations or properties of rings can be generalized by requiring them hold only up to ∗-closure, typically by replacing various instances of fractional ideals with their ∗-closure. One of the most important examples of this is in generalizing the well-known (ideal) class group Cl(D), defined as the group Inv(D) of all invertible fractional ideals of D under multiplication modulo the group

1.1 Integral Domains

15

Prin(D) of all nonzero principal fractional ideals of D, where a fractional ideal I of D is said to be invertible if I J = D for some fractional ideal J of D. The five star operation axioms imply that all nonzero principal ideals I of D are ∗-closed, that is, they satisfy I ∗ = I . More generally, the axioms imply that (I J )∗ = I J ∗ if I is invertible, so all invertible fractional ideals are ∗-closed. The set of all ∗-closed fractional ideals of D is an ordered monoid under the operation (I, J ) −→ (I J )∗ of ∗-multiplication. The group of units of this monoid, which we denote by Inv∗ (D), is an ordered group. The ∗-class group Cl∗ (D) of D is defined to be the quotient group Inv∗ (D)/ Prin(D) of Inv∗ (D) modulo its subgroup Prin(D) of all nonzero principal fractional ideals of D. If d : I −→ I d = I denotes the trivial star operation on D, then Cld (D) = Inv(D)/ Prin(D) is precisely the ideal class group of D. Clearly Cl(D) is a subgroup of Cl∗ (D), and the group Cl∗ (D)/ Cl(D) ∼ = Inv∗ (D)/ Inv(D) is a measure of the discrepancy between the ∗-class group and the ideal class group. Central to the theory of star operations also is the corresponding generalization of the notion of an invertible fractional ideal. A fractional ideal I of D is said to be ∗-invertible if I is invertible up to ∗-closure. More precisely, I is said to be ∗-invertible if (I J )∗ = D for some fractional ideal J of D, which can be seen to imply that J is uniquely determined by I up to closure and indeed that J ∗ = I −1 and therefore (I I −1 )∗ = D. Note that the ordered group Inv∗ (D) defined previously is the ordered group of all ∗-closed ∗-invertible fractional ideals of D under the operation of ∗-multiplication. The notion of ∗-closed ideal (resp., ∗-closed fractional ideal, ∗-invertible fractional ideal) generalizes the notion of an ideal (resp., fractional ideal, invertible fractional ideal). In an important sense, the theories of ideals and fractional ideals can be seen as a study of the trivial star operation d. Besides d, the three most important and useful star operations are known as v, t, and w. As is now standard, v : I −→ I v denotes the v-closure, or divisorial closure, star operation, defined by I v = (I −1 )−1 , t : I −→ I t denotes the t-closure star operation, defined by It =

 {J v : J ∈ Freg (D) is f.g. and J ⊆ I },

and w : I −→ I w denotes the w-closure operation, defined by Iw =

 {(I : K J ) : J ∈ Freg (D) is f.g. and J v = D}.

Here and hereafter, “f.g.” is an abbreviation for “finitely generated.” In general, one has d  w  t  v, where  is the natural partial ordering on the set of all star  operations on D defined by ∗  ∗ if I ∗ ⊆ I ∗ for all I ∈ Freg (D), or equivalently if every ∗-closed fractional ideal of D is ∗ -closed. As a quadrumvirate, the star operations d, v, t, and w can be used to shed enormous light on the principal ideal domains (PIDs), Dedekind domains, unique factorization

16

1 Introductory Survey of Multiplicative Ideal Theory

domains (UFDs), Krull domains, Bézout domains, Prüfer domains, GCD domains, PVMDs, among many other classes of domains. These eight important classes of integrally closed domains in particular are studied formally in Chapters 2 and 3 but can be uniquely characterized using just the operations d and t, as in the following characterization table. Domain

Alternative name

Characterization

PID Dedekind Bézout Prüfer UFD Krull GCD

d-PID d-Dedekind d-Bézout d-Prüfer t-PID t-Dedekind t-Bézout

PVMD

t-Prüfer

ideals are principal nonzero ideals are invertible f.g. ideals are principal nonzero f.g. ideals are invertible t-closed ideals are principal nonzero ideals are t-invertible t-closure of any nonzero f.g. ideal is principal nonzero f.g. ideals are t-invertible

All domains of the types above are integrally closed. Moreover, all eight classes of domains have theories that are close enough in spirit to the theory of Dedekind domains to be both amenable to and worthy of study. In Bourbaki, for example, the eight classes are characterized equivalently, without recourse to the t-operation, in terms of the ordered monoid of divisors on an integral domain [23, Chapter VII]. Typically, a Dedekind domain is defined to be an integral domain that is integrally closed, Noetherian, and of (Krull) dimension at most one, or, equivalently, a Noetherian domain that is locally a PID. Noether initiated the study of Dedekind domains in her landmark 1927 paper [158], six years after providing the first axiomatization of general commutative rings and ideals in 1921 [157]. Krull domains were first introduced by Krull in 1931 [123] and Prüfer domains by Prüfer in 1932 [167]. Both have been widely studied by the commutative algebra community. The PVMDs, or Prüfer v-multiplication domains—called pseudo-Prüferian domains in Bourbaki [23, Chapter VII]—are not as widely known. They were first introduced in 1967 by M. Griffin [91], seven years after Jaffard had introduced the operation of t-closure in 1960 [111]. While both Krull domains and Prüfer domains can be non-Noetherian and of dimension greater than one, a Dedekind domain is equivalently a Noetherian Prüfer domain, or, also equivalently, a Krull domain of dimension at most one. Thus, the Krull domains can be thought of as a higher dimensional analogue of the Dedekind domains, while the Prüfer domains can be thought of as a non-Noetherian analogue of the Dedekind domains. The PVMDs are more general than both and can be thought of as a generalization of the Dedekind domains in both directions simultaneously, namely, as a “non-Noetherian higher dimensional analogue” of the Dedekind domains. To this day I still find it to be quite remarkable that an integral domain D is a UFD if and only if every t-closed ideal I of D is principal. One direction of the implication is not difficult to prove: the ideal I t is generated by gcd(I ), for any ideal I of a UFD. The reverse direction, however, is not so obvious: try to prove it on your own! One can see that an ideal I of a domain D is t-closed if and only

1.1 Integral Domains

17

if (a1 , a2 , . . . , an )v ⊆ I for all a1 , a2 , . . . , an ∈ I . One can even express I t for any ideal I of D more explicitly as I t = {x ∈ K : (x/a1 ) ∩ · · · ∩ (x/an ) ⊆ D for some nonzero a1 , . . . , an ∈ I }, where K is the quotient field of D. It is remarkable that such a seemingly ad hoc closure operation on ideals can have such tight control over the factorization properties of an arbitrary integral domain. Informally, we say that the notion of a UFD, or t-PID, is the “t-ification” of the notion of a PID. A domain is said to be GCD domain if every finite subset has a gcd (not necessarily a linear combination, as in the case of a Bézout domain). According to our characterization table, the notion of a GCD domain is the “t-ification” of the notion of a Bézout domain. Likewise, the notion of a Krull domain, or t-Dedekind domain, is the “t-ification” of the notion of a Dedekind domain, and the notion of a PVMD, or t-Prüfer domain, is the “t-ification” of the notion of a Prüfer domain. The informal process of “t-ification” allows for a more intuitive description of the PVMDs and Krull domains, and, as we will show, it works extremely well on many different levels. Arguably, the characterization table on the previous page provides the simplest definition of the eight distinguished classes of domains that is possible. As an immediate consequence of these characterizations, the interrelationships between these eight distinguished classes of integrally closed domains can be visualized in the following beautiful diagram of implications. PID

UFD

Dedekind

Bézout

Krull

GCD

Prüfer

PVMD

Remarkably, the intersection of any of the classes in the implication diagram above is the largest class in the diagram lying above them; for example, Krull & Prüfer ⇔ Dedekind, and UFD & Prüfer ⇔ PID. We summarize this by saying that the implication diagram is a full lattice of implications, or full implication lattice. Moreover, the implication diagram is complete in the sense that none of the arrows is reversible. The implication diagram can be analyzed further, as follows. The classes on the upper face of the cube (PID, UFD, Bézout, GCD) are the corresponding classes on the lower face (Dedekind, Krull, Prüfer, PVMD) with trivial t-class group. Similarly, the classes on the upper right face of the cube (PID, Dedekind, Bézout, Prüfer) are

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1 Introductory Survey of Multiplicative Ideal Theory

the corresponding classes on the lower left face (UFD, Krull, GCD, PVMD) with trivial t-operation, that is, with t = d on Freg (D). Finally, the classes on the upper left face (PID, Dedekind, UFD, Krull) are the corresponding classes on the lower right face (Bézout, Prüfer, GCD, PVMD) that are t-Noetherian, that is, that satisfy the ascending chain condition on t-closed ideals. Thus it makes sense to think of the upward direction in the lattice as the “impose trivial t-class-group” axis, the upward-right direction as the “impose t = d” axis, and the upward-left direction as the “impose t-Noetherian” axis. To see that these relationships hold and in particular that the diagram is a full implication lattice, one may prove the following characterizations of the eight classes of integrally closed domains. Domain PID Dedekind Bézout Prüfer UFD Krull GCD PVMD

Characterization integrally closed, integrally closed, integrally closed, integrally closed, integrally closed, integrally closed, integrally closed, integrally closed,

t t t t t t t t

= d, = d, = d, =d = w, = w, = w, =w

Noetherian, trivial class group Noetherian trivial class group t-Noetherian, trivial t-class group t-Noetherian trivial t-class group

It follows from these eight characterizations, for example, that a Dedekind domain is equivalently a Noetherian Prüfer domain, or equivalently a Krull domain on which t = d, while a Krull domain is equivalently a t-Noetherian PVMD. The characterization table also suggests that the operations t and w have important roles to play in the study of these eight classes of domains. Although the star operation v is absent from the discussion above, it is distinguished among all other star operations as the largest star operation (in the sense that I ∗ ⊆ I v for all domains D, all star operations ∗ on D, and all I ∈ Freg (D)). Therefore, domains D on which v = d, that is, for which (I −1 )−1 = I for all I ∈ Freg (D), are equivalently domains on which there is only one star operation. Such domains are said to be divisorial. The divisorial condition is a very strong condition: integrally closed divisorial domains are all Prüfer domains (since v = t = d) but properly include the Dedekind domains. It follows that a Dedekind domain is equivalently a Noetherian integrally closed divisorial domain. More generally, a Noetherian divisorial domain is equivalently a Gorenstein domain of dimension at most one [19, Theorem 6.3] [98, Corollary 4.3]. According to our first characterization table, a Dedekind domain is also equivalently a divisorial Krull domain, so a divisorial UFD must be a PID. Even the Bézout domains are not all divisorial (e.g., the ring of all algebraic integers), and in fact a valuation domain, or local Bézout domain, is divisorial if and only if its maximal ideal is principal [98]. As one moves beyond the divisorial domains, the star operations v, t and w become more and more useful. The study of the Krull domains reveals the importance of v-closure, and by extension t-closure since all Krull domains satisfy t = v. A Krull domain is most commonly defined to be an intersection of a locally finite collection of DVR overrings, where an overring of a domain D is a ring lying between it and its quotient

1.1 Integral Domains

19

field, and where an indexed collection {Dλ : λ ∈ } of overrings of D is said to be locally finite if every nonzero element of D is a unit in all but finitely many λ ∈ . Krull domains are a large, distinguished, and well-studied class of domains that properly include the integrally closed Noetherian domains and the UFDs. We have noted that a Dedekind domain is equivalently a Krull domain of dimension at most one. More generally, a Noetherian Krull domain is equivalently an integrally closed Noetherian domain. Moreover, a Krull domain with trivial v-class group is equivalently a UFD. Thus, the Krull domains are a common generalization of the Dedekind domains, the integrally closed Noetherian domains, and the UFDs, and their study allows for a streamlined approach to all three of those important classes of domains. Krull domains are important even in the context of general Noetherian domains: although the integral closure of a Noetherian domain of dimension at most two is Noetherian, in dimension three or more this is not the case, yet, the integral closure of any Noetherian domain is necessarily a Krull domain. The latter fact is known as the Mori–Nagata theorem, and it provides yet another compelling motive for the study of the Krull domains. Such a study quickly reveals that the ideal class group of a Krull domain is of limited use, even in the Noetherian case, very much unlike the situation for Dedekind domains. For example, the two dimensional integrally closed Noetherian domain D = k[X, Y, Z ]/(X Y − Z 2 ), where k is any field, is not a UFD because X Y factors in D in two distinct ways, yet D has trivial ideal class group. To compensate for this deficiency, one traditionally considers an appropriate enlargement of the ideal class group of a Krull domain known as its divisor class group [23, Section VII.1.1] [69]. The divisor class group is defined for any completely integrally closed domain, where a domain D with quotient field K is said to be completely integrally closed if a ∈ D for all a ∈ K such that D[a] is a fractional ideal of D. (A domain D is integrally closed if and only if a ∈ D for all a ∈ K such that D[a] is a finitely generated fractional ideal of D.) One has the implications Krull ⇒ completely integrally closed ⇒ integrally closed, and the implications are reversible for the Noetherian domains. The divisor class group of a domain D, when it is defined, is isomorphic to its v-class group Clv (D), but the latter has the added advantage that it can be defined for any commutative ring. Both groups adequately repair the aforementioned deficiency of the ideal class group: a UFD is equivalently a Krull domain with trivial v-class group, just as a PID is equivalently a Dedekind domain with trivial ideal class group. The v-class group of k[X, Y, Z ]/(X Y − Z 2 ), for example, is cyclic of order two, and the v-class group of k[X, Y, Z , W ]/(X Y − Z W ) is infinite cyclic, for any field k of characteristic not equal to 2 [69, Proposition 11.4]. It is also noteworthy that, if D is Krull, then the group Invv (D) of v-invertible vclosed fractional ideals under v-multiplication is a free abelian group freely generated by the prime ideals of height one, which are equivalently the nonzero v-closed prime ideals, in perfect agreement with the situation for Dedekind domains. As we have mentioned already, the Dedekind domains are equivalently the Krull domains on which v = d, or, equivalently still, the Krull domains of dimension at most one, and

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1 Introductory Survey of Multiplicative Ideal Theory

the theory of Dedekind domains can be recovered from the theory of Krull domains by adding either the assumption v = d or dim D  1. Adding the Noetherian hypothesis instead, we obtain the theory of integrally closed Noetherian domains, and adding the hypothesis Clv (D) = 0 we recover the theory of unique factorization domains. We may push the analogy between Krull domains and Dedekind domains even further. A v-Noetherian, or Mori, domain, is a domain that satisfies the ascending chain condition on v-closed ideals. A domain is v-Noetherian if and only if it is t-Noetherian, in the corresponding sense defined earlier. All Noetherian domains and all Krull domains are v-Noetherian. It is not difficult to show that a domain is completely integrally closed if and only if every nonzero ideal is v-invertible. Consequently, a Noetherian domain D is integrally closed if and only if every nonzero ideal of D is v-invertible, if and only if (I I −1 )−1 = D for all nonzero ideals I of D. Moreover, just as a Dedekind domain is equivalently a completely integrally closed (or Krull) divisorial domain, a Krull domain is equivalently a completely integrally closed v-Noetherian domain. All of this shows that the v-operation is extremely natural in the setting of Krull domains. This provides a strong impetus for studying the v-operation more generally. However, for a rather large class of integrally closed domains the v-operation suffers from a rather serious deficiency. Fortunately, the t-closure operation can be used to repair that deficiency in an optimal way and, at the same time, to generalize the theories of Krull domains and Prüfer domains to that of the PVMDs. The t-closure operation is uniquely characterized as the largest star operation that is continuous when the poset Freg (D) is endowed with the Scott topology, which is a natural topology, discussed in Chapters 0 and 6, defined on any poset. The t-operation is the nearest Scott continuous approximation to v from below. Even on a one dimensional valuation domain D one might not have v = t, that is, I v may properly contain I t =  {J v : J ∈ Freg (D) is f.g. and J ⊆ I } for some fractional ideal I of D. Domains for which v = t, that is, domains for which v is Scott continuous, are called TV domains. The TV domains include the divisorial domains, the Noetherian domains, the Krull domains, and even the v-Noetherian domains. From our previous characterizations we can thus see that a Krull domain is equivalently a completely integrally closed TV domain, just as a Dedekind domain is equivalently a completely integrally closed divisorial domain. However, a Prüfer domain need not be a TV domain. In fact, since a domain is Prüfer (integrally closed and t = d) and TV (v = t) if and only if it is integrally closed and divisorial (v = t = d), a Prüfer domain is a TV domain if and only if it is divisorial. At least for Prüfer domains, the star operation d (= t), rather than the v-operation, is the more important, while for Noetherian, Krull, or t-Noetherian domains, the operations t and v coincide. Furthermore, the class of all TV PVMDs properly includes the Krull domains. All of this indicates that, at least for the study of the PVMDs, it is the t-operation, rather than the v-operation, that occupies center stage. One of the simplest examples of a PVMD that is not Krull, Prüfer, or GCD is the domain Int(Z)[X ], where Int(Z) is the ring of all polynomial functions from Z to Z with rational coefficients, called the ring of integer-valued polynomials. Another example is the subring D + X K [X ] = { f ∈ K [X ] : f (0) ∈ D} of the polynomial ring K [X ], where D is any Krull domain that is neither a Dedekind domain nor a

1.1 Integral Domains

21

√ UFD (such as D = Z[ −5][Y ]). Both of these examples, along with many others, are discussed in detail in the next section. The former example is completely integrally closed, so that a completely integrally closed PVMD need not be a Krull domain. Moreover, the domain Int(Z)[X ] satisfies the ascending chain condition on principal ideals, or ACCP, and therefore is atomic, that is, every nonzero non-unit can be written as a product of irreducible elements. By contrast, the polynomial X ∈ D + X K [X ] for any domain D with quotient field K = D cannot be written as a product of irreducible elements, whence D + X K [X ] is not atomic. One can show that a UFD is equivalently an atomic GCD domain—this follows immediately from the fact that (1) every irreducible element of a GCD domain is prime, and (2) prime factorizations, when they exist, are unique, in any integral domain. Using the process of “v-ification,” rather than “t-ification,” one can define four more important classes of domains. The “v-ification” of the notion of a Dedekind domain, or v-Dedekind domain—that is, a domain in which every nonzero ideal is vinvertible—is equivalently a completely integrally closed domain. The “v-ification” of the notion of a PID, or v-PID, is equivalently a domain in which every subset has a gcd, called a strong GCD domain, or equivalently a completely integrally closed domain with trivial v-class group. Likewise, the “v-ification” of the notion of a Bézout domain is equivalently a GCD domain. The “v-ification” of the notion of a Prüfer domain is the most general class we have considered thus far: a v-Prüfer domain, or v-domain, is a domain in which every nonzero finitely generated (fractional) ideal is v-invertible. The v-domains (or v-Prüfer domains) are the “v-analogue” of the PVMDs and properly include both the PVMDs (or t-Prüfer domains) and the completely integrally closed domains (or v-Dedekind domains), and they are all integrally closed. Analogically, the v-domains are to the PVMDs as the completely integrally closed domains are to the Krull domains. Let us say that a v-GCD domain is a v-domain D such that Clv (D) = 0. Although the implications UFD (t-PID) ⇒ strong GCD (v-PID) ⇒ v-GCD ⇒ GCD (t-Bézout) ⇒ PVMD (t-Prüfer) ⇒ v-domain (v-Prüfer)

hold, none of them is reversible. By applying the processes of “t-ification” and “v-ification” to the Dedekind domains, Prüfer domains, PIDs, and Bézout domains, one is thus led quite naturally to much larger classes of integrally closed domains that possess analogous but far more general theories. Generalizing these two processes is the process of “∗ification,” where ∗ is a star operation on an integral domain D. In particular, for any star operation ∗ on a domain D, we make the following definitions. (1) (2) (3) (4)

D is ∗-Dedekind if every I ∈ Freg (D) is ∗-invertible. D is ∗-Prüfer if every finitely generated I ∈ Freg (D) is ∗-invertible. D is a ∗-PID if I ∗ is principal for all I ∈ Freg (D). D is ∗-Bézout if I ∗ is principal for all finitely generated I ∈ Freg (D).

These four notions and their generalizations to rings with zerodivisors are the main focus of Chapters 2 and 3. Since v and t are, respectively, the largest star operation and largest Scott continuous star operation, the v-domains and PVMDs are two of

22

1 Introductory Survey of Multiplicative Ideal Theory

the largest known classes of domains that have a reasonably well behaved divisor theory, both analyzed via the v-class group and the t-class group. The next section provides a large number of examples and counterexamples. One of these is an example of an integrally closed domain that is not a PVMD, namely, V + X V [π −1 ][X ], where V is any discrete rank two valuation domain with maximal ideal πV . This example, however, is not completely integrally closed. In 1952–55, Nagata settled a famous 1936 conjecture of Krull by constructing an example of a one dimensional local completely integrally closed domain, hence a v-domain, that is not a valuation domain and consequently is not a PVMD [68, p. 159]. There also exist integrally closed domains that are not v-domains. For example, the domain K + X L[X ], where L/K is any proper field extension such that K is algebraically closed in L, is a one dimensional integrally closed v-Noetherian domain that is not a vdomain [68, p. 161]. This example also shows that an integrally closed v-Noetherian domain need not be Krull. Just as a Krull domain is equivalently a v-Noetherian v-domain, a PVMD is equivalently a v-coherent v-domain, where a domain is said to be v-coherent if for every nonzero finitely generated fractional ideal I one has I −1 =J v for some finitely generated fractional ideal J . As one might expect, every v-Noetherian domain is v-coherent. The property of v-coherence is yet another important finiteness condition to be added to the list of those mentioned thus far: Noetherian, v-Noetherian, divisorial, and TV. Using equivalences mentioned previously, one can verify the equivalences in the characterization table below. (“IC” stands for “integrally closed” and “CIC” for “completely integrally closed.”) Domain

Characterization 1

Characterization 2

Characterization 3

PID Dedekind Bézout Prüfer UFD Krull

d-Dedekind, Cl = 0 d-Dedekind d-Prüfer, Cl = 0 d-Prüfer t-Dedekind, Clt = 0 t-Dedekind

v = d, CIC, Cl = 0 v = d, CIC t = d, IC, Cl = 0 t = d, IC t = v, CIC, Clt = 0 t = v, CIC

GCD PVMD

t-Prüfer, Clt = 0 t-Prüfer

t = w, IC, Clt = 0 t = w, IC

t = d, UFD t = d, Krull t = d, GCD t = d, PVMD t = v, strong GCD v-Noetherian v-domain

strong GCD CIC v-GCD v-domain

v-Dedekind, Clv = 0 v-PID v-Dedekind v-Prüfer, Clv = 0 GCD, Clv = Clt v-Prüfer

v-coherent v-domain

Undoubtedly, then, the star operations v, t, w, and d are useful for studying various large classes of integral domains. When these tools are combined with that of localization, even more magic starts to happen. Although t-closure does not in general commute with localization, the t-closed prime ideals, or t-prime ideals, include all strong Krull primes of the D-module K /D. In other words, the strong Krull primes of K /D, and hence also the weak Bourbaki associated primes of K /D and the height one primes of D, are all t-prime. In fact, a prime ideal p of D is a strong Krull prime of K /D if and only if pDp is a t-prime ideal of Dp , and both

1.1 Integral Domains

23

conditions imply that p is a t-prime of D (but not conversely). The nonzero t-primes of a Krull domain are precisely its height one primes, while the t-primes of a Prüfer domain are precisely its prime ideals, since any Prüfer domain satisfies t = d. A domain D is said to be of finite character if the collection {Dp : p ∈ Max(D)} of localizations of D is locally finite, or equivalently if every nonzero element of D belongs to only finitely many maximal ideals of D. Likewise, a domain D is said to be of finite t-character if the collection {Dp : p ∈ t-Max(D)} of localizations of D is locally finite, where t-Max(D) denotes the set of all t-maximal ideals of D, that is, the set of ideals that are maximal among the proper t-closed ideals of D, which are necessarily t-prime. It is a surprising fact that every divisorial domain is of finite character and every TV domain is of finite t-character. Neither of these facts is easy to prove. In particular, any Krull domain is of finite t-character. A Prüfer domain, on the other hand, need not be of finite (t-)character. For example, the ring Int(Z)={ f ∈ Q[X ] : f (Z) ⊆ Z} of integer-valued polynomials is a two dimensional non-Noetherian Prüfer domain, yet X ∈ Int(Z) lies in uncountably many (t-)maximal ideals of Int(Z); in fact, the fiber of Int(Z) above a given prime p ∈ Z is in natural one-to-one correspondence with the ring of p-adic integers [27, Proposition V.2.7]. A ring-theoretic property is said to hold locally if it holds at the localizations at all prime ideals. It is convenient to say that a property holds t-locally if it holds at the localizations at all t-prime ideals. For example, it turns out that a PVMD is equivalently a domain that is t-locally a valuation domain, that is, a domain such that Dp is a valuation domain for all t-primes p of D. In fact, if p is an essential prime of D, that is, if Dp is a valuation domain, then p is t-prime (since t = d on the local Bézout domain Dp and therefore pDp is t-prime in Dp ), and the converse holds if and only if D is a PVMD. This local characterization of the PVMDs extends to our eight distinguished classes of domains, as indicated in the following table. Domain PID Dedekind Bézout Prüfer UFD Krull GCD PVMD

Characterization locally a DVR, of finite character, trivial class group locally a DVR, of finite character locally a valuation domain, trivial class group locally a valuation domain t-locally a DVR, of finite t-character, trivial t-class group t-locally a DVR, of finite t-character t-locally a valuation domain, trivial t-class group t-locally a valuation domain

 One can show that, for any domain D one has D = p∈wAss(K /D) Dp , where wAss(K /D) denotes the set of weak Bourbaki associated primes of K /D, which are all t-prime. Since every t-prime  ideal is contained in a t-maximal ideal, the latter equality implies that D = p∈t-Max(D) Dp . Moreover, it turns out that the star operation w is given by  Iw = I Dp p∈t-Max(D)

for all I ∈ Freg (D). In fact, the star operation w, like the star operations d, v, and t, can be characterized by a universal property: a star operation ∗ on D is said to be stable if (I ∩ J )∗ = I ∗ ∩ J ∗ for all I, J ∈ Freg (D), and w is characterized as

24

1 Introductory Survey of Multiplicative Ideal Theory

the largest stable Scott continuous star operation. Therefore, just as the t-operation optimally repairs a deficiency of the v-operation, the w-operation optimally repairs a deficiency of the t-operation, namely, the fact that the t-operation need not be stable. Domains for which t = w, that is, domains for which the t-operation is stable, are called TW domains. A PVMD is equivalently an integrally closed TW domain. However, there exist integrally closed domains on which d = w = t = v. Moving beyond the integrally closed domains, one may study the general classes of domains determined by setting two or more of d  w  t  v equal to one another, which comprise the following implication lattice. divisorial (v = d)

w-divisorial (v = w)

TV (t = v)

TD (t = d)

TW (t = w)

DW (d = w)

All of the implications in the lattice above are irreversible, even for integrally closed domains. All of these classes of domains have received attention in the literature [45, 98, 104, 147, 148, 186]. For example, the DW domains have been characterized as domains D for which t-Max(D) = Max(D) [148, Proposition 2.2], and the TD domains are discussed and referred to in the literature as fgv domains [186]. As we have mentioned, a PVMD is equivalently an integrally closed TW domain, a Prüfer domain is equivalently an integrally closed TD domain (or equivalently a DW (or TD) PVMD), a Krull domain is equivalently a completely integrally closed TV domain, and a Dedekind domain is equivalently a completely integrally closed divisorial domain. Now, since t  v, every t-invertible ideal is v-invertible (which is another reason that a PVMD is a v-domain and a Krull domain is CIC). A domain D is said to be an H domain if every v-invertible ideal of D is t-invertible. Clearly, every TV domain is an H domain, but the converse does not hold. A domain D is an H domain if and only if the t-maximal ideals are all Bourbaki associated primes of K /D. Moreover, a Krull domain is equivalently a completely integrally closed H domain, a PVMD is an H domain if and only if every t-maximal ideal is t-invertible, and a valuation domain is an H domain if and only if it is a TV domain, if and only if it is divisorial, and if and only if its maximal ideal is principal. In general, we have the implications Mori ⇒ TV ⇒ H and of finite t-character, which are irreversible even for Bézout domains. For example, if D is a nonlocal semilocal PID with quotient field K = D, then D + X K [X ] is a Bézout H domain of finite (t-)character that is not a TV domain, and if D is a non-DVR valuation domain with principal maximal ideal, then D is a local Bézout domain that is TV but not Mori. The Mori domains, TV domains, H domains, and domains of finite

1.1 Integral Domains

25

t-character can all be seen as generalizations of the Noetherian domains (which need not be divisorial, w-divisorial, TD, TW, or DW). We mention another well-studied class of domains lying strictly between the Krull domains (and in fact the TV PVMDs) and the PVMDs. An integral domain D is said to be of Krull type if D is an intersection of a locally finite collection of  essential localizations of D, that is, if D = p∈ Dp , where  ⊆ Spec D, each Dp is a valuation domain, and every nonzero element of D belongs to only finitely many p ∈  [92]. In that case, the set  may be taken to be canonical, namely, as the set t-Max(D) of all t-maximal ideals of D. In particular, a domain is of Krull type if and only if it is a PVMD of finite t-character. As we have mentioned already, any TV domain is of finite t-character, and therefore any Krull domain, or more generally any TV PVMD, is a domain of Krull type. Moreover, any discrete rank n valuation domain is a TV PVMD but is a Krull domain if and only if n = 1. A domain D of Krull type is a Krull domain precisely when the valuation domains Dp for p ∈  (or p ∈ t-Max(D)) can be taken to be DVRs, or, equivalently, Noetherian. In particular a Krull domain is equivalently a Krull type domain that is t-locally Noetherian. The following is the smallest full and complete implication lattice that includes the PID, Dedekind, Bézout, Prüfer, UFD, Krull, GCD, and PVMD conditions along with the divisorial, w-divisorial, TD, TV, DW, and TW conditions. PID

UFD

Dedekind

div. Bézout

Krull

TV GCD

IC divisorial

Bézout

TV PVMD

GCD

Prüfer

PVMD

divisorial

w-divisorial

TV DW

TD

TV

TW

DW

domain

The lattice collapses to the following full implication lattices, respectively, for the integrally closed domains, completely integrally closed domains, H domains, domains of finite t-character, TV domains, Mori domains, Noetherian domains, t-

26

1 Introductory Survey of Multiplicative Ideal Theory

local domains (where, equivalently, there is a unique t-maximal ideal), domains of t-dimension at most one (where, equivalently, every height one prime is t-maximal), and domains of dimension (at most) one. In the next section, we find counterexamples to verify that many of the implications in these diagrams are irreversible. PID

UFD

Dedekind

div. Bézout

Krull

TV GCD

IC divisorial

Bézout

TV PVMD

GCD

Prüfer

PVMD

IC TV DW

IC TV

IC DW

IC

PID

UFD

Dedekind

CIC Bézout

Krull

CIC GCD

CIC Prüfer

CIC PVMD

CIC DW

CIC

1.1 Integral Domains

27 PID

UFD

Dedekind

div. Bézout

Krull

TV GCD

IC divisorial

H Bézout

TV PVMD

H GCD

H Prüfer

H PVMD

divisorial

w-divisorial

TV DW

H TD

TV

H TW

H DW

H

PID

UFD

Dedekind

div. Bézout

Krull

TV GCD

IC divisorial

Bézout fin. char.

TV PVMD

GCD fin. t-char.

Prüfer fin. char.

Krull type

divisorial

w-divisorial

TV DW fin. char.

TD fin. char.

TV

TW fin. t-char.

DW fin. char.

fin. t-char.

28

1 Introductory Survey of Multiplicative Ideal Theory

PID

UFD

Dedekind

div. Bézout

Krull

TV GCD

IC divisorial

TV PVMD

divisorial

w-divisorial

TV DW

TV

PID

UFD

Dedekind

Krull

Noeth. div. = 1-dim. Goren.

Mori TW

Mori DW

Mori

1.1 Integral Domains

29 PID

Noeth. UFD

Dedekind

Noeth. IC

Noeth. divisorial

Noeth. TW

Noeth. DW

Noetherian

DVR

divisorial valuation

local divisorial

valuation

local TV DW

local TD

local DW

t-local

30

1 Introductory Survey of Multiplicative Ideal Theory PID

UFD

Dedekind

1-dim. Bézout

Krull

1-t-dim. GCD

1-dim. Prüfer

1-t-dim. PVMD

1-dim. divisorial

1-t-dim. w-divisorial

1-dim. TV DW

1-dim. TD

1-t-dim. TV

1-t-dim. TW

1-dim. DW

1-t-dim.

PID

Dedekind

1-dim. Bézout

1-dim. divisorial

1-dim. Prüfer

1-dim. TV

1-dim. TD

1-dim.

On the next page is the smallest full and complete implication lattice that, among all PVMDs, includes the PID, Dedekind, Bézout, Prüfer, UFD, Krull, GCD, and CIC conditions along with the divisorial, w-divisorial, TD, TV, DW, and TW conditions. The lattice forms a four dimensional cube.

Krull

CIC PVMD

UFD

PVMD

GCD

CIC Prüfer

TV PVMD

CIC Bézout

IC divisorial

TV GCD

CIC GCD

Dedekind

PID

Prüfer

divisorial Bézout

Bézout

1.1 Integral Domains 31

32

1 Introductory Survey of Multiplicative Ideal Theory

All of the properties of domains discussed in this section have received substantial attention in the multiplicative ideal theory literature and appear again in more general form in Chapters 2 and 3. It is useful to note that most of the properties we consider in Chapters 1–3 fall into one of the following four categories or their analogues for semistar operations. 1. The first class consists of conditions that impose star invertibility on various classes of fractional ideals. The most important of these are the ∗-Dedekind and ∗-Prüfer conditions. 2. The second class consists of conditions that impose principality on various classes of fractional ideals. The most important of these are the ∗-PIR, ∗-Bézout, and trivial ∗-class group conditions. 3. The third class consists of properties that impose finiteness conditions on various classes of fractional ideals. The most important of these are the ∗-Noetherian, ∗-coherent, and finite ∗-character conditions. 4. The fourth class consists of conditions that impose relations on various star operations. The most important of these conditions are the divisorial, TD, wdivisorial, TW, DW, and TV conditions. Understanding the interactions between various properties of the types above is one of the main goals of multiplicative ideal theory. We close this section with some further tables and implication lattices that may help the reader assimilate some of the terminology used in the literature and in this book. The tables in the next few pages summarize many of the characterizations that were mentioned in this chapter, along with many others that are also proved in Chapters 2 and 3. In the tables, c D ( f ) for any f ∈ K [X ] denotes the content of f in D, which is the finitely generated fractional ideal generated by the coefficients of f . Domain D

Characterizations

IC

(I : K I ) = D for all nonzero f.g. ideals I c D ( f g)v = (c D ( f )c D (g))v for all nonzero f, g ∈ D[X ] v-Prüfer, i.e., all nonzero f.g. ideals are v-invertible (I v : K I v ) = D for all nonzero f.g. ideals I if I v ⊇ J v with I f.g., then there exists H ⊆ D with J v = (H I )v all nonzero two-generated ideals are v-invertible ((a, b)((a) ∩ (b)))v = (ab) for all nonzero a, b ∈ D ((I + J )(I ∩ J ))v = (I J )v for all nonzero f.g. ideals I, J ((I + J )(I ∩ J ))v = (I J )v for all I, J ∈ Freg (D) if I, J ∈ Freg (D) are v-invertible, then I + J and I ∩ J are v-invertible if I, J ∈ Freg (D) are v-invertible, then I + J is v-invertible t-Prüfer, i.e., all nonzero f.g. ideals are t-invertible v-coherent v-domain Dp is a valuation domain for all p ∈ t-Max(D) integrally closed and t = w integrally closed and (I ∩ J )t = I t ∩ J t for all nonzero ideals I, J c D ( f g)w = (c D ( f )c D (g))w for all nonzero f, g ∈ D[X ]

v-domain

PVMD

1.1 Integral Domains

Domain D

Prüfer

GCD

Bézout

Valuation

CIC Krull

Dedekind

33

Characterizations if I t ⊇ J t with I f.g., then there exists H ⊆ D with J t = (H I )t all nonzero two-generated ideals are t-invertible ((a, b)((a) ∩ (b)))t = (ab) for all nonzero a, b ∈ D ((I + J )(I ∩ J ))t = (I J )t for all finitely generated ideals I, J ((I + J )(I ∩ J ))t = (I J )t for all I, J ∈ Freg (D) if I, J ∈ Freg (D) are t-invertible, then I + J is t-invertible d-Prüfer, i.e., all nonzero f.g. ideals are invertible PVMD and t = d integrally closed and t = d Dp is a valuation domain for all p ∈ Max(D) c D ( f g) = c D ( f )c D (g) for all nonzero f, g ∈ D[X ] if I ⊇ J with I f.g., then there exists H ⊆ D with J = H I (a, b)((a) ∩ (b)) = (ab) for all nonzero a, b ∈ D (I + J )(I ∩ J ) = I J for all f.g. ideals I, J (I + J )(I ∩ J ) = I J for all I, J ∈ Freg (D) if I, J ∈ Freg (D) are invertible, then I + J is invertible v-Bézout, i.e., v-closure of any nonzero f.g. ideal is principal t-Bézout, i.e., t-closure of any nonzero f.g. ideal is principal PVMD with trivial t-class group gcd(S) exists for every finite subset S the gcd of any two elements exists the lcm of any two elements exists the intersection of any two principal ideals is principal d-Bézout, i.e., all f.g. ideals are principal Prüfer with trivial class group GCD and t = d local and Bézout local and Prüfer set of all ideals is totally ordered by inclusion x ∈ D or x −1 ∈ D for all nonzero x ∈ K v-Dedekind, i.e., all nonzero ideals are v-invertible (I : K I ) = D for all nonzero ideals I t-Dedekind, i.e., all nonzero ideals are t-invertible locally finite intersection of DVR overrings t-locally a DVR, and of finite t-character Mori PVMD Mori v-domain CIC (v-Dedekind) and Mori (v-Noetherian) CIC and t = v CIC H domain if I t ⊇ J t , then there exists H ⊆ D with J t = (H I )t d-Dedekind, i.e., all nonzero ideals are invertible locally a DVR, and Noetherian locally a DVR, and of finite character Noetherian, integrally closed, and of dimension at most one Noetherian, integrally closed, and divisorial (v = d) CIC and divisorial Noetherian and Prüfer Mori and Prüfer Krull and Prüfer Krull of dimension at most one

34

1 Introductory Survey of Multiplicative Ideal Theory

Domain D

Strong GCD

UFD

PID

Noetherian

Mori

H v-Coherent Finite character Finite t-character Divisorial w-divisorial TD TV TW DW Krull type

Characterizations Krull and divisorial Krull and t = d if I ⊇ J , then there exists H ⊆ D with J = H I v-PID, i.e., all v-closed ideals are principal gcd(S) exists for every subset S CIC with trivial v-class group t-PID, i.e., all t-closed ideals are principal Krull with trivial t-class group Mori PVMD with trivial t-class group Mori GCD ACCP and GCD ACCP and every irreducible element is prime atomic GCD atomic and every irreducible element is prime strong GCD and v = t d-PID, i.e., all ideals are principal Dedekind with trivial class group UFD of dimension at most one UFD and divisorial UFD and t = d strong GCD and divisorial GCD, divisorial, and Noetherian Noetherian and Bézout ACC on ideals every ideal is f.g. d-Noetherian ACC on v-closed ideals v-Noetherian t-Noetherian every v-invertible ideal is t-invertible for all f.g. I one has I −1 = J v for some f.g. J every nonzero elt. lies in only fin. many max. ideals every nonzero elt. lies in only fin. many t-max. ideals v=d v=w t =d t =v t =w d=w every maximal ideal is t-maximal PVMD of finite t-character

1.2 Some Examples and Counterexamples

35

1.2 Some Examples and Counterexamples In this section we provide, without proofs but with references, some examples and counterexamples. In the previous section, we discussed 12 important properties that an integral domain can possess: field, PID, Dedekind, UFD, Krull, Bézout, Prüfer, GCD, PVMD, completely integrally closed, v-domain, and integrally closed. Below is the smallest full and complete lattice containing these 12 classes of domains. The 12 properties therefore partition the class of all integral domains into the 17 disjoint subclasses listed in Table 1.1. Our first goal in this section is to explain the examples provided in Table 1.2 of integral domains within each of those 17 subclasses. field

PID

UFD

Dedekind

CIC Bézout

Krull

CIC GCD

CIC Prüfer

Bézout

CIC PVMD

GCD

Prüfer

CIC

PVMD

v-domain

IC domain

domain

Let R be a ring with total quotient ring K . Rings lying between R[X ] and K [X ] provide an ample source of examples and counterexamples. Three of the most interesting and well studied of these are the rings R[X ] ⊆ Int(R) ⊆ R + X K [X ],

36

1 Introductory Survey of Multiplicative Ideal Theory

Table 1.1 Partition of the class of integral domains Class

Field PID Ded. UFD Krull Bézout

F



P

Prüfer

GCD

PVMD

CIC

v-dom.

IC 





















































D

 

U









K



















B-C









B









P-C





P





G-C





G





PV-C



PV



   

C







































V

 

I N

Table 1.2 Examples of integral domains within each class Class F P D U K B-C

Contains D[X ] for D in Class . . .

Contains D+ X K [X ] for D in Class . . .

F

F

P, U D, K

B P-C P G-C

P, B-C, B

B-C, G-C

G PV-C PV C V I N

B, G P-C, PV-C P, PV C V I N

D, P-C, P U, G-C, G K, PV-C, PV C, V I N

Example Q Z √ Z[ −5] Z[X ] √ Z[ −5, X ] OQ

Z + X Q[X ] Int(Z) √ √ Z[ −5] + X Q( −5)[X ] OQ [X ]

Z[X ] + Y Q(X )[Y ] Int(Z)[X ] √ √ Z[ −5, X ] + Y Q( −5, X )[Y ] C C + X K [X ] Q + X Q(T )[X ] √ Z[ 5]

C is any one dimensional local completely integrally closed domain that is not a valuation domain

1.2 Some Examples and Counterexamples

37

where R + X K [X ] = { f ∈ K [X ] : f (0) ∈ R} and Int(R) = { f ∈ K [X ] : f (R) ⊆ R}. When R is a domain, the t-closure operation has proved to be quite effective for studying all three of these rings of univariate polynomials. The ring Int(R), known as the ring of integer-valued polynomials on R, has been the subject of intense study in the case where R is a domain [27] and is by far the most difficult of the three to grapple with. Remark 1.2.1. Let R be a ring. Using the notion of the symmetric algebra of an R-module, one may construct the ring R[X ] as the symmetric algebra S R (R) of the R-module R, and the ring R + X K [X ] as the symmetric algebra S R (K ) of the R-module K . In this section, as in the previous, we only consider integral domains. We base our first examples on the following theorem. Theorem 1.2.2 ([37, 148]). Let D be an integral domain with quotient field K . (1) (2) (3) (4)

D[X ] is a PVMD (resp., Krull, GCD, a UFD) if and only if D is. D[X ] is DW (resp., Prüfer, Dedekind, Bézout, a PID) if and only if D is a field. D + X K [X ] is a PVMD (resp., Prüfer, GCD, Bézout) if and only if D is. D + X K [X ] is completely integrally closed (resp., Krull, Dedekind, a UFD, a PID) if and only if D is a field. In fact, the complete integral closure of D + X K [X ] is K [X ].

Proofs of statements (1) and (2) of the theorem are outlined in Exercises 2.9.10 and 2.9.11; statement (2) also follows from [148, Proposition 2.12]; and proofs of statements (3) and (4) are outlined in Exercise 2.9.13. By the theorem, the two classes on the left edge of the implication lattice below are preserved under the association D −→ D[X ] but exclude D + X K [X ] for D not a field, the two classes on the far right edge are preserved under D −→ D + X K [X ] but exclude D[X ] for D not a field, while the two classes in the middle edge are preserved under both D  −→ D[X ] and D −→ D + X K [X ]. This allows us to produce examples of integral domains in each of the first nine subclasses, all as listed in Table 1.2, and it further motivates the study of PVMDs and GCD domains. UFD

Krull

Bézout

GCD

PVMD

Prüfer

38

1 Introductory Survey of Multiplicative Ideal Theory

In the proof of Theorem 1.2.2, one uses t-class groups to compare the domains D[X ] and D + X K [X ] with the domain D. Although in general the t-class group Clt (D) is not functorial in D, it is functorial for flat extensions, and both D[X ] and D + X K [X ] are flat extensions of D. More generally, we have the following proposition, which is proved and generalized to rings with zerodivisors in Section 3.3. Proposition 1.2.3 (cf. [190, p. 442–443]). Let D ⊆ D  be an extension of integral domains. Each of the following conditions implies the next. D  is flat as a D-module. I −1 D  = (I D  )−1 for all finitely generated I ∈ Freg (D). (I −1 D  )v = (I D  )−1 for all finitely generated I ∈ Freg (D). If I is any t-closed ideal of D  , then I ∩ D is a t-closed ideal of D. D  is a t-compatible extension of D, that is, I t ⊆ (I D  )t , or equivalently (I D  )t = (I t D  )t , for all I ∈ Freg (D). (6) D  is a t-linked extension of D, that is, I t = D implies (I D  )t = D  for all I ∈ Freg (D). (7) There is a group homomorphism Invt (D) −→ Invt (D  ) induced by the map I −→ (I D  )t from Freg (D)t to Freg (D  )t . (8) There is a group homomorphism Clt (D) −→ Clt (D  ) induced by the map I −→ (I D  )t from Freg (D)t to Freg (D  )t .

(1) (2) (3) (4) (5)

In fact, (4) and (5) are equivalent. Here, Freg (D)∗ for a star operation ∗ on D denotes the image of ∗, that is, the set of all ∗-closed fractional ideals of D. Since D[X ] is free as a D-module, there is an induced group homomorphism Clt (D) −→ Clt (D[X ]). Theorem 1.2.4 ([4, 5, 9, 71, 73, 105, 139]). Let D be an integral domain. (1) The induced group homomorphism Clt (D) −→ Clt (D[X ]) is injective, and it is an isomorphism if and only if D is integrally closed. (2) D[X ] is integrally closed (resp., a v-domain, completely integrally closed, H, of finite t-character, of Krull type) if and only if D is. (3) Suppose that D is integrally closed. Then D[X ] is TV (resp., TW, w-divisorial, Mori) if and only if D is. (4) D[X ] is of finite character if and only if D is a field. A proof of Theorem 1.2.4 is outlined in Exercise 2.9.12. Unfortunately, if D is Mori, then D[X ] need not be [170, 171], and if D is a TV domain, then it is not known whether or not D[X ] need be a TV domain. The following examples follow from Theorems 1.2.2 and 1.2.4.

1.2 Some Examples and Counterexamples

39

Example 1.2.5. (1) Class U: If D is a Noetherian UFD that is not a PID (e.g., D = Z[T ]), then D[X ] is a Noetherian UFD that is not Prüfer. (2) Class K: If D is a Dedekind domain that is not a PID, √ or equivalently a Dedekind domain with nontrivial class group (e.g., D = Z[ −5]), then D[X ] is a Noetherian Krull domain that is neither GCD nor Prüfer. (3) Class G: If D is a Bézout domain that is not a PID, or equivalently a nonNoetherian Bézout domain (e.g., the ring OQ of all algebraic integers or the ring of all entire functions), then D[X ] is GCD but neither Krull nor Prüfer. In fact, the ring OQ is a one dimensional Bézout domain and therefore completely integrally closed, so OQ [X ] is a completely integrally closed GCD domain that is neither Krull nor Prüfer, and therefore neither H nor TD. The domain D + X K [X ] is a special case of what is known as the pullback construction in the category of rings. Let ϕ : S −→ R and ψ : T −→ R be two ring homomorphisms with a common codomain R. The pullback, or fiber product, of S and T (or more precisely of ϕ and ψ) is the subring S × R T = {(x, y) ∈ S × T : ϕ(x) = ψ(y)} of S × T , which those familiar with category theory may recognize as a limit of ϕ

ψ

the diagram S −→ R ←− T in the (bicomplete) category of rings. For example, consider the two homomorphisms ϕ : D −→ K and ψ : K [X ] −→ K , given by ϕ(x) = x and ψ( f ) = f (0), respectively. Then the pullback D × K K [X ] = {(x, f ) ∈ D × K [X ] : f (0) = x} is isomorphic to the ring D + X K [X ] via the projection (x, f ) −→ f onto the second coordinate. Pullbacks are an excellent source of examples and counterexamples in commutative algebra. See [4, 37, 66, 72, 133], for example. Theorem 1.2.6 ([4, 37, 66, 138, 147, 148]). Let D be an integral domain with quotient field K .  (1) D + X K [X ] is flat as a D-module (isomorphic to D ⊕ ∞ n=1 K ), and the induced group homomorphism Clt (D) −→ Clt (D + X K [X ]) is an isomorphism. (2) D + X K [X ] is integrally closed (resp., a v-domain, TW, DW, TD, H) if and only if D is. (3) Suppose that D is not a field. Then D + X K [X ] is not atomic since X cannot be written as a product of irreducible elements of D + X K [X ], and it is not completely integrally closed since the complete integral closure of D + X K [X ] is K [X ]. (4) D + X K [X ] is of finite character if and only if D is semilocal. (5) D + X K [X ] is of finite t-character if and only if D has finitely many t-maximal ideals, if and only if D is semilocal and DW.

40

1 Introductory Survey of Multiplicative Ideal Theory

(6) If D + X K [X ] is TV, then D is semilocal, TV, and DW. (7) D + X K [X ] is a PVMD of finite t-character if and only if D + X K [X ] is a Prüfer domain of finite character, if and only if D is a PVMD with finitely many t-maximal ideals, and if and only if D is a semilocal Prüfer domain. (8) D + X K [X ] is a TV PVMD if and only if D + X K [X ] is a divisorial Prüfer domain, if and only if D is a divisorial valuation domain, and if and only if D is a valuation domain with principal maximal ideal. (9) D + X K [X ] is divisorial if and only if D is a local divisorial domain such that every proper D-submodule of K containing D is a fractional ideal of D. A proof of the theorem is outlined in Exercise 3.11.10. Statements (5)–(8) of the theorem follow from results in [37] and are due to M. Zafrullah (private communication), and statement (9) is due to E. Houston (private communication). Open Problem 1.2.7. Find equivalent conditions on a domain D with quotient field K so that D + X K [X ] is TV. The following examples follow from Theorems 1.2.2 and 1.2.6. Example 1.2.8. (1) Class B: If D is a nonlocal semilocal PID with quotient field K = D, then D + X K [X ] is a Bézout H domain of finite (t-)character that is neither TV nor completely integrally closed. (2) Class PV: Let D be a Krull domain, with quotient field K , that is neither a UFD nor a Dedekind domain (e.g., D = D0 [T ], where D0 is any Dedekind domain that is not a PID); then D + X K [X ] is an H PVMD that is not Krull, completely integrally closed, Mori, TV, atomic, GCD, or Prüfer. Moreover, since D is Krull but not a UFD, D must have infinitely many height one (t-maximal) primes, and therefore D + X K [X ] is not of finite t-character, hence not of Krull type. Proposition 1.2.9 ([15]). Let D ⊆ D  be an extension of integral domains. Each of the following conditions implies the next. I −1 D  = (I D  )−1 for all I ∈ Freg (D). (I −1 D  )v = (I D  )−1 for all I ∈ Freg (D). If I is any v-closed ideal of D  , then I ∩ D is a v-closed ideal of D. D  is a v-compatible extension of D, that is, I v ⊆ (I D  )v , or equivalently (I D  )v = (I v D  )v , for all I ∈ Freg (D). (5) D  is a v-linked extension of D, that is, I v = D implies (I D  )v = D  for all I ∈ Freg (D). (6) There is a group homomorphism Invv (D) −→ Invv (D  ) induced by the map I −→ (I D  )v from Freg (D)v to Freg (D  )v . (7) There is a group homomorphism Clv (D) −→ Clv (D  ) induced by the map I −→ (I D  )v from Freg (D)v to Freg (D  )v .

(1) (2) (3) (4)

1.2 Some Examples and Counterexamples

41

In fact, (3) and (4) are equivalent. Moreover, the extensions D[X ] and D + X K [X ] satisfy all of the conditions above for D  , the extension Int(D) satisfies conditions (2) through (7), and the induced group homomorphisms Clv (D) −→ Clv (D[X ]), Clv (D) −→ Clv (D + X K [X ]), and Clv (D) −→ Clv (Int(D)) are injective. The following proposition is due to M. Zafrullah (private communication); see Exercises 2.9.11 and 2.9.13. Proposition 1.2.10. If D is an integrally closed domain, then D[X ] has trivial vclass group if and only if D has trivial v-class group. If D is a domain with quotient field K , then D + X K [X ] has trivial v-class group if and only if D has trivial v-class group. Consider now the following six properties that an integral domain may possess: divisorial, TD, w-divisorial, TW, DW, and TV. These six properties partition the class of all integral domains into eight disjoint subclasses, as listed in Table 1.3. Equivalent characterizations of the integrally closed domains in each of the eight subclasses are provided in Table 1.4, and examples in each class, as well as completely integrally closed examples, when they exist, are provided in Table 1.5. They are based on Example 1.2.11 below. In the implication diagram following Table 1.4, the DW and TD properties on the right are preserved under the association D −→ D + X K [X ] for all domains D but exclude D[X ] for D not a field; the TV and w-divisorial properties on the left are preserved under the association D −→ D[X ] for integrally closed domains D but may exclude D + X K [X ]; and the TW property is preserved by both D −→ D[X ] for integrally closed domains D and D −→ D + X K [X ] for all domains D.

Table 1.3 Partition of the class of integral domains divisorial w-divisorial TD Class vd wv td tvdw tv dw tw n



 



TV  

  

DW   

TW   

 

42

1 Introductory Survey of Multiplicative Ideal Theory

Table 1.4 Partition of the class of integrally closed domains

Class

Characterization

IC vd IC td IC wv IC tvdw IC dw IC tv IC tw IC n

Prüfer and divisorial Prüfer but not divisorial TV PVMD but not Prüfer IC, TV, and DW but not PVMD IC and DW but neither TV nor PVMD IC and TV but neither DW nor PVMD PVMD but neither TV nor Prüfer IC but not TV, DW, or PVMD

divisorial (v = d)

w-divisorial (v = w)

TV (t = v)

TD (t = d)

TW (t = w)

DW (d = w)

The integrally closed domains satisfying the TD, TW, w-divisorial, divisorial, and DW properties, respectively, are characterized as follows. (1) An integrally closed TD domain is equivalently a Prüfer domain. (2) An integrally closed TW domain is equivalently a PVMD. (3) An integrally closed w-divisorial domain is equivalently a TV PVMD, or equivalently a domain D of Krull type such that D is an H domain (or each t-maximal ideal of D is t-invertible) and each nonzero t-prime of D is contained in a unique t-maximal ideal of D [45, Theorem 3.3]. (4) By statement (3), an integrally closed divisorial domain is equivalently a Prüfer domain D of finite character such that D is an H domain (or each maximal ideal of D is finitely generated) and each nonzero prime ideal of D is contained in a unique maximal ideal of D. It follows that a valuation domain is divisorial if and only if it is w-divisorial, if and only if its maximal ideal is principal. (5) A domain D (integrally closed or not) is DW if and only if every maximal ideal of D is t-maximal [148, Proposition 2.2], if and only if t-Max(D) = Max(D). Consequently, any one dimensional domain is DW since every height one prime is t-prime. A Dedekind domain is equivalently a DW Krull domain, and a Prüfer domain is equivalently a DW PVMD. There is no known useful characterization of the integrally closed TV domains. However, a completely integrally closed TV (or Mori or H) domain is equivalently a Krull domain. An integrally closed TV or Mori domain need not be a v-domain. Moreover, any TV domain is an H domain of finite t-character, but a Bézout H domain of finite (t-)character need not be TV.

1.2 Some Examples and Counterexamples

43

Table 1.5 Examples of integrally closed domains within each class Class

Example: D[X ] for IC Example: D+ X K [X ] D in Class . . . for IC D in Class . . .

IC vd IC td IC wv

td vd, wv

IC tvdw IC dw

dw

CIC example

IC non-CIC example

Z

V + X K [X ]

Int(Z)

Z + X Q[X ]

Z[X ]

(V + X K [X ])[Y ]

does not exist

I = Q + X Q(T )[X ]

C

I + Y L[Y ]

does not exist

I [Y ]

IC tv

tvdw, tv

IC tw

td, tw

wv, tw

Int(Z)[X ]

(Z + X Q[X ])[Y ]

IC n

dw, n

tv, n

C[X ]

(I + Y L[Y ])[Z ]

C is any one dimensional local completely integrally closed domain that is not a valuation domain, V is any DVR with quotient field K = V , and L = Q(T, X ) is the quotient field of I

Example 1.2.11. (1) Class vd: Any Dedekind domain is divisorial. A Noetherian divisorial domain is equivalently a one dimensional Gorenstein domain. Every number field of degree greater than two has infinitely many orders that are divisorial and infinitely many orders that are not divisorial [113]. (2) Class td: Any valuation domain with non-principal maximal ideal is a nondivisorial (TD) Prüfer domain. (3) Class wv: If D is any TV PVMD that is not a field, then D[X ] is also a TV PVMD but is neither DW nor Prüfer. (4) Class tvdw: The domain K + X L[X ], where L/K is a proper field extension such that K is algebraically closed in L, is an example of a one dimensional integrally closed Mori domain that is not a v-domain [68, p. 161]. By [148, Theorem 3.1(3)], the domain K + X L[X ] is DW, and therefore K + X L[X ] is Mori, TV, and DW but not TW. (5) Class tw: If D is any non-divisorial Prüfer domain, then D[X ] is a PVMD that is neither TV nor Prüfer. (6) Class tv: Let D be an integrally closed Mori domain that is not a PVMD, such as the domain K + X L[X ] of example (4); then the domain D[Y ] is an integrally closed Mori domain, hence TV domain, that is neither TW nor DW. (7) Class dw: By [148, Example 2.1(1)], the domain Z + X Q(T )[[X ]] is an integrally closed DW domain that is not a PVMD and is neither TW nor TV. In 1952–55, Nagata constructed an example of a one dimensional local completely integrally closed domain that is not a valuation domain and consequently is neither TV nor a PVMD. Any such domain is also in Class dw. (8) Class n: If D is any integrally closed domain that is neither TV nor a PVMD, then both D[X ] + Y K (X )[Y ] and (D + Y K [Y ])[X ] are not DW, TV, or TW. Moreover, if D is in Class dw, then D[X ] is in Class n.

44

1 Introductory Survey of Multiplicative Ideal Theory

A domain D is said to be almost Dedekind if Dp is a DVR for every maximal ideal p of D. One has the irreversible implications Dedekind ⇒ almost Dedekind ⇒ Prüfer, which are reversible for the Noetherian domains. Clearly, every almost Dedekind domain is completely integrally closed and of dimension at most one. Example 1.2.12. (1) The domains Z + X Q[X ] and Int(Z) are both two dimensional Prüfer domains that are not almost Dedekind. The domain Z + X Q[X ] is Bézout but not completely integrally closed, while the domain Int(Z) is completely integrally closed but not Bézout. (The ring E of all entire functions, on the other hand, is an infinite dimensional completely integrally closed Bézout domain.) (2) A local domain is almost Dedekind if and only if it is a DVR, if and only if it is a Noetherian valuation domain. In particular, any non-Noetherian valuation domain is a Prüfer domain that is not almost Dedekind. In fact, any one dimensional valuation domain that is not a DVR is a one dimensional completely integrally closed Prüfer domain that is not almost Dedekind. (3) The existence of a non-Noetherian (non-Dedekind) almost Dedekind domain was conjectured by Krull, and an example was first constructed in 1953 by Nakano [152], the example being the integral closure of Z in the smallest field containing Q and all complex pth roots of unity for all primes p. In the early 1990s, it was shown that there exist such examples with all residue fields finite [31, 80]. Further examples can be obtained as rings of integer-valued polynomials. The articles [48, 49, 179], among others, provide applications of t-closure to integervalued polynomial rings. One may have Int(Dp )  Int(D)p if p is a prime ideal of a domain D, even if D is almost Dedekind, so the technique of localization is not always helpful when studying Int(D). Nevertheless, one has the following. Theorem 1.2.13 ([27, 48]). Let D be an integral domain with quotient field K . (1) For any prime ideal p of D, one has Int(Dp ) = Int(D)p = Dp [X ] if p is not a weak Bourbaki associated prime of K /Dp , hence if p is not a t-maximal ideal of D, with finite residue field. (2) If D is an H domain (or TV or Mori or Noetherian), then Int(D) = D[X ] if and only if there is a t-maximal ideal (or equivalently, a maximal conductor ideal) of D with finite residue field. (3) If D is of finite character or of finite t-character (or TV or Mori or Noetherian or of Krull type), or more generally if D is an intersection of a locally finite collection {Dp : p ∈ } of localizations of D for some subset  of Spec(D) (which holds if and only if every nonzero proper conductor ideal of D is contained in a finite and nonzero number of primes in ), then Int(U −1 D) = U −1 Int(D) for every multiplicative subset U of D.

1.2 Some Examples and Counterexamples

45

(4) If D is Dedekind, then Int(D) is free as a D-module. (5) If D is a PVMD such that Int(Dp ) = Int(D)p for all p (e.g., if D is of Krull type), then Int(D) is locally free, hence flat, as a D-module. Generally, then, when studying Int(D) it often suffices to consider localizations of Int(D) at the t-maximal ideals of D with finite residue field. Although it is unknown whether or not Int(D) is necessarily free (or flat, or locally free) as a D-module, by [179, Proposition 2.1] it is always a t-compatible extension of D, and in particular there is an induced group homomorphism Clt (D) −→ Clt (Int(D)). For any integral domain D, we let X 1 (D) denote the set of all height one primes of D. Theorem 1.2.14 ([27, 28, 74, 179]). Let D be an integral domain with quotient field K = D. The induced group homomorphism Clt (D) −→ Clt (Int(D)) is injective. Moreover, one has the following. (1) Int(D) is integrally closed (resp., is completely integrally closed, satisfies ACCP), if and only if D is integrally closed (resp., is completely integrally closed, satisfies ACCP). (2) If Int(D) is a PVMD (resp., Krull, GCD, Prüfer, UFD, Dedekind, Bézout, PID, Noetherian, Mori), then so is D. (3) If D is Noetherian, or more generally if D is Jaffard, that is, if the Krull dimension dim D of D is equal to the supremum of the Krull dimensions of the valuation overrings of D, then dim(Int(D)) = dim(D[X ]) = 1 + dim D. (4) If Int(D) is a Prüfer domain, then D is an almost Dedekind domain with all residue fields finite. (5) If Int(Dp ) = Int(D)p for every prime ideal p of D, then Int(D) is a Prüfer domain if and only if D is an almost Dedekind domain with all residue fields finite. (6) Int(D) is Krull if and only if D is Krull and Int(D) = D[X ]. (7) If D is a Krull domain, then Int(D) is a PVMD, and the converse holds if D is Noetherian. More generally, if D is a domain of Krull type, then Int(D) is a PVMD if and only if all prime ideals p of D such that Int(D)p = Dp [X ] are of height one (or equivalently are such that Dp is completely integrally closed). (8) If D is an H domain of Krull type (e.g., if D is a TV PVMD), then Int(D) is a PVMD if and only if Dp is one dimensional, or equivalently completely integrally closed, for all t-maximal ideals of D with finite residue field. If D also has all residue fields finite, then Int(D) is a PVMD if and only if wAss(K /D) ⊆ X 1 (D), if and only if Ass(K /D) = X 1 (D), if and only if t-Max(D) = X 1 (D), if and only if D is t-locally one dimensional, and if and only if D is t-locally completely integrally closed. (9) If D is of finite t-character, then Int(D) is a GCD domain if and only if D is a GCD domain and Int(D) = D[X ]. (10) Let D be a one dimensional Noetherian domain.

46

1 Introductory Survey of Multiplicative Ideal Theory

(a) If Int(D) is Mori, then Int(D) = D[X ]. (b) Let p be any maximal ideal of D with finite residue field. Then the prime ideals of Int(D) lying above p are in one-to-one correspondence with the p via the correspondence α −→ pα = { f ∈ Int(D) : f (α) ∈ completion D p }, and all such prime ideals are maximal with residue field D/p. In fact, pD p is transcendental or pα has height one or height two according as α ∈ D algebraic over K . (c) If D is Dedekind and has a maximal ideal p with finite residue field, then the maximal ideals pα are all t-maximal, and therefore X ∈ Int(D) lies in uncountably many t-maximal ideals of Int(D), so in particular Int(D) is not of finite t-character. The theorem above yields the following examples. Example 1.2.15. Let D be a Dedekind domain that is not a PID. (1) If D has all residue fields finite, then Int(D) = D[X ] is a two dimensional Prüfer domain that is neither Krull nor GCD. Moreover, X ∈ Int(D) lies in uncountably many (t-)maximal ideals of Int(D) with finite residue field, so in particular Int(D) is not of finite (t-)character. It follows, then, that Int(D) is also not divisorial, hence not a TV domain. (2) If D has all residue fields infinite, then Int(D) = D[X ] is a two dimensional Krull domain that is neither GCD nor Prüfer. (3) If D has both finite and infinite residue fields, then Int(D) = D[X ] is a two dimensional completely integrally closed PVMD satisfying ACCP, but it is not Prüfer, GCD, Krull, of Krull type, Mori, TV, H, or of finite t-character. Several more counterexamples can be found among integer-valued polynomial rings using the following theorem. Theorem 1.2.16 98, 179]). Let V be a valuation domain that is not a field,  ([27, n m , where m is the maximal ideal of V . and let p = ∞ n=1 (1) V is Noetherian if and only if V is a DVR. (2) V [X ] is of finite t-character. (3) V is completely integrally closed if and only if V is one dimensional, if and only if Int(V ) is completely integrally closed, if and only if Int(V ) is a PVMD, and if and only if Int(V ) = V [X ] or V is a DVR with finite residue field. (4) Int(V ) is of Krull type if and only if Int(V ) = V [X ]. (5) V is divisorial if and only if the maximal ideal m of V is principal (or finitely generated). In either case, V /p is a DVR, and V is one dimensional if and only if V is a DVR, if and only if p = (0).

1.2 Some Examples and Counterexamples

47

(6) Int(V ) = V [X ] if and only if the maximal ideal of V is principal with finite residue field. In either case, one has Int(V ) ⊆ Vp [X ], and the prime ideals of Int(V ) lying above m are in one-to-one correspondence with the completion of the DVR V /p at m/p via the correspondence α −→ { f ∈ Int(V ) : f (α) ∈  m/p}, and all such prime ideals are maximal and t-maximal with residue field V /m. In particular, X ∈ Int(V ) lies in uncountably many t-maximal ideals of Int(V ), so Int(V ) is not of finite t-character. Example 1.2.17. Let V be a valuation domain that is not a DVR. (1) Since V is Bézout but not a UFD, it is not atomic and therefore does not satisfy ACCP and is not Mori. Moreover, dim(Int(V )) = 1 + dim V . (2) If V is one dimensional, then V is a one dimensional completely integrally closed valuation domain, hence local Bézout and of Krull type, that is not Krull, atomic, Mori, TV, H, or divisorial, and Int(V ) = V [X ] is a two dimensional completely integrally closed GCD domain of finite t-character that is not Krull, Prüfer, atomic, Mori, TV, TD, H, or of finite character. (3) If the maximal ideal of V is principal with finite residue field, then Int(V ) = V [X ] is an integrally closed domain that is not a PVMD (hence not TW), of finite t-character (hence not TV), or completely integrally closed. Moreover, Int(V ) is not DW since by [179, Proposition 2.4] and [27, Lemma V.1.3] it has maximal ideals that are not t-maximal. In particular, one has v = t = w = d on Int(V ), while Int(V ) is integrally closed. On the next page is the smallest full and complete implication lattice that includes the field, PID, Dedekind, Bézout, Prüfer, UFD, Krull, GCD, PVMD, strong GCD, CIC, v-GCD, v-domain, and integrally closed conditions along with the divisorial, w-divisorial, TD, TV, DW, and TW conditions. The lattice has 37 vertices, and the 20 properties partition the class of all integral domains into the 37 disjoint nonempty subclasses characterized in Table 1.6.

Krull

PVMD

v-domain

CIC

TW

IC

divisorial

IC TV

CIC DW

CIC PVMD

w-divisorial

DW v-domain

GCD

CIC Prüfer

TV PVMD

domain

TV

TV DW

Prüfer

IC divisorial

v-GCD

strong GCD

CIC GCD

v-GCD Bézout

div. Bézout

CIC Bézout

Dedekind

strong GCD Bézout

TV GCD

UFD

PID

field

TD

IC DW

IC TV DW

Bézout

DW

48 1 Introductory Survey of Multiplicative Ideal Theory



 

    

  



F P D U K B-S B-C B-V B-tv B P-C P-tv P G-S G-C G-V G-tv G PV-C PV-tv PV

 

Dedekind UFD Krull

Field PID

Class

    

 

Bézout

Table 1.6 Partition of the class of integral domains

       

  

    

    



 

Prüfer GCD                      





 

PVMD str. GCD



 



      

CIC

 



 





 

                    

                    

vvIC GCD dom.





  

div.









    

wdiv.

       

  

TD









    

TV

                    

TW

(continued)

       

  

DW

1.2 Some Examples and Counterexamples 49

C-dw C V-dw V Itvdw I-tv I-dw I N-vd N-wd N-td Ntvdw N-tv N-dw N-tw N

Class

Field PID

Table 1.6 (continued)

Dedekind UFD Krull

Bézout

Prüfer GCD

PVMD str. GCD    

 

  

    

vvIC GCD dom.

CIC



div.

 

wdiv.





TD





 





TV



 











DW



  

TW

50 1 Introductory Survey of Multiplicative Ideal Theory

1.2 Some Examples and Counterexamples

51

Table 1.7 Examples of integral domains within each class Class Contains D[X ] Contains Example for D in Class . . . D+ X K [X ] for D in Class . . . F P D U K B-S

F P, U D, K

Q Z D Z[X ] D[X ] OQ

B-C B-V

V B-S, B-V

B-tv B P-C P-tv P G-S

B-S, G-S

G-C G-V

B-C, G-C B-V, G-V

G-tv G PV-C PV-tv PV

B-tv, G-tv B, G P-C, PV-C P-tv, PV-tv P, PV

C-dw C V-dw V I-tvdw I-tv I-dw I N-vd N-wv N-td N-tvdw N-tv N-dw N-tw N

F

B-C, B

D, P-C, P-tv, P

G-S, G-V U, G-C, G-tv, G

K, PV-C, PV-tv, PV

C-dw, C V-dw, V

C-dw, V-dw C, V

I-tvdw, I-tv I-dw, I

I-dw I

N-td

N-dw, N

N-dw N-wv, N-tw N-tv, N

OQ + X Q[X ]

W V + X K [X ] Int(Z) P D + X K [X ] OQ [X ]

V [X ] OQ [X ] + Y Q(X )[Y ] W [X ] (V + X K [X ])[Y ] Int(Z)[X ] P[X ] (D + X K [X ])[Y ]

C C[X ] C + X K [X ] (C + X K [X ])[Y ] I = Q + T C[T ] I [X ] I + X K [X ] (I + X K [X ])[Y ] N N [X ] N + X K [X ] M M[X ] M + X K [X ] N [X ] + Y K (X )[Y ] (M + X K [X ])[Y ]

52

1 Introductory Survey of Multiplicative Ideal Theory

Examples of domains in each of the 37 classes are provided in Table 1.7. In the table, domains D, V , W , P, C, N , and M can be chosen as follows. √ (1) D is any non-PID Dedekind domain (e.g., Z[ −5]). (2) V is any valuation domain with value group isomorphic to a proper dense subgroup of R. (3) W is any non-DVR valuation domain with principal maximal ideal. (4) P is any non-Dedekind non-Bézout divisorial Prüfer domain. (5) C is any one dimensional local completely integrally closed domain that is not a valuation domain. (6) N √ is any order, in a number field, that is Gorenstein but not Dedekind, e.g., Z[ 5] [113, Proposition 3.6]. √ √ (7) M is any order, in a number field, that is not Gorenstein, e.g., Z[2 3 2, 2 3 4] [113, Example 7.3]. In the table, K denotes the quotient field of each of the respective domains.

1.3 A Preview of Rings with Zerodivisors The goal of Chapters 2 and 3 is to prove the results mentioned in Section 1.1 and to generalize them to rings with zerodivisors. Rather than working with an integral domain and its quotient field, we work with a commutative ring and its total quotient ring. For greater generality we work with semistar operations rather than star operations. A (unital) semistar operation on a commutative ring R with total quotient ring K is a closure operation  on the poset K(R) of all R-submodules of K such that R  = R and I  J  ⊆ (I J ) for all I, J ∈ K(R). An R-submodule I of K is regular if I contains a regular element (that is, a non-zerodivisor) of R, and I is fractional if I −1 = (R : K I ) is regular. Furthermore, I is -invertible if there exists an R-submodule J of K such that (I J ) = R. We are primarily concerned in Chapters 2 and 3 with the regular R-submodules of K , and we are particularly interested in studying conditions on a commutative ring R that guarantee that various classes of regular fractional R-submodules of K are -invertible or principal. A simple example of the type of generalization we are interested is the notion of a Dedekind ring. Recall that a Dedekind domain is equivalently an integral domain in which every nonzero ideal is invertible. More generally, a Dedekind ring is a ring in which every regular ideal is invertible. Notice that the word “nonzero” has been replaced with the word “regular.” “Regular” is not the only way to generalize “nonzero”—another is “semiregular,” and yet another is “dense”—but it is the notion most naturally associated to the total quotient ring of a ring. Chapter 5 of this book considers alternatives to the total quotient ring, the largest of which is called the complete ring of quotients, where the “dense” R-submodules play the role of “nonzero.” Such alternative notions of “ring of quotients” and “nonzero” lead to analogous but more difficult (and, surprisingly, less general) theories.

1.3 A Preview of Rings with Zerodivisors

53

A superb example of a theorem that can be generalized from integral domains to commutative rings concerns the integral closure of a Noetherian ring. As noted in Section 1.1, the integral closure of a Noetherian integral domain need not be Noetherian, but it must always be a Krull domain. In 1976, Huckaba introduced the notion of a Krull ring that is used today and proved that the integral closure of any Noetherian ring is a Krull ring [106, Corollary 2.3]. This, along with many related results, provides a strong motivation for studying Krull rings with zerodivisors. Given that an integral domain D is a Krull domain if and only if every nonzero ideal of D is t-invertible, it is no surprise that a commutative ring R is a Krull ring if and only if every regular ideal of R is t-invertible, where the t-operation is the largest semistar operation on R of finite type. In Chapters 2 and 3, we take this as the definition of a Krull ring and from there deduce the theory of Krull domains and Krull rings. With this approach the t-operation occupies a central role from the start and takes precedence, even, over the methods of localization and valuations. In Chapters 2 and 3, we are interested in studying the following conditions for any semistar operation , each defined as in the following table: -PIR, -Bézout, -Dedekind, and -Prüfer. Ring -PIR -Bézout -Dedekind -Prüfer

Characterization every -closed regular ideal is principal -closure of every f.g. regular ideal is principal every regular ideal is -invertible every f.g. regular ideal is -invertible

If  and  are finite type semistar operations on R with   , then one has the following full lattice of implications.  -PIR

-PIR

 -Dedekind

 -Bézout

-Dedekind

-Bézout

 -Prüfer

-Prüfer

The upward direction in the lattice is the “impose Cl (R) = 0” axis, the upward-right direction is the “impose    ” axis, and the upward-left direction is the “impose  Noetherian” axis. Here, one writes    if I  = I  for all regular ideals I of R, and one says that R is -Noetherian if R satisfies the ascending chain condition on the regular -closed ideals of R. Moreover, the -class group Cl (R) of R is the group of -invertible regular fractional ideals of R under the operation (I, J ) −→ (I J ) of -multiplication modulo the subgroup of regular principal fractional ideals of R.

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1 Introductory Survey of Multiplicative Ideal Theory

Letting  be the trivial semistar operation d, the conditions mentioned above are as follows: r-PIR, r-Bézout, Dedekind, and Prüfer, respectively. Letting  be the t-operation, they are r-UFR, r-GCD, Krull, and PVMR, respectively. It follows that the full lattice of implications above yields the following full lattice of implications. r-PIR

r-UFR

Dedekind

r-Bézout

Krull

r-GCD

Prüfer

PVMR

Moreover, for integral domains, the full implication lattice above is equivalent to the following full implication lattice (which is an implication diagram that was discussed at great length in Section 1.1). PID

UFD

Dedekind

Bézout

Krull

GCD

Prüfer

PVMD

The tables on the next four pages summarize many of the characterizations of these classes of integrally closed rings that are proved in Chapters 2 and 3. In the tables, R is a commutative ring with total quotient ring T (R) and  is a semistar operation on R. Each class of rings in the table is defined via the first characterization in the table for that class. Although some of the terms used in the table have not been defined

1.3 A Preview of Rings with Zerodivisors

55

yet, they will be in the course of the next two chapters. The tables are here to provide a reference guide for the reading and a preview of what is to come. Ring

Characterization

IC

every element of T (R) that is integral over R lies in R for all a ∈ T (R), if R[a] is f.g. as an R-module, then a ∈ R (I :T (R) I ) = R for all f.g. regular ideals I c R ( f g)v = (c R ( f )c R (g))v for all f, g ∈ R[X ] with c R ( f ) regular all f.g. regular ideals are v-invertible if I v ⊇ J v with I, J reg. ideals and I f.g., then J v = (H I )v for some reg. ideal H generalized v-Prüfer, and ((I + J )(I ∩ J ))v = (I J )v for all f.g. regular ideals I, J generalized v-Prüfer, and ((I + J )(I ∩ J ))v = (I J )v for all regular ideals I, J generalized v-Prüfer, and if I, J are regular and v-invertible, then I + J is v-invertible all f.g. regular ideals are t-invertible t-Prüfer w-Prüfer if I t ⊇ J t with I, J reg. ideals and I f.g., then J t = (H I )t for some reg. ideal H v-coherent and v-Prüfer integrally closed and w  t integrally closed and (I : R J )t = (I t : R J ) for all reg. ideals I, J with J f.g. (R[p] , [p]R[p] ) is a valuation pair of T (R) for all p ∈ t -Max(R) R[p] is a PVMR (or a valuation ring) for all p ∈ t-Max(R) and R is v-coherent c R ( f g)w = (c R ( f )c R (g))w for all f, g ∈ R[X ] with c R ( f ) regular every element of T (R) that is almost integral over R lies in R v-Dedekind for all a ∈ T (R), if R[a] is a fractional ideal of R, then a ∈ R (I :T (R) I ) = R for all regular ideals I all regular ideals are v-invertible if I v ⊇ J v with I, J reg. ideals, then J v = (H I )v for some reg. ideal H all regular ideals are t-invertible t-Dedekind CIC and Mori CIC and t  v CIC and H t-Prüfer and t-Noetherian v-Prüfer and v-Noetherian v-Dedekind and v-Noetherian (R[p] , [p]R[p] ) is a discrete rk. one valuation pair of T (R) for all p ∈ t -Max(R) and R is of finite t-character R[p] is Krull (or a Krull valuation ring) for all p ∈ wAss(T (R)/R) and every reg. elt. of R lies in only finitely many p ∈ wAss(T (R)/R) all f.g. regular ideals are invertible d-Prüfer PVMR and t  d integrally closed and t  d if I ⊇ J are reg. ideals with I f.g., then J = H I for some reg. ideal H quasi-Marot, and (I + J )(I ∩ J ) = I J for all f.g. regular ideals I, J quasi-Marot, and (I + J )(I ∩ J ) = I J for all regular ideals I, J quasi-Marot, and if I, J are regular and invertible, then I + J is invertible (R[p] , [p]R[p] ) is a valuation pair of T (R) for all p ∈ Max(R) R[p] is Prüfer (or a Prüfer valuation ring) for all p ∈ Max(R) c R ( f g) = c R ( f )c R (g) for all f, g ∈ R[X ] with c R ( f ) regular

v-Prüfer

PVMR

CIC

Krull

Prüfer

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1 Introductory Survey of Multiplicative Ideal Theory

Ring

Characterization

Dedekind

all regular ideals are invertible d-Dedekind Krull and t  d Krull and every regular prime ideal is maximal Krull and every regular t-maximal ideal is maximal integrally closed, d-Noetherian, and every regular prime ideal is maximal CIC and v  d Prüfer and d-Noetherian Prüfer and Krull if I ⊇ J are reg. ideals, then J = H I for some reg. ideal H (R[p] , [p]R[p] ) is a discrete rk. one valuation pair of T (R) for all p ∈ Max(R) and R is of finite character R[p] is Dedekind (or a Dedekind valuation ring) for all p ∈ Max(R) and R is of finite character v-closure of every f.g. regular ideal is principal r-GCD, that is, PVMR with trivial t-class group t-Prüfer with trivial t-class group t-Marot GCD t-Bézout every f.g. regular ideal is principal d-Bézout Prüfer with trivial class group every regular v-closed ideal is principal CIC with trivial v-class group v-Dedekind with trivial v-class group v-Marot strong GCD t-Marot strong GCD every regular t-closed ideal is principal r-UFR, that is, Krull with trivial t-class group Krull and every regular t-maximal ideal is principal t-Dedekind with trivial t-class group v-PIR and v  t t-Marot factorial t-Marot strong GCD and t  v every regular non-unit is a product of primes factorial and every r-prime element is prime every regular ideal is principal d-PIR r-UFR and t  d Dedekind with trivial class group Dedekind and every regular maximal ideal is principal v-PIR and v  d Marot strong GCD and v  d

v-Bézout

r-Bézout v-PIR

t-PIR

r-PIR

1.3 A Preview of Rings with Zerodivisors

Ring

Characterization

-Prüfer

all f.g. regular ideals are -invertible -Prüfer if I  ⊇ J  with I, J reg. ideals and I f.g., then J  = (H I ) for some reg. ideal H generalized -Prüfer, and ((I + J )(I ∩ J )) = (I J ) for all f.g. regular ideals I, J generalized -Prüfer, and ((I + J )(I ∩ J )) = (I J ) for all regular ideals I, J generalized -Prüfer, and if I, J are regular and -invertible, then I + J is -invertible all f.g. regular ideals are t -invertible PVMR and t  t w -Prüfer -coherent and -Prüfer integrally closed and w  t c R ( f g)w = (c R ( f )c R (g))w for all f, g ∈ R[X ] with c R ( f ) regular (R[p] , [p]R[p] ) is a valuation pair of T (R) for all p ∈ t -Max(R) all regular ideals are -invertible CIC and   v if I  ⊇ J  with I, J reg. ideals, then J  = (H I ) for some reg. ideal H all regular ideals are t -invertible Krull and t  t -Dedekind and   t -Dedekind and -Noetherian -Prüfer and -Noetherian t -Prüfer and t -Noetherian (R[p] , [p]R[p] ) is a discrete rk. one valuation pair of T (R) for all p ∈ t -Max(R), R is of finite t-character, and t  t -closure of every f.g. regular ideal is principal t -Prüfer with trivial t -class group PVMR with trivial t-class group and t  t t-Marot GCD and t  t t-Bézout and t  t t -Bézout every regular -closed ideal is principal -Dedekind with trivial -class group CIC with trivial v-class group and   v v-Marot strong GCD and   v v-PIR and   v every regular t -closed ideal is principal t -Dedekind with trivial t -class group Krull with trivial t-class group and t  t v-Marot strong GCD and t  v t-PIR and t  t -PIR and   t every regular non-unit is a product of primes, and t  t factorial ring in which every r-prime element is prime, and t  t

t -Prüfer

-Dedekind

t -Dedekind

-Bézout

-PIR

t -PIR

57

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1 Introductory Survey of Multiplicative Ideal Theory

Ring

Characterization

r-atomic factorial

every regular non-unit is a product of irreducible elements every regular non-unit is a product of irreducible elements and any such factorization is unique up to reordering and associates every regular non-unit is a product of r-prime elements r-atomic and every regular irreducible element is r-prime r-atomic and weak GCD pt -closure of every regularly generated ideal is principal Krull with trivial t-class group t-PIR, that is, every regular t-closed ideal is principal Krull and every regular t-maximal ideal is principal t-Dedekind with trivial t-class group v-PIR and v  t t-Marot factorial t-Marot strong GCD and t  v every regular non-unit is a product of primes factorial and every r-prime element is prime PVMR with trivial t-class group v-Bézout t-Bézout t-closure of every f.g. regular ideal is principal t-Prüfer with trivial t-class group t-Marot GCD gcd(I ) exists for all f.g. regular ideals I gcd(I ) exists for all regular ideals I gcd(I ) exists for all f.g. regularly generated ideals I gcd(I ) exists for all regularly generated ideals I ACC on the regular -closed ideals ∀ regular ideal I ∃ a f.g. ideal J ⊆ I with I  = J  t -Noetherian t-Noetherian v-Noetherian ACC on the regular divisorial ideals d-Noetherian ACC on regular ideals every regular ideal is finitely generated all reg. ideals are regularly generated (a, b) is regularly generated for all a, b with a regular ∀ reg. ideal I ∃ a regularly generated J with I  = J  ∀ a, b with a regular ∃ a regularly generated J with (a, b) = J  I −1 is strictly -finite for every f.g. regular ideal I of R any regular element lies in only finitely many maximal ideals any regular element lies in only finitely many t-maximal ideals vd vw t d t v t w dw

r-UFR

r-GCD

GCD strong GCD weak GCD weak strong GCD -Noetherian

Mori

r-Noetherian

Marot -Marot -coherent fin. character fin. t-character divisorial w-divisorial TD TV TW DW

Exercises

59

Exercises Section 1.1 1. Let D be an integral domain with quotient field K . Show that K is a fractional ideal of D if and only if K = D. 2. Let D be an integral domain. Prove the following. a) ((I −1 )−1 )−1 = I −1 for any nonzero fractional ideal I of D. b) v = ((−)−1 )−1 is a star operation on D. c) v is the largest star operation on D. 3. Show that a domain D is a TD domain (that is, t = d on Freg (D)) if and only if every nonzero finitely generated ideal of D is divisorial. 4. Let D be an integral domain with quotient field K . Show the following. a) t and w are star operations on D. b) I t = {x ∈ K : (x/a1 ) ∩ · · · ∩ (x/an ) ⊆ D for some nonzero a1 , . . . , an ∈ I } for any fractional ideal I of D. 5. Show by example and direct computation that not every ideal of Z[X ] is v-closed. 6. Let D be an integral domain with quotient field K . a) Let ∗ be a star operation on D. Show that (I J )∗ = I J ∗ for all I, J ∈ Freg (D) with I invertible. b) Let ∗ be a self-map of Freg (D). Consider the following axioms. (4 ) (a I )∗ = a I ∗ for all I ∈ Freg (D) and all nonzero a ∈ K . (4 ) (I ∗ J ∗ )∗ = (I J )∗ for all I, J ∈ Freg (D). Show that, assuming axioms (1)–(3) for star operations, axiom (4) for star operations and axioms (4 ) and (4 ) are all equivalent. In the existing literature, the star operation axioms are usually taken to be (1)–(3), (4 ), and (5). 7. Let D be an integral domain, and let ∗ be a self-map of Freg (D). Show that ∗ is a star operation on D if and only if D ∗ = D and (I ∗ : K J ) = (I ∗ : K J ∗ ) for all I, J ∈ Freg (D). 8. Let I and J be fractional ideals of an integral domain D. Prove the following. a) If I J = D, then J = I −1 . b) If ∗ is a star operation on D and (I J )∗ = D, then J ∗ = I −1 . 9. Let D be an integral domain with quotient field K . Let {Dλ : λ ∈ } be a collection  of rings lying between D and K whose intersection is D. Show that I −→ {I Dλ : λ ∈ } is a star operation on D. 10. a) Show that, if p is a prime ideal of a Noetherian integral domain D, then one need not have I v (D/p) ⊆ (I (D/p))v for every ideal I of D. b) Let D = Z[X ] and D  = Z[X/2], and let I = (2, X )D. Show that I v = D and (I D  )v = 2D  and therefore I v D   (I D  )v . Conclude that, if D  is a Noetherian overring of a Noetherian integral domain D, then one need not have I v D  ⊆ (I D  )v for every ideal I of D. 11. Find an example of a Noetherian domain that is not of finite character.

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12. Complete the following implication diagram to a full implication lattice. Dedekind

1-dim. Goren.

IC Noetherian

IC divisorial

CIC Prüfer

Krull

Noetherian

divisorial

Prüfer

CIC PVMD

Mori

TD

PVMD

CIC

IC

13. Draw the smallest full and complete implication lattice that includes the PID, Dedekind, Bézout, Prüfer, UFD, Krull, GCD, PVMD, v-Noetherian, and vcoherent conditions. 14. In the following implication diagram, try to find as many pairs of classes as possible whose supremum in the lattice is not the intersection of the two classes. As a challenge, try to complete the diagram to a full implication lattice.

Exercises

61 Dedekind

1-dim. Goren.

IC Noetherian

Noeth. div.

IC divisorial

Krull

divisorial

H Prüf. fin. char.

TV PVMD

H Prüfer

Mori

H Krull type

Prüfer

Prüf. fin. char.

TV

H PVMD

TD

TD fin. char.

H TW fin. t-char.

PVMD

Krull type

H TW

TW

TW fin. t-char.

H fin. t-char.

H

fin. t-char.

domain

Section 1.2 1. 2. 3. 4. 5. 6. 7.

Using Theorems 1.2.2 and 1.2.4, verify the examples in Example 1.2.5. Using Theorems 1.2.2 and 1.2.6, verify the examples in Example 1.2.8. Using Theorems 1.2.2 and 1.2.6, verify the examples in Example 1.2.12. Using Theorem 1.2.14, verify the examples in Example 1.2.15. Using Theorem 1.2.16, verify the examples in Example 1.2.17. In Proposition 1.2.3, verify that (4) ⇔ (5) ⇒ (6) ⇒ (7) ⇒ (8). Using Exercise 1.1.1.3, show that the overring Z[X/2] of Z[X ] is not t-linked.

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8. Prove the following using results from the section. a) If D is a Dedekind domain that is not a PID, then D + X K [X ] is an H Prüfer domain that is not TV, Bézout, Dedekind, or completely integrally closed. b) If D is a Prüfer domain that is neither Bézout nor Dedekind, then D[X ] is a PVMD that is not Krull, GCD, or Prüfer. c) If D is a Dedekind domain that is not a PID, then both (D + X K [X ])[Y ] and D[Y ] + X K (Y )[X ] are H PVMDs that are not TV, Krull, GCD, Prüfer, or completely integrally closed. 9. Let D be an integral domain with quotient field K , and let R = D + X K [X ]. Let I be an ideal of R. Show that the following conditions are equivalent. 1) I ∩ D = (0). 2) X K [X ]  I . 3) I K [X ] = K [X ]. 10. Let D be an integral domain with quotient field K , and let R = D + X K [X ]. Prove the following. a) Every ideal of R is of the form f I R = f (I + X K [X ]), where f ∈ K [X ] and I is a nonzero D-submodule of K such that f (0)I ⊆ D. b) The finitely generated ideals of R are the ideals f J R, where f ∈ R and J is a finitely generated ideal of D. 11. Let D be an integral domain with quotient field K , and let R = D + X K [X ]. Using the results of the previous exercise, prove that R is a Bézout domain (resp., Prüfer domain) if and only if D is. 12. Let D be an integral domain with quotient field K , and let R = D + X K [X ]. Prove the following. a) The nonzero prime ideals of R are the ideals p + X K [X ], where p is a prime ideal of D, along with the principal ideals f R, where f ∈ K [X ] is irreducible in K [X ] and f (0) = 1. b) The maximal ideals of R are the ideals p + X K [X ], where p is a maximal ideal of D, along with the principal ideals f R, where f ∈ K [X ] is irreducible in K [X ] and f (0) = 1. 13. Let D be an integral domain with quotient field K . Let I [X ] = I D[X ] for all ideals I of D. Prove the following. a) I [X ] ∩ D = I for all ideals I of D. b) (J : K I )[X ] = (J [X ] : K (X ) I [X ]) for all nonzero ideals I, J of D. In particular, I v [X ] = I [X ]v . c) I t [X ] = I [X ]t for all nonzero ideals I of D. d) The map I −→ I [X ] is an inclusion preserving injection from the set of t-closed ideals of D to the set of t-closed ideals of D[X ], and its left inverse is the map J −→ J ∩ D. e) I is t-invertible in D if and only ifI [X ] is t-invertible in D[X ]. 14. (∗∗) A domain D is essential if D = p∈Ess(D) Dp , where Ess(D) is the set of all essential primes of D. Extend Tables 1.5, 1.6, and 1.7 to include any or all of the following properties: essential, Noetherian, Mori, H, of finite character, of finite t-character, coherent, and v-coherent.

Chapter 2

Semistar Operations on Commutative Rings

In this chapter, we develop the theory of semistar operations on commutative rings and apply the theory to study many important classes of integral domains, such as the Dedekind domains, Prüfer domains, UFDs, Krull domains, and PVMDs, along with natural generalizations of these classes of domains to rings with zerodivisors. We postpone the use of local methods until Chapter 3. Convention 2.0.1. All rings in Chapters 2–4 are assumed commutative with identity.

2.1 Total Quotient Rings and Kaplansky Fractional Ideals In this section, we study various generalizations of the notion of an ideal of a ring, as well as various generalizations of the notion of a field to rings with zerodivisors. The notion of a field generalizes to rings with zerodivisors in several inequivalent ways. In Chapters 2–4, the most important of these is as follows. Definition 2.1.1. A total quotient ring is a ring in which every non-zerodivisor is a unit. Example 2.1.2. (1) (2) (3) (4) (5)

A field is equivalently an integral domain that is a total quotient ring. Any finite ring is a total quotient ring. If R and S are total quotient rings, then so is R × S. If K 1 , K 2 , . . . , K n are fields, then K 1 × K 2 × · · · × K n is a total quotient ring. If R is a ring and m is a maximal ideal of R, then R/mn is a local total quotient ring for any positive integer n. (6) k[X 1 , X 2 , . . .]/mn is a non-Noetherian local total quotient ring for any integer n > 1, where k is any field and m = (X 1 , X 2 , . . .).

© Springer Nature Switzerland AG 2019, corrected publication 2020 J. Elliott, Rings, Modules, and Closure Operations, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-24401-9_2

63

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2 Semistar Operations on Commutative Rings

Recall that an R-module M is R-torsion-free if am = 0 implies m = 0 for all m ∈ M and all non-zerodivisors a of R. For example, a ring S containing a ring R is R-torsion-free (as an R-module) if and only if every non-zerodivisor of R is a non-zerodivisor of S. We note the following equivalent characterizations of the total quotient rings. Proposition 2.1.3. Let R be a ring. The following conditions are equivalent. (1) (2) (3) (4)

R is a total quotient ring. R is the only ideal of R containing a non-zerodivisor. Every maximal (or prime) ideal of R consists only of zerodivisors. Every R-module is R-torsion-free.

Corollary 2.1.4. A local ring is a total quotient ring if and only if its maximal ideal consists only of zerodivisors. Just as every integral domain is a subring of some field, every ring is a subring of some total quotient ring. This is because, for any ring R, the localization U −1 R of R at the multiplicative set U of all non-zerodivisors of R is a total quotient ring containing R. In fact, it is a “minimal” R-torsion-free total quotient ring containing R, in the sense of Proposition 2.1.5 below. Proposition 2.1.5. Let R be a ring, U the multiplicative set of all non-zerodivisors of R, and K  an R-torsion-free total quotient ring containing R. Then K = {au −1 ∈ K  : a ∈ R and u ∈ U } is the smallest total quotient ring containing R and contained in K  . Moreover, the map U −1 R −→ K  acting by a/u −→ au −1 is a well-defined and injective ring homomorphism with image K , and therefore U −1 R and K are isomorphic. Furthermore, the map U −1 R −→ K  is the only ring homomorphism from U −1 R to K  that is the identity on R. The proposition above justifies the following definition. Definition 2.1.6. Every ring R is a subring of a smallest R-torsion-free total quotient ring K that is unique up to a unique isomorphism that is the identity on R. More precisely, for any ring R, there exists an R-torsion-free total quotient ring K containing R such that, if R is contained in any other R-torsion-free total quotient ring K  , then there exists a unique injective ring homomorphism K −→ K  that is the identity on R. Any such ring K is unique up to a unique ring isomorphism that is the identity on R and is called the total quotient ring, or the total ring of fractions, of R (though it is only unique up to isomorphism). We denote by T (R) the total quotient ring U −1 R of R, where U is the multiplicative set of all non-zerodivisors of R.

2.1 Total Quotient Rings and Kaplansky Fractional Ideals

65

Example 2.1.7. (1) (2) (3) (4)

The total quotient ring of a domain is its field of fractions, or quotient field. If R and S are rings, then T (R × S) is isomorphic to T (R) × T (S). The total quotient ring of Z × Z is Q × Q. The total quotient ring of Z[X ]/(X 2 ) is Q[X ]/(X 2 ).

The total quotient ring of a ring R can also be uniquely characterized as the largest localization of R containing R as a subring, in the sense that any localization of R containing R as a subring is contained in the total quotient ring of R. In a similar manner, the total quotient ring of R can be uniquely characterized as a maximal ring of fractions of R in the sense of Exercise 2.1.24. Remark 2.1.8. Chapter 5 of this book considers larger alternatives to the total quotient ring, the largest of which is called the complete ring of quotients Q(R). Such alternative notions of “ring of quotients” lead to analogous but more difficult theories. Definition 2.1.9. Let R be a ring. An overring of R is a ring lying between R and its total quotient ring. Example 2.1.10. The overrings of Z are precisely the rings Z[ p −1 : p ∈ ] = −1 Z, where  is an arbitrary set of prime numbers and  denotes the submonoid of the monoid of all positive integers under multiplication generated by . Thus there is an isomorphism between the poset of all overrings of Z and the poset 2P of all subsets of the set P of all prime numbers (both posets ordered by inclusion). In particular, the set of all overrings of Z is uncountable, with cardinality 2ℵ0 . Proposition 2.1.11. Let R be a ring. (1) (2) (3) (4)

T (R) is R-torsion-free. Any subring of an R-torsion-free ring extension of R is R-torsion-free. Any overring of R is R-torsion-free. A ring extension S of R is R-torsion-free if and only if T (R) is a subring of T (S).

Example 2.1.12. Any field containing an integral domain D is D-torsion-free and therefore, by the proposition, contains the quotient field T (D) of D. However, also by the proposition, a non-R-torsion-free total quotient ring containing a ring R does not contain the total quotient ring T (R) of R. Here is an example. Consider the unique ring homomorphism Z −→ Z/(4) × Q, which acts by a −→ (a, a). Its image R is isomorphic to Z, and Z/(4) × Q is a total quotient ring containing R, but Z/(4) × Q is not R-torsion-free because (2, 2) is a non-zerodivisor of R but is a zerodivisor of Z/(4) × Q since (2, 2)(2, 0) = (0, 0). Since R is isomorphic to Z, the total quotient ring T (R) of R is isomorphic to Q. However, T (R) ∼ = Q is not isomorphic to a subring of Z/(4) × Q because there is no ring homomorphism from Q to Z/(4) × Q. Therefore, even though Z/(4) × Q is a total quotient ring containing R, it does not contain the total quotient ring of R.

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Definition 2.1.13. Let R be ring. (1) Mod R (T ), for any ring T containing R, denotes the poset of all R-submodules of T , ordered by the subset relation. (2) K(R) denotes the poset Mod R (T (R)), where T (R) is the total quotient ring of R. Elements of K(R), that is, R-submodules of T (R), are called Kaplansky fractional ideals of R. (3) I(R) denotes the partially ordered set Mod R (R) of all ideals of R. The ideals of a ring R are the elements of Mod R (T ) contained in R. In particular, I(R) is a subposet of Mod R (T ). The notion of a Kaplansky fractional ideal therefore generalizes the notion of an ideal. Like the poset of all ideals of R, the poset Mod R (T ) for any ring T containing R is complete, or a complete lattice, that is, it is a poset that possesses suprema and of Mod R (T ), then the infimum of infima of all its subsets: if {Iλ : λ ∈ } is a subset {Iλ : λ ∈ } in Mod R (T ) is the intersection  λ∈ Iλ , and the supremum of R-module sum which by definition is {Iλ : λ ∈ } in ModR(T ) is the λ∈ Iλ ,   the R-submodule R λ∈ Iλ of T generated by the union λ∈ Iλ . In fact, the complete lattice Mod R (T ) is not just a poset: it also has the structure of an ordered monoid. Definition 2.1.14. An ordered semigroup is a semigroup M equipped with a partial ordering  on M such that x  x  and y  y  together imply x y  x  y  , for all x, y, x  , y  ∈ M. An ordered monoid is an ordered semigroup that is also a monoid. Ordered semigroups combine both algebraic structure and order-theoretic structure in a mutually compatible way. They arise naturally in many branches of mathematics, especially in algebra, topology, and logic. The monoid structure of Mod R (T ) (and K(R) and I(R)) comes from the operation of multiplication, defined as follows. Definition 2.1.15. Let R be a subring of a ring T . If I and J are R-submodules of T , then we define their product I J to be the R-submodule of T generated by the subset {ab : a ∈ I, b ∈ J } of T . More explicitly, for all I, J ∈ Mod R (T ) one has  IJ =

n 

ak bk : n ∈ Z>0 and ak ∈ I and bk ∈ J for all k .

k=0

It is easy to show that this commutative, associative, and unital product on Mod R (T ) gives the poset Mod R (T ) the structure of an ordered commutative monoid. In other words, for all H, I, J, I  , J  ∈ Mod R (T ), one has the following. (1) (2) (3) (4)

H (I J ) = (H I )J . I J = J I. RI = I. If I ⊆ I  and J ⊆ J  , then I J ⊆ I  J  .

2.1 Total Quotient Rings and Kaplansky Fractional Ideals

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In fact, the ordered monoid Mod R (T ) is a commutative unital quantale, or multiplicative lattice. Such structures and various generalizations thereof are studied in Chapter 6. The fact that Mod R (T ) is a multiplicative lattice basically amounts to the fact that Mod R (T ) is an ordered commutative monoid and a complete lattice and satisfies the following distributive law:

J

 λ∈

Iλ =



J Iλ , for all J ∈ Mod R (T ) and all subsets {Iλ : λ ∈ } of Mod R (T ).

λ∈

In general, the R-submodules I + J and I ∩ J of T can be defined ordertheoretically, as the supremum and infimum, respectively, of {I, J } in the poset Mod R (T ). By contrast, the product I J cannot be defined order-theoretically. In other words, the ordered monoid structure of Mod R (T ) is usually not definable from its poset structure alone, even if T = R or T = T (R). Thus, in order to more completely represent its structure, we regard Mod R (T ) as an ordered monoid, not just as a poset. Of course, the same comments apply to K(R) = Mod R (T (R)) and to I(R) = Mod R (R). The following definition introduces further important operations on Mod R (T ). Definition 2.1.16. Let R be a subring of a ring T , and let I, J ∈ Mod R (T ). (1) Let (I :T J ) = {x ∈ T : x J ⊆ I }. (2) Let (I : S J ) = (I :T J ) ∩ S = {x ∈ S : x J ⊆ I } for any ring S with R ⊆ S ⊆ T. (3) Let I −1T = (R :T I ) = {x ∈ T : x I ⊆ R}. (4) If T = T (R), then let I −1 = I −1T (R) = (R :T (R) I ) = {x ∈ T (R) : x I ⊆ R}. The proof of the following proposition is left as an exercise. Proposition 2.1.17. Let R be a subring of a ring T , let I, J, J  ∈ Mod R (T ), and let {Iλ }λ∈ and {Jμ }μ∈M be indexed subsets of Mod R (T ). (1) (2) (3) (4) (5) (6) (7) (8) (9)

(I :T J ) is the largest H ∈ Mod R (T ) such that H J ⊆ I . ((I :T J ) :T J  ) = (I :T J J  ).  :T J  ) ⊆ (I (I:T J )(J :T J ). ( λ Iλ :T μ Jμ ) = λ,μ (Iλ :T Jμ ). −1T I is the largest  H ∈ Mod R (T ) such that H I ⊆ R. ( μ Jμ )−1T = μ (Jμ−1T ). −1 If T = T (R), then I is the largest  H ∈ K(R) such that H I ⊆ R. If T = T (R), then ( μ Jμ )−1 = μ (Jμ−1 ). (I : R J ) is the largest ideal H of R such that H J ⊆ I .

Definition 2.1.18. An ordered monoid M is said to be residuated if for all x, y ∈ M there is a largest z ∈ M such that zy  x and a largest z  ∈ M such that yz   x. By statement (1) of the proposition above, the R-submodule (I :T J ) is characterized as the largest R-submodule H of T such that H J ⊆ I . It follows that the

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ordered commutative monoid Mod R (T ) is residuated. In fact, the ordered monoid Mod R (T ) is a coherent multiplicative lattice, a condition (implying both complete and residuated) that is worthwhile to study at some depth: see Chapter 6. The following definition singles out two important classes of Kaplansky fractional ideals. Definition 2.1.19. Let R be a ring with total quotient ring K . (1) An element of R is regular if it is a non-zerodivisor of R. (2) An R-submodule I of K is regular if I contains a regular element of R. (3) An R-submodule I of K is fractional, or a fractional ideal of R, if the R-submodule I −1 = (R : K I ) of K is regular, or equivalently if a I ⊆ R for some regular element a of R. A ring R is a total quotient ring if and only if (1) is the only regular ideal of R, while a ring R is an integral domain if and only if (0) is the only non-regular ideal of R. Thus, the notion of an integral domain and the notion of a total quotient ring are in some sense complementary notions. Note that a Kaplansky fractional ideal need not be a fractional ideal, and a fractional ideal need not be an ideal. Rather, the notion of a Kaplansky fractional ideal is a generalization of the notion of a fractional ideal, and the notion of a fractional ideal is a generalization of the notion of an ideal. Note also that any fractional ideal of R is of the form a −1 a for some ideal a and regular element a of R, and conversely any such Kaplansky fractional ideal of R is a fractional ideal. Proposition 2.1.20. Let R be a ring and I a Kaplansy fractional ideal of R. The following conditions are equivalent. (1) I is fractional. (2) I is contained in a principal Kaplansky fractional ideal of R. (3) I is contained in a finitely generated Kaplansky fractional ideal of R. Consequently, any finitely generated Kaplansky fractional ideal of R is fractional. Moreover, the converse holds if and only if R is Noetherian, that is, R is Noetherian if and only if every fractional Kaplansky fractional ideal of R is finitely generated. The following definition generalizes the notions of regular and fractional to any extension of R. Definition 2.1.21. Let R be a subring of a ring T . An R-submodule I of T is said to be T -regular if I T = T . An R-submodule I of T is said to be T -fractional if the R-submodule I −1T = (R :T I ) of T is T -regular, or equivalently if I −1T T = T . We note the following. Proposition 2.1.22. Let R be a ring with total quotient ring K . An R-submodule of K is regular if and only if it is K -regular, and it is fractional if and only if it is K -fractional.

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Proof. Let I be an R-submodule of K . If I is regular, say, a ∈ I is regular, then a is a unit of K , so that K = a K ⊆ I K , whence I K = K and thus I is K -regular. Conversely, suppose that n I is Kci -regular, that is, suppose that I K = K . Then 1 ∈ I K , ai di , where ai ∈ I and ci , di ∈ R with di regular for all i. so we may write 1 = i=1 n Then it is clear that d = d1 d2 · · · dn is a regular element of R and d = i=1 ai bi with ci ai ∈ I and bi = di d ∈ R for all i, whence d ∈ I and thus I is regular. Thus we have shown that I is regular if and only if I is K -regular. It follows immediately, then, that I is fractional if and only if I is K -fractional (if and only if I −1 is regular, or K -regular).  The following proposition and corollary, respectively, collect some important properties of the T -regular and T -fractional (resp., regular and fractional) R-submodules of T (resp., of T (R)). Proposition 2.1.23. Let R be a subring of a ring T , and let I, J ∈ Mod R (T ) be R-submodules of T . If J is T -regular and J ⊆ I , then I is T -regular. If I is T -fractional and J ⊆ I , then J is T -fractional. I is T -fractional if and only if I H ⊆ R for some T -regular H ∈ Mod R (T ). I J is T -regular if and only if I and J are T -regular. If I and J are T -fractional, then I J is T -fractional. If I J is T -fractional and I is T -regular, then J is T -fractional. I J is T -regular and T -fractional if and only if I and J are T -regular and T -fractional. (8) If I is T -regular and J is T -fractional, then (I :T J ) is T -regular. (9) If I is T -fractional and J is T -regular, then (I :T J ) is T -fractional. (10) If I and J are T -regular and T -fractional, then (I :T J ) is T -regular and T -fractional. (1) (2) (3) (4) (5) (6) (7)

Proof. Proofs of statements (1)–(7) are left as an exercise. To prove (8), note that, if I is T -regular and J is T -fractional then T = T T = I T J −1T T = I J −1T T ⊆ (I :T J )T ⊆ T , so (I :T J ) is T -regular. Similarly, to prove (9), note that, if I is T -fractional and J is T -regular, then (I :T J )J I −1T ⊆ R and J I −1T is T -regular, and therefore (I :T J ) is T -fractional. Finally, (10) follows immediately from (8) and (9).  Corollary 2.1.24. Let R be a ring with total quotient ring K , let I, J ∈ K(R) be Kaplansky fractional ideals of R, and let a, b ∈ R. (1) (2) (3) (4) (5) (6) (7)

If J is regular and J ⊆ I , then I is regular. If I is fractional and J ⊆ I , then J is fractional. I is fractional if and only if I H ⊆ R for some regular H ∈ K(R). I J is regular if and only if I and J are regular. a is regular if and only if the principal ideal (a) = a R of R is regular. ab is regular if and only if a and b are regular. I is fractional if and only if I = c−1 a for some ideal a and regular element c of R.

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(8) (9) (10) (11) (12) (13) (14)

If I and J are fractional, then I J is fractional. If I J is fractional and I is regular, then J is fractional. I J is regular and fractional if and only if I and J are regular and fractional. If I is fractional and J is regular, then (I : K J ) is fractional. If I is regular and J is fractional, then (I : K J ) is regular. If I and J are regular and fractional, then (I : K J ) is regular and fractional. If I and J are regular (resp., fractional), then so are I + J , I ∩ J , and I J .

Because of statement (7) of the corollary above, statements about the ideals of a ring can often be generalized to include its fractional ideals as well. Definition 2.1.25. Let R be a ring. (1) S reg for any subset S of K(R) denotes the poset of all regular Kaplansky fractional ideals in S. (2) Kreg (R) = K(R)reg denotes the poset of all regular Kaplansky fractional ideals of R. (3) Ireg (R) = I(R)reg denotes the poset of all regular ideals of R. (4) F(R) denotes the poset of all fractional ideals of R. (5) Freg (R) = F(R)reg denotes the poset of all regular fractional ideals of R. We note the following inclusions of posets:

⊆

⊆

⊆

I(R) ⊆ F(R) ⊆ K(R) Ireg (R) ⊆ Freg (R) ⊆ Kreg (R). Since all six of the posets above are closed under multiplication and contain the identity element R, they all inherit the structure of an ordered monoid from the ordered monoid K(R). Convention 2.1.26. Hereafter, we consider all six of the posets above as ordered monoids. Since (I : K J ) ∈ Freg (R) for all I, J ∈ Freg (R) and (I : R J ) ∈ Ireg (R) for all I, J ∈ Ireg (R), it follows that, like K(R) and I(R), the ordered monoids Freg (R) and Ireg (R) are residuated. However, the ordered monoids F(R) and Kreg (R) need not be residuated. Nevertheless, if I ∈ F(R) and J ∈ Kreg (R), then (I : K J ) ∈ F(R); and, likewise, if I ∈ Kreg (R) and J ∈ F(R), then (I : K J ) ∈ Kreg (R). Unlike K(R) and I(R), the posets Kreg (R), F(R), Freg (R), and Ireg (R) are not necessarily complete. For example, one can show that the poset F(R) is complete if and only if R is a total quotient ring, if and only if F(R) = K(R). Nevertheless, the posets Kreg (R), F(R), Freg (R), and Ireg (R) are lattice-ordered, that is, they possess suprema and infima of all nonempty finite subsets, and they are also bounded complete, that is, they possess suprema of all nonempty subsets that are bounded above, or, equivalently, they possess infima of all nonempty subsets that are bounded below. Suprema and infima are computed in Kreg (R), F(R), Freg (R), I(R), and Ireg (R) exactly as they are in K(R), namely, as module sums and intersections, respectively.

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The ordered monoid K(R) and the total quotient ring T (R) of a ring R play a central role in Chapters 2–4. Familiarity with the classes of zero dimensional rings and von Neumann regular rings can lead to a better appreciation of how comparatively rich the ideal-theoretic structure of a total quotient ring can be. Definition 2.1.27. A ring is said to be zero dimensional if its Krull dimension is 0, that is, if every prime ideal of the ring is maximal (or, equivalently, if every prime ideal of the ring is minimal). Example 2.1.28. (1) A field is equivalently a zero dimensional integral domain. Thus, the notion of a zero dimensional ring generalizes the notion of a field to rings with zerodivisors. (2) Any finite ring is a zero dimensional ring. (3) If R and S are zero dimensional rings, then so is R × S. (4) If K 1 , K 2 , . . . , K n are fields, then K 1 × K 2 × · · · × K n is zero dimensional. (5) If R is a ring and m is a maximal ideal of R, then R/mn is zero dimensional for any positive integer n; in fact, m/mn is the only prime ideal of R/mn . (6) k[X 1 , X 2 , . . .]/mn is a non-Noetherian local zero dimensional ring for any integer n > 1, where k is any field and m = (X 1 , X 2 , . . .). We wish to show that any zero dimensional ring is a total quotient ring. Lemma 2.1.29. Let R be a ring and S a multiplicative subset of R such that 0 ∈ / S. (1) There exists an ideal of R that is maximal among the ideals of R that do not meet S. (2) Any ideal of R that is maximal among the ideals of R that do not meet S is prime. Proof. Let I denote the set of all ideals of R that do not meet S. The set I is nonempty because (0) ∈ I. Moreover, if X is any totally ordered subset of I, then   X is a proper ideal of R that does not meet S, whence X ∈ I. Therefore, by Zorn’s lemma, there exists an ideal of R that is maximal among the ideals of R that do not meet S. Let p be any such ideal. We claim that p is prime. Let a, b ∈ R with ab ∈ p but a, b ∈ / p. Then p + a R is an ideal of R properly containing p, whence by maximality the ideal p + a R meets S, that is, there exists an s1 ∈ S of the form r1 + ac1 , where r1 ∈ p and c1 ∈ R. Likewise, there exists an s2 ∈ S of the form r2 + bc2 , where r2 ∈ p and c2 ∈ R. It follows that s1 s2 = (r1 + ac1 )(r2 + bc2 ) = r1r2 + ac1r2 + bc2 r1 + abc1 c2 lies in p ∩ S, which is a contradiction. Therefore p is prime.  Proposition 2.1.30. Every minimal prime ideal of a ring consists only of zerodivisors. Proof. Let p be a minimal prime of a ring R. Let S = {ab : a ∈ R − p, b ∈ R reg } denote the monoid generated by the monoid R − p and the monoid R reg of all regular elements of R. Note that 0 ∈ / S because 0 = ab and b ∈ R reg implies a = 0 ∈ p, for all a ∈ R. By the lemma, then, there exists a prime ideal q of R that is maximal

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among the ideals of R that do not meet S. Since q does not meet S, one has q ⊆ p and q ⊆ R − R reg . Therefore q = p by minimality of p, whence also p = q ⊆ R − R reg , so p consists only of zerodivisors.  From Propositions 2.1.30 and 2.1.3, we obtain the following. Corollary 2.1.31. Every zero dimensional ring is a total quotient ring. Proof. If R is zero dimensional, then every maximal ideal of R is minimal, and therefore by Proposition 2.1.30 every maximal ideal of R consists only of zerodivisors, whence by Proposition 2.1.3 R is a total quotient ring.  The following example shows that total quotient rings can have infinite Krull dimension. Example 2.1.32. Let R = k[Y, X 1 , X 2 , X 3 , . . .]/(Y X 1 , Y X 2 , Y X 3 , . . .), where k is a field, and let m = (y, x1 , x2 , x3 , . . .)R, where xi (resp., y) is the image of X i (resp., Y ) in R. Then m is a maximal ideal of R, and m is the set of all zerodivisors of R. Consequently, the localization Rm of R at m is the total quotient ring of R. Moreover, Rm is non-Noetherian and has infinite Krull dimension. As we have seen, every ring can be embedded in a total quotient ring. By contrast, not every ring can be embedded in a zero dimensional ring [107, p. 14]. Thus the zero dimensional rings cannot play the same role for rings as do the total quotient rings. A ring R is said to be von Neumann regular if (a 2 ) = (a) for all a ∈ R, or equivalently if R is reduced and zero dimensional. (See Exercises 2.1.25 and 2.1.27.) In fact, by [17, Exercise 2.27], a ring R is von Neumann regular if and only if it is absolutely flat, that is, if and only if every R-module is flat. Such rings are also said to be of weak global dimension zero [87]. Example 2.1.33. (1) (2) (3) (4)

A field is equivalently a von Neumann regular integral domain. If R and S are von Neumann regular rings, then so is R × S. If K 1 , K 2 , . . . , K n are fields, then K 1 × K 2 × · · · × K n is von Neumann regular. A finite ring is von Neumann regular if and only if it is reduced, if and only if it is isomorphic to a finite direct product of finite fields. (5) If R is a ring, m is a maximal ideal of R, and n is any positive integer, then R/mn is von Neumann regular if and only if mn = m, if and only if R/mn is a field. (6) A ring R is Boolean if a 2 = a for all a ∈ R. Any Boolean ring is a von Neumann regular ring

of characteristic 2. (7) The ring ∞ n=1 (Z/2Z) is a non-Noetherian Boolean (von Neumann regular) ring. A good sense of the relationships between the total quotient rings, the zero dimensional rings, and the von Neumann regular rings can be acquired from the two characterization theorems below. Since we do not need these characterizations, we omit the proofs, which can be found in [16] or in [107, Chapter I].

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Theorem 2.1.34 ([16, Theorem 4]). Let R be a ring. The following conditions are equivalent. (1) (2) (3) (4)

R is√zero dimensional. R/ 0 is von Neumann regular. For every a ∈ R there exists a positive integer n such that (a n )2 = (a n ). R is a total quotient ring and for every a ∈ R there exists an idempotent e ∈ R such that a + (1 − e) is a unit and a(1 − e) is nilpotent. (5) R is a total quotient ring and for every a ∈ R there exists a b ∈ R such that a + b is a unit and ab is nilpotent. (6) R is a total quotient ring and for every a ∈ R there exists a positive integer n such that a n = ue for some unit u and idempotent e of R. Theorem 2.1.35 ([16, Theorem 2]). Let R be a ring. The following conditions are equivalent.

(1) R is von Neumann regular. (2) R is reduced and zero dimensional. (3) R is a total quotient ring and for every a ∈ R there exists an idempotent e ∈ R such that a + (1 − e) is a unit and a(1 − e) = 0. (4) R is a total quotient ring and for every a ∈ R there exists a b ∈ R such that a + b is a unit and ab = 0. (5) R is a total quotient ring and for every a ∈ R one has a = ue for some unit u and idempotent e of R. Remark 2.1.36. A ring R can be embedded in a field if and only if T (R) is a field, if and only if R is a integral domain. Analogously, by [107, Theorem 2.10], a ring R can be embedded in a von Neumann regular ring if and only if the complete ring of quotients Q(R) of R is von Neumann regular, if and only if R is reduced. It is also known which rings can be embedded in a zero dimensional ring, but the answer is more complicated: see [82] or [107, Theorem 3.5], for example.

2.2 Dedekind Rings, Prüfer Rings, and Invertibility If R is a ring but not a total quotient ring, then one has K(R)  F(R)  Freg (R). Nevertheless, the invertible elements of the monoids K(R), Kreg (R), F(R), and Freg (R) are the same and are characterized by Definition 2.2.1, Corollary 2.2.6, and Proposition 2.2.14 of this section. Definition 2.2.1. Let R be a ring. A Kaplansky fractional ideal I of R is said to be invertible if I is invertible in the monoid K(R), that is, if I J = R for some Kaplansky fractional ideal J of R. Example 2.2.2. Let R be a ring with total quotient ring K , and let a ∈ K . Then the principal fractional ideal (a) of R is invertible if and only if a is a unit of K , if and only if a = c/d for some regular elements c and d of R.

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Definition 2.2.3. Let R be a subring of a ring T . An R-submodule I of T is said to be T -invertible if I is invertible in the monoid Mod R (T ), that is, if I J = R for some R-submodule J of T . Proposition 2.2.4. Let R be a subring of a ring T , and let I ∈ Mod R (T ) be T -invertible. Then I is finitely generated, T -regular, and T -fractional. n ai bi for Proof. Suppose that I J = R for some J ∈ Mod R (T ), so that 1 = i=1 someai ∈ I and bi ∈ J . Then one has I = (a1 , . . . , an )R. Indeed, if x ∈ I , then n (xbi )ai and xbi ∈ I J ⊆ R for all i, so x ∈ (a1 , . . . , an )R. Finally, since x = i=1 I J = R is T -regular, it follows that I and J are T -regular and therefore are T -fractional as well.  Note that, if I J = R, then J ⊆ I −1T , so that R = I J ⊆ I I −1T = R and therefore = R and J = J (I I −1T ) = (J I )I −1T = I −1T . Consequently, I is T -invertible II if and only if I I −1T = R. −1T

Corollary 2.2.5. Let R be a ring, and let I ∈ K(R) be invertible. Then I is finitely generated and regular. Corollary 2.2.6. Let R be a ring and I ∈ K(R). The following conditions are equivalent. (1) I is invertible. (2) I ∈ Freg (R) is invertible in the monoid Freg (R). (3) I I −1 = R. In particular, the set Inv(R) of all invertible (Kaplansky) fractional ideals of R is the largest subgroup of the monoid K(R) and is also the largest subgroup of the monoid Freg (R). Moreover, if I J = R for some J ∈ K(R), then J = I −1 . Recall that a ring is local if it has a unique maximal ideal, and a ring is semilocal if it has only finitely many maximal ideals. In any semilocal (or local) ring, the invertible fractional ideals take on a very simple form. Proposition 2.2.7. A Kaplansky fractional ideal of a semilocal ring is invertible if and only if it is principal and generated by a regular element. Proof. Let R be a ring. If a is a regular element of R, then (a) is invertible with inverse (a −1 ). Conversely, suppose that I ∈ K(R) is invertible. Let the distinct maximal ideals

of R be m1 , m2 , . . . , mn . For i = j we may choose ci j ∈ mi − m j . Setting c j = i= j ci j gives c j ∈ mi if and only if i = j. As I I −1  mi , there exist ai ∈ I and / mi . Let a = c1 a bi ∈ I −1 such that ai bi ∈ 1 + . . . + cn an and b = c1 b1 + · · · + cn bn , so that a ∈ I and b ∈ I −1 , and ab = i, j ci c j ai b j . Now, ai b j ∈ I I −1 ⊆ R, so = j. On the other hand, ck ck ak bk ∈ / mk . Consequently, ci c j ai b j ∈ mk if k = i or k  / mk . Therefore exactly one term in the sum i, j ci c j ai b j does not lie in mk , so ab ∈ ab does not lie in any maximal ideal of R, so ab must be a unit of R. Therefore a is a regular element of R and a R ⊆ I = abI ⊆ a I −1 I ⊆ a R, so I = a R, as required. 

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Recall that a module M over a ring R is said to be flat if the functor M ⊗ R − from the category of R-modules to itself preserves exact sequences [17, Chapter 2]. Lemma 2.2.8. Let R be a ring and M a flat R-module. Then I M ∩ J M = (I ∩ J )M for all ideals I and J of R. Proof. Tensoring the exact sequence 0 −→ I ∩ J −→ R −→ R/I ⊕ R/J of R-modules with the flat R-module M, we get the exact sequence 0 −→ (I ∩ J )M −→ M −→ M/I M ⊕ M/J M. Moreover, the kernel of the map from M into M/I M ⊕ M/J M is equal to I M ∩ J M. Hence the desired equality follows.  Let K be the total quotient ring of R. For any R-module M, the kernel of the canonical R-module homomorphism M −→ K ⊗ R M is equal to the R-torsion submodule Tors M = {m ∈ M : am = 0 for some a ∈ R reg } of M. It follows that an R-module M is R-torsion-free if and only if the canonical homomorphism M −→ K ⊗ R M is injective. Consequently, we have the following. Lemma 2.2.9. Let S be a commutative algebra over a ring R. Let K be the total quotient ring of R and L the total quotient ring of S. (1) S is R-torsion-free if and only if every regular element of R is a regular element of S, if and only if the canonical ring homomorphism S −→ K ⊗ R S is an inclusion. (2) If S is a flat R-algebra, then S is R-torsion-free. (3) Suppose that S is R-torsion-free. (a) The inclusion S −→ L induces an inclusion K ⊗ R S −→ K ⊗ R L ∼ = L. (b) The homomorphism R −→ S induces a natural homomorphism K −→ L of rings, so L is naturally a K -algebra. (c) K ⊗ R S is isomorphic to the compositum K S of S and (the image of) K in L. (d) One has I S ∈ K(S) (resp, I S ∈ F(S), I S ∈ Freg (S)) for all I ∈ K(R) (resp, I ∈ F(R), I ∈ Freg (R)). (e) If S is an R-torsion-free extension of R, then the inclusion R ⊆ S extends to an inclusion K ⊆ L. If S is a flat R-algebra, then Lemma 2.2.8 implies that I S ∩ J S = (I ∩ J )S for all ideals I and J of R. We need the following extension of this result to Kaplansky fractional ideals. The proof follows the same method used in the proof of Lemma 2.2.8.

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Proposition 2.2.10. Let R be a ring with total quotient ring K and S a flat R-algebra with total quotient ring L, so L is a K -algebra. Then I S ∩ J S = (I ∩ J )S in K(S) for all I, J ∈ K(R). Proof. Tensoring the exact sequence 0 −→ I ∩ J −→ K −→ K /I ⊕ K /J of R-modules with the flat R-module S, we get the exact sequence 0 −→ (I ∩ J ) ⊗ R S −→ K ⊗ R S −→ (K /I ⊕ K /J ) ⊗ R S, which since K ⊗ R S is isomorphic to the compositum K S of K and S in L is equivalent to the exact sequence 0 −→ (I ∩ J )S −→ K S −→ K S/I S ⊕ K S/J S. The desired equality follows.



The next result proves to be very useful in subsequent sections. Proposition 2.2.11. Let R be a ring with total quotient ring K and S a flat R-algebra with total quotient ring L, so L is a K -algebra. One has the following. (1) (I : K J )S = (I S : K S J S) for all I, J ∈ K(R) with J finitely generated. (2) (I : K J )S = (I S : L J S) for all I, J ∈ K(R) with J finitely generated and regular. (3) (I S)−1 = I −1 S in L for all finitely generated I ∈ Freg (R). Proof. Let I ∈ K(R) and a ∈ K . Consider the multiplication by a map K −→ K of R-modules followed by the projection K −→ K /I . The kernel of the composed map K −→ K /I is the R-module (I : K a R), so we have an exact sequence 0 −→ (I : K a R) −→ K −→ K /I of R-modules. Since S is flat over R, tensoring with S yields the exact sequence 0 −→ (I : K a R) ⊗ R S −→ K ⊗ R S −→ (K /I ) ⊗ R S, which is equivalent to the exact sequence 0 −→ (I : K a R)S −→ K S −→ K S/I S. Therefore the kernel (I S : K S aS) of the multiplication by a map K S −→ K S/I S is equal to (I : K a R)S. This proves statement (1) when J = a R is principal. Now suppose that J ∈ K(R) is finitely generated, say, J = (a1 , . . . , an ), where ai ∈ K for all i. Then, using also Proposition 2.2.10, we see that (I : K J )S = ((I : K a1 ) ∩ · · · ∩

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(I : K an ))S = (I : K a1 )S ∩ · · · ∩ (I : K an )S = (I S : K S a1 S) ∩ · · · ∩ (I S : K S an S) = (I S : K S a1 S + · · · + an S) = (I S : K S J S). This proves statement (1). Now, suppose furthermore that J is regular, say, a ∈ J is regular. Let x ∈ (I S : L J S). Then ax ∈ I S, so x = a −1 (ax) ∈ K (I S) = (K I )S ⊆ K S and therefore x ∈ K S. Thus we have (I S : L J S) ⊆ (I S : K S J S), so equality holds. This proves (2), and then (3) follows immediately.  Corollary 2.2.12. Let R be a ring with total quotient ring K and S a flat overring of R. One has (I : K J )S = (I S : K J S) for all I, J ∈ K(R) with J finitely generated. In particular, one has (I S)−1 = I −1 S for all finitely generated I ∈ K(R). Note that, if p is a prime ideal of a ring R, then the localization Rp of R at p is a flat R-algebra. However, Rp may not contain R as a subring. For example, the localization map Z/6Z −→ (Z/6Z)(2) ∼ = Z/3Z is not injective. It is clear that the localization map R −→ Rp is injective if and only if p contains all of the zerodivisors of R, in which case Rp is an overring of R. If R is a total quotient ring, for example, then R −→ Rp is injective if and only if it is an isomorphism, which holds if and only if R is local with maximal ideal p. At any rate, since Proposition 2.2.11 requires only the flatness of S as an R-algebra, we obtain the following. Corollary 2.2.13. Let R be a ring and p a prime ideal of R. Then (I Rp )−1 ⊇ I −1 Rp for all I ∈ K(R), and equality holds if I ∈ Freg (R) is finitely generated and regular. The following proposition, provides a convenient characterization of the invertible fractional ideals of a ring. Proposition 2.2.14. Let R be a ring and I ∈ Freg (R). The following conditions are equivalent. (1) I is invertible. (2) I is finitely generated and I Rp is a principal (or invertible) fractional ideal of Rp for every prime ideal p of R. (3) I is finitely generated and I Rm is a principal (or invertible) fractional ideal of Rm for every regular maximal ideal m of R. Proof. Suppose that I is invertible. Then there exists a J ∈ Freg (R) such that I J = R. Then (I Rp )(J Rp ) = I J Rp = Rp , so I Rp is invertible and therefore principal in the local ring Rp , for every prime ideal p of R. Moreover, by Corollary 2.2.5, I is finitely generated. Thus (1) ⇒ (2) ⇒ (3). Suppose, then, that (3) holds. Since I Rm is invertible, it is also principal and regular, in Rm , so by Corollary 2.2.13 one has I I −1 Rm = (I Rm )(I Rm )−1 = Rm , for every regular maximal ideal m of R. If I I −1 = R, then there exists a maximal ideal m of R containing I I −1 . But I I −1 is regular, so m is regular and Rm = I I −1 Rm ⊆ mRm , which is a contradiction.  Therefore I I −1 = R. Next, we provide an important application of Proposition 2.2.14 to the Dedekind domains.

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Definition 2.2.15. Let D be an integral domain. (1) D is a principal ideal domain, or PID, if every ideal of D is principal. (2) D is a discrete rank (at most) one valuation domain, or DVR, if D is a local PID. (3) D is Dedekind if every nonzero (fractional) ideal of D is invertible. (4) D is Bézout if every finitely generated (fractional) ideal of D is principal. (5) D is Prüfer if every nonzero finitely generated (fractional) ideal of D is invertible. Proposition 2.2.16. An integral domain D is a Dedekind domain if and only if D is Noetherian and Dm is a DVR for every maximal ideal m of D. Proof. Let D be an integral domain. Suppose that D is Noetherian and Dm is a DVR for every maximal ideal m of D. Let I be any nonzero fractional ideal of D. Then I is finitely generated since D is Noetherian, and I Dm is principal in Dm since Dm is a PID, for every maximal ideal m of D. Therefore, by Proposition 2.2.14, I is invertible. Therefore D is Dedekind. Conversely, suppose that D is Dedekind, that is, suppose that every nonzero fractional ideal of D is invertible. Then D is Noetherian since invertible ideals are finitely generated. Moreover, if m is a maximal ideal of D, then, since every ideal of Dm is extended from D and every nonzero ideal of D is invertible, it follows from Proposition 2.2.14 that every nonzero ideal of Dm is  principal and therefore Dm is a DVR. Thus D is Dedekind. The following result is clear. Proposition 2.2.17. One has the following. (1) A Dedekind domain is equivalently a Noetherian Prüfer domain. (2) A PID is equivalently a Noetherian Bézout domain. Thus, we have the following implications among integral domains. DVR

PID

Dedekind

Bézout

Prüfer All of the classes of domains above are important in algebraic number theory, especially the DVRs, PIDs, and Dedekind domains. This is supported by the following well-known examples.

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Example 2.2.18. (1) The ring Z of integers is a PID. (2) If K is a field, then the polynomial ring K [X ] is a PID, and the ring K [[X ]] of formal power series is a DVR. (3) A DVR is equivalently a unique factorization domain (UFD) with at most one irreducible element up to associates. (4) A DVR is equivalently a local Dedekind domain. (5) A valuation domain is a local Bézout domain, or equivalently a local Prüfer domain. Valuation domains are studied in Section 3.1. (6) The ring O K of all algebraic integers in any finite field extension K of the field Q of rational numbers is a Dedekind domain [23, Corollary VII.5.2], [191, Theorem V.19]. It may or may not be a PID, depending in a very intricate way on K . (It is a PID if and only if the ideal class group of O K is trivial, if and only if K has class number equal to 1.) (7) The ring of all algebraic integers (in C) is a non-Noetherian Bézout domain [70, Example III.1.3]. (8) The ring Int(Z) = { f ∈ Q[X ] : f (Z) ⊆ Z} of all integer-valued polynomials is a non-Dedekind, non-Bézout Prüfer domain. (9) More generally, the ring Int(O K ) = { f ∈ K [X ] : f (O K ) ⊆ O K } of all integervalued polynomials on O K is a non-Dedekind, non-Bézout Prüfer domain for any finite extension K of Q [27, Theorem VI.1.7]. Moreover, Int(O K ) is of Krull dimension two and is free as an O K -module. The classes of integral domains defined in Definition 2.2.15 can be generalized to rings with zerodivisors in several inequivalent ways [20, 87, 120], including the following. Definition 2.2.19. Let R be a ring. (1) R is an r-PIR if every regular (fractional) ideal of R is principal. (2) R is Dedekind if every regular (fractional) ideal of R is invertible. (3) R is r-Bézout if every finitely generated regular (fractional) ideal of R is principal. (4) R is Prüfer if every finitely generated regular (fractional) ideal of R is invertible. Example 2.2.20. (1) A domain D is an r-PIR (resp., Dedekind, r-Bézout, Prüfer) in the sense of Definition 2.2.19 if and only if D is a PID (resp., Dedekind domain, Bézout domain, Prüfer domain) in the sense of Definition 2.2.15. (2) A finite direct product R1 × R2 × · · · × Rn of rings R1 , R2 , . . . , Rn is an r-PIR (resp., Dedekind, r-Bézout, Prüfer) if and only if Ri is an r-PIR (resp., Dedekind, r-Bézout, Prüfer) for all i. (See Exercise 2.2.10.) We note the following implications.

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r-PIR

Dedekind

r-Bézout

Prüfer Moreover, from Proposition 2.2.7, we obtain the following. Proposition 2.2.21. Any semilocal Dedekind domain is a PID, and any semilocal Prüfer domain is Bézout. More generally, any semilocal Dedekind ring is an r-PIR, and any semilocal Prüfer ring is r-Bézout. Convention 2.2.22. The prefix “r-” is short for “regularly” and is used for many other terms introduced in this text, including “r-Noetherian,” “r-UFR,” “r-GCD,” “r-atomic,” and “r-prime.” Remark 2.2.23. (1) r-Bézout rings are also referred to as quasi-Bézout rings, almost Bézout rings, and regular Bézout rings. (2) Just as every PID is a UFD, every regular non-unit of an r-PIR factors, uniquely up to reordering and associates, as a product of prime elements: see Exercise 2.2.11. (3) In Section 3.10 we extend the notion of a DVR to rings with zerodivisors in several inequivalent ways. (4) As shown in [120], which builds on the prior work of Klingler, Lucas, Sharma, Glaz, Schwarz, Boynton, and others [87], the notion of a Prüfer domain extends to rings with zerodivisors in at least 17 inequivalent ways. The notion of a Prüfer ring is the most general of those 17 conditions. (5) A ring R is said to be a PIR, or a principal ideal ring, if every ideal of R is principal, and R is said to be Bézout if every finitely generated ideal of R is principal. These are much stronger conditions, respectively, than the r-PIR and r-Bézout conditions. Examples of PIRs include the quotient of any Dedekind domain by any nonzero ideal (or in fact the quotient of any Dedekind ring by any regular ideal, by Exercise 3.7.11). Remark 2.2.24. A regular fractional ideal of a ring R is projective if and only if it is invertible, and it is free if and only if it is principal. Thus, a ring R is Dedekind (resp., an r-PIR) if and only if every regular fractional ideal of R is projective (resp., free), and R is Prüfer (resp., r-Bézout) if and only if every finitely generated regular fractional ideal of R is projective (resp., free). A ring R is said to be hereditary if every (fractional) ideal of R is projective, and R is said to be semihereditary if every finitely generated (fractional) ideal of R is projective. Thus, every hereditary ring is Dedekind, and every semihereditary ring is Prüfer.

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As we show in Example 2.7.13, Dedekind rings, unlike Dedekind domains, need not be Noetherian. However, every regular ideal of a Dedekind ring is (invertible and therefore) finitely generated. This motivates the following definition. Definition 2.2.25. A ring R is r-Noetherian if every regular ideal of R is finitely generated. Note that a domain is r-Noetherian if and only if it is Noetherian. Therefore the following result, whose proof is clear, generalizes Proposition 2.2.17. Proposition 2.2.26. One has the following. (1) A Dedekind ring is equivalently an r-Noetherian Prüfer ring. (2) An r-PIR is equivalently an r-Noetherian r-Bézout ring. Example 2.2.27. (1) If K is a total quotient ring, then K = (1) is the only regular ideal of K , and therefore K is an r-PIR and is therefore r-Noetherian, Dedekind, and Prüfer. (2) Let Q[ε] = Q[X ]/(X 2 ), where ε is the image of X in Q[X ]/(X 2 ). Let R = Z + εQ[ε] = {a + bε : a ∈ Z, b ∈ Q}. Then R is both an r-PIR and Bézout (that is, every regular or finitely generated ideal of R is principal), but R is nonNoetherian, hence not a PIR, since the ideal εQ[ε] of R is not finitely generated. Nevertheless, the total quotient ring Q[ε] of R is an Artinian local ring that is isomorphic to the quotient Q[[X ]]/(X 2 ) of the DVR Q[[X ]] and therefore has only three ideals, namely, (0)  (ε)  Q[ε]. The non-regular ideals of an r-Noetherian ring, or even a total quotient ring, can be poorly behaved. One might be led to suspect that the r-Noetherian, Dedekind, and Prüfer conditions for rings with zerodivisors are not strong enough to banish pathological examples or yield interesting theorems. However, as we reveal in subsequent sections, quite the opposite is the case. While, granted, these conditions control only the regular ideals, the total quotient ring to some extent controls the non-regular ideals, and this balance between the two is what makes the conditions so natural. Indeed, the regular ideals of a ring R are those that extend to the unit ideal K of the total quotient ring K , or, in other words, that are rendered “irrelevant” in K . Those are the ideals controlled by conditions on regular ideals like the r-Noetherian, Dedekind, and Prüfer conditions, which are automatic for total quotient rings. In the meantime, the non-regular ideals of R extend to all of the proper ideals of K , thus potentially allowing one to study properties of the non-regular ideals by studying the total quotient ring. (For example, the non-regular prime ideals of R are in one-to-one correspondence with the prime ideals of K , and consequently the minimal prime ideals of R, which are by necessity non-regular, are in one-to-one correspondence with the minimal prime ideals of K .) In other words, controlling the non-regular ideals of a ring can often be achieved by imposing natural conditions on its total quotient ring, such as the semilocal, reduced, zero dimensional, von Neumann regular, and field conditions. See [85] for an excellent survey of such methods.

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2.3 Integral Closure and Complete Integral Closure In this section, we study the integral closure and complete integral closure of a ring and introduce the classes of integrally closed, completely integrally closed, and v-Prüfer rings. Definition 2.3.1. Let T be an extension of a ring R. An element a of T is integral over R if a is a root of some monic polynomial in R[X ]. Example 2.3.2. A complex number that is integral over Z is known as an algebraic integer. The only rational numbers that are algebraic integers are the integers. Recall that if a1 , . . . , an are elements of an extension T of a ring R, then R[a1 , . . . , an ] denotes the smallest subring of T containing R and a1 , a2 , . . . , an and is given by R[a1 , . . . , an ] = { f (a1 , . . . , an ) : f ∈ R[X 1 , . . . , X n ]}. The following result is well-known: see any of [17, Proposition 5.1], [43, Corollary 4.6], [145, Theorem 9.1], [178, Lemma 2.1.9], [191, Section V.1]. Proposition 2.3.3. Let R be a ring and T an extension of R, and let a ∈ T . The following conditions are equivalent. (1) (2) (3) (4)

a is integral over R. R[a] is finitely generated as an R-module. R[a] is contained in a subring of T that is finitely generated as an R-module. There exists a faithful R[a]-module that is finitely generated as an R-module.

Proof. If a is integral over R, say, if a is a root of a monic polynomial in R[X ] of degree n, then one can show that 1, a, a 2 , . . . , a n−1 ∈ T generate R[a] as an Rmodule. Thus (1) ⇒ (2). Clearly one has (2) ⇒ (3) ⇒ (4). Suppose that (4) holds, so that there exists a faithful R[a]-module M that is finitely generated as an R-module, say, with generators x1 , x2 , . . . , xn . For each i, we may write axi =

n 

ai j x j ,

j=1

where ai j ∈ R for all i, j. For all r ∈ M let r · : M −→ M be the R-module endomorphism x −→ r x of M. The map R −→ End R (M) acting by r −→ r · is a homomorphism of noncommutative rings whose image R· in End R (M) is commutative. Consider the n × n matrix A = (δi j a · −ai j ·) with entries in the ring R·, where δi j is the Kronecker delta. Then the matrix linear transformation A : M n −→ M n satisfies A(x1 , x2 , . . . , xn )T = 0 since n  (δi j a · −ai j ·)x j = 0 j=1

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for all i  n. By multiplying on the left by the adjoint of A, it follows that the endomorphism det A of M annihilates each xi and is therefore the zero endomorphism of M. Expanding out the determinant, we obtain an equation of the form (a·)n + b1 (a·)n−1 + · · · + bn−1 (a·) + bn · = 0·, where bi ∈ R for all i, whence a n + b1 a n−1 + · · · + bn = 0 since M is faithful. Therefore (4) implies (1).  The notion of an integral element generalizes as follows. Definition 2.3.4. Let T be an extension of a ring R. An element a of T is almost integral over R if R[a] is contained in a finitely generated R-submodule of T . As corollaries of Proposition 2.3.3, we have the following. Corollary 2.3.5. Let T be an extension of a ring R. If an element of T is integral over R, then it is almost integral over R. Corollary 2.3.6. Let R be ring with total quotient ring K . An element a of K is integral over R if and only if R[a] is a finitely generated fractional ideal of R. An element a of K is almost integral over R if and only if R[a] is a fractional ideal of R, if and only if there is a regular element c of R such that ca n ∈ R for all positive integers n. Example 2.3.7. Let k be a field and let R = k[X, X Y, X Y 2 , X Y 3 , . . .] and T = k[X, Y ]. Then Y ∈ T is almost integral over T but not integral over R. The ring R in the example above is non-Noetherian. For Noetherian rings, integrality and almost integrality coincide. Corollary 2.3.8. Let T be an extension of a Noetherian ring R. Then an element a of T is integral over R if and only if a is almost integral over R. Proof. Suppose that a ∈ T is almost integral over R, so there exists a finitely generated R-submodule I of T containing R[a]. Since R is Noetherian, every finitely generated R-module is Noetherian, so I is Noetherian. Therefore R[a] ⊆ I is finitely generated, so a is integral over R. Conversely, if a is integral over R, then a is almost integral over R by Corollary 2.3.5.  Definition 2.3.9. Let R be a ring, and let T be an extension of R. (1) The integral closure of R in T is the set of all elements of T that are integral over R. (2) The integral closure of R, denoted R, is the integral closure of R in its total quotient ring T (R). (3) R is integrally closed if R equals its integral closure R. (4) The complete integral closure of R in T is the set of all elements of T that are almost integral over R.

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(5) The complete integral closure of R, denoted R  , is the complete integral closure of R in its total quotient ring T (R). (6) R is completely integrally closed, or CIC, if R equals its complete integral closure R  . Remark 2.3.10. Unfortunately, none of the notations used in the literature to denote the integral closure or the complete integral closure of a ring are standard; some authors use “R  ” to denote the integral closure of a ring R and R ∗ to denote the complete integral closure of R. There are also no standard notations to denote the integral closure of a ring R in an extension T . Note that a ring R with total quotient ring K is integrally closed (that is, R = R) if and only if a ∈ R for every a ∈ K that is integral over R, and R is completely integrally closed (that is, R = R  ) if and only if a ∈ R for every a ∈ K that is almost integral over R. Example 2.3.11. Let D be an integral domain. (1) Any UFD is completely integrally closed. (2) It is well-known that D[X ] integrally closed if and only if D is integrally closed, and D[X ] is completely integrally closed if and only if D is completely integrally closed [23, Chapter V]. The following result is fundamental. Proposition 2.3.12. Let T be an extension of a ring R. (1) The integral closure of R in T is a subring of T containing R. (2) The complete integral closure of R in T is a subring of T containing the integral closure of R in T . (3) If R is Noetherian, then the integral closure of R in T is equal to the complete integral closure of R in T . Proof. To prove (1), let a, b ∈ T be integral over R. Then R[a] and R[b] are finitely generated R-modules, so R[a, b] = R[a]R[b] is also finitely generated as an Rmodule. But R[a + b], R[a − b], and R[ab] are all contained in R[a, b], which is a subring of T , and therefore a + b, a − b, and ab are integral over R by Proposition 2.3.3. This proves (1). Next, we prove (2). Suppose that a, b ∈ T are almost integral over R. Then there exist finitely generated R-submodules I and J of T containing R[a] and R[b], respectively. It follows that I J is a finitely generated R-submodule of T containing R[a, b] and thus containing R[a + b], R[a − b], and R[ab]. Therefore a + b, a − b, and ab are almost integral over R. Thus the complete integral closure of R is a subring of T . Since integrality implies almost integrality, this proves (2). Finally, (3) is a restatement of Corollary 2.3.8.  Corollary 2.3.13. Let R be a ring. Then R and R  are overrings of R with R ⊆ R  . Moreover, equality holds if R is Noetherian.

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85

Example 2.3.14.

√ (1) Let d be a squarefree √ of Z[ d] is √ integer. As is well-known, the integral closure equal to Z[(1 + d)/2] if d ≡ 1 (mod 4), and it is equal to Z[ d] otherwise. (2) Let Z[ε] = Z[X ]/(X 2 ), where ε is the image of X in Z[X ]/(X 2 ). The integral closure of Z[ε] is equal to Z + εQ[ε]. In particular, Z[ε] is a one dimensional Noetherian ring with non-Noetherian integral closure. (3) The ring R = k[X, X Y, X Y 2 , X Y 3 , . . .], where k is any field, is integrally closed but not completely integrally closed. Its complete integral closure is R  = k[X, Y ]. Although integral closures are integrally closed, this fact requires proof. Lemma 2.3.15. Let R be a ring and T an R-algebra that is finitely generated as an R-module. If N is any finitely generated T -module, then N is finitely generated as an R-module. Proof. Let a1 , a2 , . . . , an generate T as an R-module and x1 , x2 , . . . , xm generate N as a T -module. Then one readily verifies that the mn products ai x j generate N as an R-module.  Proposition 2.3.16. Let R be a ring and T an extension of R. Then the integral closure of R in T is integrally closed in T . In particular, the integral closure R of R is integrally closed. Proof. Let S be the integral closure of R in T , and let a ∈ T be integral over S. Then a n + s1 a n−1 + · · · + sn−1 a + sn = 0 for some elements s1 , s2 , . . . , sn of S. Let S0 = R and Sk = R[s1 , s2 , . . . , sk ] for all k  n. Then the R-algebra Sk is finitely generated as an Sk−1 -module for all k  n, by Proposition 2.3.3. From the lemma, it then follows by induction that Sn is finitely generated as an R-module. Now, a is integral over Sn , and therefore Sn [a] is finitely generated as an Sn -module. It then follows, again from the lemma, that Sn [a] is finitely generated as an R-module. Therefore R[a] is contained in a subring Sn [a] of T that is finitely generated as an R-module, and so a is integral over R by Proposition 2.3.3. Thus a ∈ S. Therefore, S is integrally closed in T .  It is important to note that the analogue of the proposition above for complete integral closure is false: complete integral closures are not necessarily completely integrally closed. In other words, while R is necessarily integrally closed, R  need not be completely integrally closed. Equivalently, one has R = R, yet (R  ) may properly contain R  . (See [81, Example 1] or Example 4.6.20 for an example.) However, it is true that complete integral closures are integrally closed. (See Exercise 2.3.10.) The next proposition provides a useful method for proving results about integral and complete integral closures. Note first that, if I is an ideal of R, then (I : K I ) is the largest overring of R containing I as an ideal. Indeed, (I : K I ) is an overring of R with (I : K I )I = I , and if S I = I for some overring S of R then S ⊆ (I : K I ).

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More generally, if I is a Kaplansky fractional ideal of R, then (I : K I ) is the largest overring of R containing I as a Kaplansky fractional ideal. Proposition 2.3.17. Let R be a ring with total quotient ring K . The complete integral closure of R is equal to R =

 {(I : K I ) : I ∈ Freg (R)},

and the integral closure of R is equal to R=

 {(I : K I ) : I ∈ Freg (R) is f.g.}.

In particular, R is completely integrally closed if and only if R = (I : K I ) for all I ∈ Freg (R), and R is integrally closed if and only if R = (I : K I ) for all finitely generated I ∈ Freg (R). Proof. Let I ∈ Freg (R) and a ∈ (I : K I ). Then R[a] ⊆ (I : K I ) ∈ Freg (R). Choosing a regular c ∈ R such that c(I : K I ) ⊆ R, we have c R[a] ⊆ R, and therefore a is almost integral over R. Conversely, if a is almost integral over R, then I = R[a] is a regular fractional ideal of R such that a ∈ (I : K I ). Next, let I ∈ Freg (R) be finitely generated and a ∈ (I : K I ). Then R[a]I ⊆ I , so I is an R[a]-module, and it is a faithful R[a]-module since I is regular. Since also I is finitely generated as an R-module, it follows from Proposition 2.3.3 that a is integral over R. Conversely, if a is integral over R, then I = R[a] ∈ Freg (R) is  finitely generated and a ∈ (I : K I ). Corollary 2.3.18. Every Prüfer ring is integrally closed, and every Dedekind ring is completely integrally closed. Proof. Observe that (I : K I ) = R if I is invertible.



We note the following partial generalization of Proposition 2.3.17. Proposition 2.3.19.  Let T be an extension of a ring R. Then the integral closure of R in T is equal to {(I :T I ) : I ∈ Mod T (R) is f.g. and T -regular}. In particular, R is integrally closed in T if and only if R = (I :T I ) for all finitely generated T -regular I ∈ Mod R (T ). The following is another important tool for studying integral and complete integral closures. Definition 2.3.20. Let R be a ring. Define v : K(R) −→ K(R) by I v = (I −1 )−1 for all I ∈ K(R). The operation v is called the v-operation, v-closure, or divisorial closure, on R. An element I of K(R) is divisorial, or v-closed, if I v = I . Remark 2.3.21. (1) According to [68], the “v” in “v-operation” comes from the German “Vielfachenideale” or “V-Ideale” (“ideal of multiples”), terminology used by Prüfer in 1932 [167, Section 7].

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(2) It is well-known that, if  R is an integral domain, then I v for any I ∈ K(R) is equal to the intersection {x R : x ∈ T (R), x R ⊇ I } of the principal fractional ideals of R containing I . It is also known that this equality does not necessarily hold if R is a ring with zerodivisors. (See Example 2.7.20.) (3) Let R be a ring. An R-module M is said to be reflexive if the natural R-module homomorphism M −→ Hom R (Hom R (M, R), R) acting by a −→ (ϕ −→ ϕ(a)) is an isomorphism. If I is a regular Kaplansky fractional ideal of R, then the natural R-module homomorphism I −1 −→ Hom R (I, R) acting by a −→ (x −→ ax) is an isomorphism. Consequently, a regular fractional ideal of R is reflexive if and only if it is divisorial. As shown in Section 2.2, a Kaplansky fractional ideal I of a ring R is invertible if and only if I is invertible in the monoid K(R), if and only if I I −1 = R. Thus, if I is invertible, then I −1 is actually the unique inverse of I in the monoid K(R). In that case, one has (I −1 )−1 = I by a very basic rule for monoids, namely, that if x ∈ M is invertible, where M is a monoid, then (x −1 )−1 = x. Thus, every invertible Kaplansky fractional ideal is divisorial. However, the converse need not hold, even in a local Noetherian domain of dimension one. Moreover, an ideal in a local Noetherian domain of dimension one or in a local Noetherian UFD of dimension two need not be divisorial. Example 2.3.22. Let k be a field. (1) Consider the domain D = k[[X 2 , X 3 ]] ⊆ k[[X ]], which is a local Noetherian domain of dimension one with maximal ideal m = (X 2 , X 3 ) = X 2 k[[X ]]. The field of fractions K of D is the same as that of k[[X ]], namely, K = k((X )) = k[[X ]][1/ X ]. One can show that m−1 = (D : K X 2 k[[X ]]) = k[[X ]] and

(m−1 )−1 = (D : K k[[X ]]) = X 2 k[[X ]] = m.

Therefore m is divisorial. (In fact, one can show that every ideal of D is divisorial.) However, m is not invertible since mm−1 = X 2 k[[X ]]k[[X ]] = X 2 k[[X ]] = m. Note also that (m : K m) = (X 2 k[[X ]] :k((X )) X 2 k[[X ]]) = k[[X ]], and therefore by Proposition 2.3.17 the ring k[[X ]] is contained in the integral closure of D. (Alternatively, note that X is integral over R since it is a root of the monic polynomial t 2 − X 2 ∈ R[t], and therefore D[X ] = k[[X ]] is contained in the integral closure of D.) Moreover, k[[X ]] is a DVR and is therefore integrally closed, so that the integral closure of D is k[[X ]]. (2) Consider the domain D = k[[X, Y ]], which is a local Noetherian UFD of dimension two with maximal ideal m = (X, Y ). The field of fractions k((X, Y )) of D

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is obtained from D by inverting all nonzero power series in k[[X, Y ]] (and those not in m are already invertible). One can show that m−1 = (D :k((X,Y )) (X, Y )) = k[[X, Y ]] = D and therefore (m−1 )−1 = D −1 = D. Therefore m is not divisorial. (3) In Section 2.9 we show that, if D is a UFD, then I v for any ideal I of D is principal and is generated by the gcd of the elements of I . This explains the previous example, since the gcd of X and Y in the UFD k[[X, Y ]] is 1. Proofs of the following lemma and proposition are left as an exercise (Exercises 2.1.14 and 2.1.15). Lemma 2.3.23. Let R be a ring and I, J ∈ K(R). (1) (2) (3) (4) (5)

I J ⊆ R if and only if J ⊆ I −1 . If I ⊆ J , then I −1 ⊇ J −1 . I ⊆ (I −1 )−1 . ((I −1 )−1 )−1 = I −1 . I −1 J −1 ⊆ (I J )−1 .

Proposition 2.3.24. Let R be a ring and I, J ∈ K(R). (1) (2) (3) (4) (5) (6) (7)

If I ⊆ J , then I v ⊆ J v . I ⊆ I v. (I v )v = I v . I v J v ⊆ (I J )v . R v = R. (I v )−1 = (I −1 )v = I −1 . (I J )v = (I J v )v = (I v J )v = (I v J v )v .

Statements (1)–(3) of Proposition 2.3.24 say that the map v : K(R) −→ K(R) is a closure operation on the poset K(R). Statements (1)–(4) say that the map v is a nucleus on the ordered monoid K(R), and statement (5) says that the nucleus v is unital. A (unital) semistar operation on a ring R is a unital nucleus on the ordered monoid K(R). Thus, v-closure is a semistar operation, defined on any ring R. In Section 2.4 we undertake a study of semistar operations. In the remainder of this section, we are concerned mainly with the operation of v-closure. Remark 2.3.25. If D  is a Noetherian overring of a Noetherian domain D, then one need not have I v D  ⊆ (I D  )v for every ideal I of D. Similarly, if p is a prime ideal of a Noetherian domain D, then one need not have I v (D/p) ⊆ (I (D/p))v for every ideal I of D. (See Exercise 1.1.10.) Thus, the operation of v-closure is highly dependent on the ambient ring one is working in. As we have seen, “divisorial,” or “v-closed,” is a generalization of “invertible.” Another useful generalization of “invertible” is “v-invertible.” Definition 2.3.26. Let R be a ring. An element I of K(R) is v-invertible if (I J )v = R for some J ∈ K(R).

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Clearly, if I ∈ K(R) is invertible then I is v-invertible, since I J = R implies R = I J ⊆ (I J )v ⊆ R v = R. Lemma 2.3.27. Let R be a ring with total quotient ring K , and let I ∈ K(R). Then (I : K I )v ⊆ (I v : K I ) = (I v : K I v ) = (I −1 : K I −1 ) = (I I −1 )−1 . Moreover, the following conditions are equivalent. (1) (2) (3) (4)

I is v-invertible. (I I −1 )v = R. (I −1 : K I −1 ) = R. (I v : K I v ) = R.

Furthermore, if (I J )v = R for some J ∈ K(R), then J v = I −1 and I v = J −1 . Proof. We first prove conditions (1) and (2) are equivalent. Clearly, if (I I −1 )v = R then I is v-invertible. Conversely, if I is v-invertible, say, (I J )v = R, then I J v ⊆ (I J )v = R, so that J v ⊆ I −1 , and likewise I v ⊆ J −1 so that I −1 = (I v )−1 ⊇ (J −1 )−1 = J v , and therefore J v = I −1 and likewise I v = J −1 . In particular, one has (I I −1 )v = (I J v )v = (I J )v = R. Therefore conditions (1) and (2) are equivalent. Now, one has (I −1 : K I −1 ) = ((R : K I ) : K I −1 ) = (R : K I I −1 ) = (I I −1 )−1 . Similarly, one has (I v : K I v ) = ((R : K I −1 ) : I v ) = (R : K I −1 I v ) = (I −1 I v )−1 = ((I −1 I v )v )−1 = ((I −1 I )v )−1 = (I −1 I )−1 . Since also J −1 = R if and only if J v = R for any J ∈ K(R), statements (2)–(4) are equivalent. The lemma follows.  Corollary 2.3.28. A ring R is completely integrally closed if and only if every I ∈ Freg (R) is v-invertible. Proof. Suppose that R is completely integrally closed. Then, by Proposition 2.3.17, for any I ∈ Freg (R) one has (I v : K I v ) = R, and therefore I is v-invertible by Lemma 2.3.27. Conversely, if every I ∈ Freg (R) is v-invertible, then R ⊆ (I : K I ) ⊆ (I : K I )v ⊆ (I v : K I v ) = R, so R = (I : K I ), for all I ∈ Freg (R), and therefore R is completely integrally closed by Proposition 2.3.17.  Example 2.3.29. (1) Let D be the local domain of Example 2.3.22(1). Then (mm−1 )v = mv = m and therefore the maximal ideal m of D is not v-invertible. (2) Let D be the local domain of Example 2.3.22(2). Then (mm−1 )v = (mD)v = mv = D, and therefore the maximal ideal m of D is v-invertible but neither divisorial nor invertible. In fact, since D is Noetherian and integrally closed, D is completely integrally closed, and therefore by Corollary 2.3.28 every nonzero fractional ideal of D is v-invertible. The following definition provides a natural common generalization of the completely integrally closed rings and the Prüfer rings. Definition 2.3.30. (1) A ring R is v-Prüfer if every finitely generated I ∈ Freg (R) is v-invertible. (2) A v-domain is a v-Prüfer domain, or equivalently, a domain in which every nonzero finitely generated (fractional) ideal is v-invertible.

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Corollary 2.3.31. Any completely integrally closed ring or Prüfer ring is a v-Prüfer ring, and any v-Prüfer ring is integrally closed. Proof. The first implication follows from Corollary 2.3.28. Let I be a finitely generated regular ideal of a v-Prüfer ring R. Since I is v-invertible, we have (I −1 I )v = R. Therefore, since also I (I : K I ) = I , we have (I : K I )v = ((I −1 I )v (I : K I ))v = (I −1 I (I : K I ))v = (I −1 I )v = R, and therefore R ⊆ (I : K I ) ⊆ (I : K I )v = R, whence (I : K I ) = R. Therefore, by Proposition 2.3.17, R is integrally closed.  Remark 2.3.32. In Bourbaki, a v-domain is said to be regularly integrally closed [23, Chapter VII]. The class of v-Prüfer rings is very large. It is one of the most general class of integrally closed rings that we consider in this book. By Corollaries 2.3.31 and 2.3.18, we have the following full implication lattice.

Dedekind

CIC Prüfer

CIC

Prüfer v-Prüfer

integrally closed The following examples show that none of the implications above is reversible. Example 2.3.33. (1) A valuation domain (that is, a local Prüfer domain) is CIC if and only if it is at most one dimensional. See Section 3.1 for the definition of a valuation domain, and see Exercises 3.1.17–3.1.19 for details on the construction of such domains. (2) If D is a Dedekind domain but not a field, then D[X ] is CIC but not Prüfer. (See Theorem 1.2.4.) (3) There exist one dimensional local CIC Prüfer domains that are not Dedekind. A local CIC Prüfer domain is equivalently a valuation domain whose value group is isomorphic to a subgroup of R, and any such domain is necessarily one dimensional. However, a valuation domain is Noetherian (or Dedekind) if and only if it is a DVR.

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(4) There exist one dimensional local CIC domains that are not Prüfer. In 1952–1955 Nagata settled a famous 1936 conjecture of Krull by producing an example of a one dimensional local CIC domain that is not a valuation domain and consequently is not Prüfer [68, p. 159]. (5) There exist local Prüfer domains that are not CIC, or equivalently, valuation domains whose value group is not isomorphic to a subgroup of R, or, equivalently still, valuation domains that are of dimension greater than one. (6) There exist v-Prüfer domains (that is, v-domains) that are neither CIC nor Prüfer. For example, if D is any Prüfer domain that is not CIC, then D[X ] is a v-domain that is neither CIC nor Prüfer. (See Theorem 1.2.4.) (7) There exist one dimensional integrally closed domains that are not v-Prüfer. For example, the domain K + X L[X ], where L/K is a proper field extension such that K is algebraically closed in L, is an example of a one dimensional integrally closed (Mori) domain that is not a v-domain [68, p. 161]. The following result provides an equivalent characterization of the Dedekind rings. Proposition 2.3.34. A Dedekind ring is equivalently a completely integrally closed ring in which every regular (fractional) ideal is divisorial. Proof. Let R be a Dedekind ring. Then R is completely integrally closed by Corollary 2.3.18. Moreover, if I is a regular fractional ideal of R, then I is invertible and therefore I v = I . Conversely, suppose that R is a completely integrally closed ring such that I v = I for every regular (fractional) ideal I of R. It follows that, if I is a regular fractional ideal of R, then I is v-invertible, whence I I −1 = (I I −1 )v = R, so that I is invertible. Therefore R is Dedekind.  Remark 2.3.35. The term “divisorial” comes from the word “divisor.” Let R be a ring. For any I, J ∈ K(R), write I ∼ J if I v = J v (or equivalently if I −1 = J −1 ). The relation ∼ is an equivalence relation on K(R). Any ∼-equivalence class contains a unique divisorial Kaplansky fractional ideal, which is the largest element in that class. A divisor of R is a ∼-equivalence class div I of some regular fractional ideal I of R. (This generalizes the definition in [23, Chapter VII] of a divisor of an integral domain.) A divisor div I of R may then be identified with the divisorial regular fractional ideal I v . One writes div I + div J = div(I J ), which is well-defined since I1v = I2v and J1v = J2v implies (I1 J1 )v = (I2 J2 )v . The set Div(R) = Freg (R)/ ∼ of all divisors of R is a commutative monoid under the operation + with identity element 0 = div R, and it is an ordered monoid under the relation  defined by div I  div J if I v ⊆ J v . A divisor div I is said to be principal if div I = div a R for some regular a ∈ T (R), or equivalently if I v is principal. A divisor div I is said to be invertible if div I + div J = 0 for some divisor div J , or equivalently if I is v-invertible. The divisor class group Clv (R) of R is the group of all invertible divisors of R modulo the subgroup of all principal divisors of R. Alternatively, write div I ∼ = div J if some ideal I  ∈ div I is isomorphic as an R-module to some ideal J  ∈ div J (or equivalently if I v is isomorphic to J v as an R-module, or equivalently if I v = a J v for some regular a ∈ T (R)). Then Clv (R) is the group of isomorphism classes, or ∼ =-equivalence classes, of invertible divisors of R.

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The ordered monoid Div(R) may be identified with the ordered monoid Freg (R)v of all divisorial regular fractional ideals of R under the operation (I, J ) −→ (I J )v of v-multiplication. Under this identification, the units of the monoid Div(R) are precisely the v-invertible regular fractional ideals of R, which form an ordered group, denoted Invv (R). The divisor class group of R is then identified with the group Invv (R)/ Prin(R), where Prin(R) is the group of regular principal fractional ideals of R. The divisor class group is an important tool for studying rings. For example, it is well-known that a UFD is equivalently a Krull domain with trivial divisor class group. Krull domains, and various subgroups of the divisor class group, are introduced in Section 2.5. We end this section by stating, with references, some well-known deeper theorems concerning integral closure. (These results will not be used at all later in the text.) First is the celebrated Krull–Akizuki theorem. See [23, Proposition VII.2.5], [145, Theorem 11.7], or [178, Theorem 4.9.2] for a proof. Theorem 2.3.36 (Krull–Akizuki theorem). Let D be a one dimensional Noetherian domain with quotient field K , let L be a finite extension of K , and let B be a subring of L containing D. Then B is a Noetherian domain of dimension at most one, and for every nonzero ideal b of B, the D-algebra B/b is finitely generated as a D-module. We show in Section 3 that a domain is a Dedekind domain if and only if it is Noetherian, integrally closed, and of dimension at most one. Thus we have the following. Corollary 2.3.37. Let D be a one dimensional Noetherian domain with quotient field K , let L be a finite extension of K , and let D  be the integral closure of D in L. Then D  is a Dedekind domain. Example 2.3.38. Since the ring Z of integers is a PID, hence Dedekind, the integral closure of Z in any finite extension of Q is also Dedekind. Such a Dedekind domain is called an algebraic number ring. However, the integral closure of Z in C (or in any algebraic closure of Q) is a one dimensional non-Noetherian Bézout domain, called the ring of algebraic integers. The following well-known theorem is due to Nagata [153, Proposition 2], [154, Theorem 3], [155, (33.2) and (33.12)]. Theorem 2.3.39 (Nagata, 1953–1955). The integral closure of a Noetherian domain of dimension at most two is Noetherian. However, there exist Noetherian domains of dimension three with non-Noetherian integral closure. For any domain D, one has D[X ] = D[X ] by [23, Proposition V.1.12]. Moreover, for any ring R the ring R[X ] is Noetherian if and only if R is Noetherian, in which case dim R[X ] = 1 + dim R. It follows, then, from Nagata’s result that there exist Noetherian domains of any finite dimension n  3 whose integral closure is not Noetherian.

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Nagata also found an example of a non-Noetherian domain of dimension two with Noetherian integral closure. In fact, he found an example of a two dimensional Noetherian integral domain D and a non-Noetherian domain D  with D ⊆ D  ⊆ D, proving that the analogue of the Krull–Akizuki theorem for two dimensional domains is false [153, Proposition 1]. Note, here, that D  = D is Noetherian by Nagata’s theorem and yet D  is not Noetherian. Krull domains, introduced in Section 2.5, are a higher dimensional analogue of the Dedekind domains. (A Dedekind domain is equivalently a Krull domain of dimension at most one.) The celebrated Mori–Nagata theorem states that the integral closure of a Noetherian domain is a Krull domain. This shows that the integral closure of a Noetherian domain, though not necessarily Noetherian, nevertheless possesses at least some of the crucial properties of the Dedekind domains. In 1976, Huckaba generalized Nagata’s theorem above to rings with or without zerodivisors as follows [106, Theorem B and Proposition 3.1]. Theorem 2.3.40 (Huckaba, 1976). The integral closure of a Noetherian ring of dimension at most two is r-Noetherian. Moreover, there exist Noetherian rings of all positive dimensions with non-Noetherian integral closure. We also note the following analogue of Corollary 2.3.37 for Prüfer domains. Theorem 2.3.41 ([70, Theorem III.1.2], [79, Theorem 22.3]). Let D be a Prüfer domain with quotient field K , let L be an algebraic extension of K , and let D  be the integral closure of D in L. Then D  is a Prüfer domain. This result was generalized to Prüfer rings by Griffin in 1970; however, his proposition omits a necessary hypothesis. We provide a proof of the following theorem in Section 3.9. Theorem 2.3.42 (cf. [93, Proposition 14]). Let R be a Prüfer ring with total quotient ring K , let L be any total quotient ring that is integral over K , and let S be the integral closure of R in L. Then S is a Prüfer ring. Given Theorem 2.3.42 and Corollary 2.3.37, it seems reasonable to make the following conjecture. Conjecture 2.3.43. Let R be a Dedekind ring with total quotient ring K , let L be any total quotient ring that is a finite extension of K , and let S be the integral closure of R in L. Then S is a Dedekind ring.

2.4 Semistar Operations A semistar operation on a ring R is a type of closure operation on the ordered monoid K(R) of all Kaplansky fractional ideals of R. The notion of a semistar operation is a special case of that of a nucleus on an ordered monoid. Semistar operations are one of the main tools employed in Chapters 2 and 3.

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Definition 2.4.1. A nucleus on an ordered monoid M is a sub-multiplicative closure operation on M, that is, it is a self-map  : x −→ x  of M such that  is: (1) (2) (3) (4)

order-preserving: x  y implies x   y  , expansive: x  x  , idempotent: (x  ) = x  , sub-multiplicative: x  y   (x y) ,

where each of these axioms holds for all x, y ∈ M. A nucleus  on M is unital if 1 = 1. The set Nucl(M) of all nuclei on an ordered monoid M is a partially  ordered set, where one defines    if x   x  for all x ∈ M, or equivalently if  every -closed element of M is  -closed. A (unital) semistar operation on a ring R is a unital nucleus on the ordered monoid K(R) of all Kaplansky fractional ideals of R. Semistar operations are a powerful tool for understanding and controlling the ideals of a ring. In this section, we provide a concise but thorough introduction to the theory of semistar operations—a “crash course,” so to speak. Proofs of many of the results are left to the reader, who is alternatively invited to consult Chapter 6, which generalizes all of these results to nuclei on various classes of ordered monoids. Definition 2.4.2. Let R be a ring. A (unital) semistar operation on R is a unital nucleus on the ordered monoid K(R), that is, it is a self-map  : I −→ I  of the set K(R) such that  is: (1) (2) (3) (4) (5)

order-preserving: I ⊆ J implies I  ⊆ J  , expansive: I ⊆ I  , idempotent: (I  ) = I  , sub-multiplicative: I  J  ⊆ (I J ) , unital: R  = R,

where each of these axioms holds for all I, J ∈ K(R). The set Semistar(R) of all semistar operations on a ring R is a partially ordered set, where one defines     if I  ⊆ I  for all I ∈ K(R), or equivalently if every -closed Kaplansky fractional ideal of R is  -closed. The sub-multiplicative axiom (4) of semistar operations may be replaced with the following axiom: (4 ) a I  ⊆ (a I ) for all I ∈ K(R) and all a ∈ K . Example 2.4.3. By statements (1)–(5) of Proposition 2.3.24, for any ring R the operation v : I −→ I v = (I −1 )−1 of divisorial closure on K(R) is a semistar operation on R. In fact, as we show in Proposition 2.4.10 below, it has the distinction of being the largest (unital) semistar operation on R in the sense that   v for any semistar operation  on R. The smallest semistar operation on a ring also has a name and a symbol.

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95

Definition 2.4.4. Let R be a ring. Define d : K(R) −→ K(R) by I d = I for all I ∈ K(R). The operation d is called the trivial semistar operation on R. Definition 2.4.5. A semistar operation  on a ring is reduced if (0) = (0). Convention 2.4.6. For the following reasons, caution must be exercised, in this text and in the literature, when interpreting the term “semistar operation.” (1) Semistar operations on rings are not assumed to be reduced. On any field there are exactly two semistar operations, only one of which is reduced. However, any semistar operation on an integral domain that is not a field is necessarily reduced; indeed, if R is a domain with quotient field K and  is a semistar operation on R, then (0) K ⊆ (0K ) = (0) implies that (0) = (0) or (0) = K , and the latter case implies that R = K since  is unital. (2) In the literature, semistar operations are mostly considered in the context of integral domains, where unfortunately (0) is usually left undefined. Effectively, this means that semistar operations in the context of integral domains traditionally have been assumed to be reduced. By (1), the only situation where the difference matters is the case where R is a field. (3) In Chapters 2 and 3 of this book, semistar operations are assumed to be unital. However, in the literature, and in Chapters 4 and 5 of this book, semistar operations are not assumed to be unital. A semistar operation in this more general sense is a ∗-operation in the sense of Huckaba [107, Section 20]. In Chapters 2 and 3, we have assumed that semistar operations are unital in order to avoid incessant repetition of the word “unital.” (4) In the literature, reduced unital semistar operations on integral domains, as we have defined them, are called (semi)star operations. Thus, if R is an integral domain but not a field, then by a “semistar operation on R” in Chapters 2 and 3 we mean a “(semi)star operation on R.” (5) In [54], N. Epstein defines a (not necessarily unital) semistar operation on a ring R using the axiom (4 ) a I  = (a I ) for all I ∈ K(R) and all units a ∈ T (R), instead of using the axioms (4) or (4 ). While (4 ) follows from axioms (1)–(4), axiom (4 ) is not strong enough to replace axiom (4). The work herein indicates, however, that there are compelling reasons to assume axiom (4). The following proposition and corollary provide a convenient recipe for constructing examples of semistar operations. Proposition 2.4.7. Let R be a ring and O any collection of overrings of R, and let  T = S∈O S. Then the self-map O : I −→

S∈O

of K(T ) ⊆ K(R) is a semistar operation on T .

IS

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Corollary 2.4.8. Let R be a ring and O any collection of overrings of R whose intersection is R. Then the self-map O : I −→



IS

S∈O

of K(R) is a semistar operation on R. One easily verifies the following elementary relationships between a given semistar operation on R and the various operations on K(R) introduced in Section 2.1. Lemma 2.4.9. Let  be a semistar operation on a ring R, let I, J ∈ K(R), and let {Iλ : λ ∈ } be a subset of K(R).   (1) ( λ∈ Iλ ) = ( λ∈ Iλ ) .     (2) λ∈ Iλ = ( λ∈ Iλ ) . (3) The intersection of a collection of -closed elements of K(R) is -closed. (4) (I J ) = (I  J ) = (I J  ) = (I  J  ) . (5) (I : K J ) ⊆ (I  : K J ) = (I  : K J  ) = (I  : K J ) , and therefore (I  : K J ) is -closed. (6) (I  )−1 = (I −1 ) = I −1 , and therefore I −1 is -closed. (7) If I is invertible, then I  = I and (I J ) = I J  . (8) a I  ⊆ (a I ) for all a ∈ T (R). (9) (a I ) = a I  and (a R) = a R for all regular elements a of T (R). Statements (1)–(3) of the lemma are consequences of the fact that  is a closure operation on K(R) and K(R) is a complete lattice. Proposition 2.4.10. For any ring R, the divisorial closure operation v : I −→ I v = (I −1 )−1 on K(R) is the largest (unital) semistar operation on R. Proof. Let  be any semistar operation on R, and let I ∈ K(R). Since I I −1 ⊆ R, one has I  I −1 ⊆ (I I −1 ) ⊆ R  = R and therefore I  ⊆ (I −1 )−1 = I v . Therefore   v.  Remark 2.4.11. If we omit semistar axiom (5) and thus do not assume that all semistar operations are unital, then the largest semistar operation on a ring R is the operation acting by I −→ T (R) for all I ∈ K(R). A possibly nonunital semistar operation  is unital if and only if   v. Another useful consequence of the fact that the poset K(R) is complete is that the poset of all semistar operations on R is also complete. In fact, we have the following. Proposition 2.4.12. Let R be a ring. The poset of all semistar operations on R is complete. The infimum inf  of a set  of semistar operations acts by I −→ I inf  =

{I  :  ∈ }

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97

and the supremum sup  acts by I −→ I sup  =

{J ∈ K(R) : J ⊇ I and J  = J for all  ∈ }.

Moreover, for all I ∈ K(R), one has I sup  = I if and only if I  = I for all  ∈ . See Proposition 6.2.7 for a generalization of the proposition to nuclei. Alternatively, the proposition may be proved directly by appealing to the following lemma. Lemma 2.4.13. Let R be a ring and C ⊆ K(R). There exists a (unique)  semistar operation  on R such that C = {I ∈ K(R) : I  = I } if and only if X ∈ C for all X ⊆ C and for all I ∈ C and all J ∈ K(R)one has (I : K J ) ∈ C. If these two equivalent conditions hold, then one has I  = {J ∈ C : J ⊇ I } for all I ∈ K(R). Definition 2.4.14. For any semistar operation  on a ring R, let K(R) denote the ordered monoid of all -closed fractional ideals of R under the operation (I, J ) −→ I  J = (I J ) of -multiplication. To see that the operation of -multiplication is indeed associative, note that ((H I ) J ) = (H I J ) = (H (I J ) ) for all H, I, J ∈ K(R). If S is an overring of R, then S  is an overring of S since S  S  ⊆ (SS) = S  . An important class of semistar operations is singled out by the following definition. Definition 2.4.15. A semistar operation  on a ring R is of finite type if I =



{J  : J ∈ K(R) is f.g. and J ⊆ I }

for all I ∈ K(R). As discussed in Chapter 6, a semistar operation is of finite type if and only if it is continuous in a certain topology on the poset K(R) known as the “Scott topology.” Thus, the finite type condition is a continuity condition. Another important regularity condition on semistar operations is known as “stability.” Definition 2.4.16. A semistar operation  on a ring R with total quotient ring K is stable if (I ∩ J ) = I  ∩ J  for all I, J ∈ K(R) and (I : K J ) = (I  : K J ) for all I, J ∈ K(R) with J finitely generated. Unfortunately, the v-operation need not be of finite type, nor need it satisfy either condition required for stability. Remark 2.4.17. The expert on semistar operations will recognize the fact a semistar operation on an integral domain R is stable if and only if the first condition holds, that is, if and only if (I ∩ J ) = I  ∩ J  for all I, J ∈ K(R), for in that case the second condition, on colons, follows from the first. However, over a general commutative ring R this is no longer the case. Since both conditions are required for the theory

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of stable semistar operations to work as it does over integral domains, one is forced to include both conditions in the definition of stability. This is one of the valuable lessons one learns by considering nuclei more generally on ordered monoids; see Definition 6.6.1 and Theorem 6.6.7. Proposition 2.4.18. Let R be a ring and  a set of semistar operations on R. (1) If  is of finite type for all  ∈  and  is finite, then inf  is of finite type. (2) If  is stable for all  ∈ , then inf  is stable. Proof. See Proposition 6.2.7 and Theorem 6.4.16.



Proposition 2.4.19. Let R be a ring with total quotient ring K and S an R-torsionfree R-algebra with total quotient ring L, so that L is a K -algebra, and suppose that S ∩ K = R. Let  be any semistar operation on S. Then the map  R : I −→ (I S) ∩ K is a semistar operation on R. Moreover, if  is of finite type, then  R is of finite type, and if  is stable and S is flat over R, then  R is stable. Proposition 2.4.20. Let R be a ring and O a collection of overrings of R whose intersection  is R. Let O denote the semistar operation on K(R) acting by I −→ I O = S∈O I S for all I ∈ K(R). If S is flat over R for all S ∈ O, then O is stable. Proof. Let  = O . Then  is a semistar operation on R by Corollary 2.4.7. Suppose that S is flat over R  for all S ∈ O. Then,by Proposition 2.2.10, for all I, J ∈ K(R), we I S ∩ have (I ∩ J ) = S∈O (I ∩ J )S = S∈O (I S ∩ J S) = S∈O S∈O J S 2.2.12, for all I, J ∈ K(R) with J finitely = I  ∩ J  . Moreover, by Corollary    generated, we have (I : K J ) = S∈O (I : K J )S = S∈O (I S : K J S) = (( S∈O I S)  : K J ) = (I  : K J ). Thus,  is stable. Even though a semistar operation need not be of finite type or stable, for any semistar operation  on a ring there is a largest finite type semistar operation t less than or equal to , and there is also a largest stable semistar operation  = s less than or equal to . The semistar operation t can be thought of as the optimal Scott continuous approximation of , and the semistar operation  can be thought of as the optimal stable approximation of  (both from below). These are constructed as follows. Definition 2.4.21. Let R be a ring with total quotient ring K . (1) For any semistar operation  on R, we let  {J  : J ∈ K(R) is f.g. and J ⊆ I },  I  = I s = {(I : K J ) : J ∈ I(R), J  = R},  I w = {(I : K J ) : J ∈ I(R) is f.g. and J  = R}, I t =

for all I ∈ K(R). The operations t ,  = s , and w are semistar operations on R.

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99

(2) The t-operation is the semistar operation t = vt . (3) The s-operation is the semistar operation s = v = vs (4) The w-operation is the semistar operation w = vw . Remark 2.4.22. t depends only the operation  restricted to the set {J ∈ K(R) : J is f.g.}.  depends only on the set {J ∈ I(R) : J  = R}. The unions in Definition 2.4.21 may be replaced with sums, or suprema, in K(R). The two occurrences of “I(R)” in Definition 2.4.21 may be replaced with “K(R).” (5) For all I ∈ Kreg (R), one has

(1) (2) (3) (4)

 {J  : J ∈ Kreg (R) is f.g. and J ⊆ I },  {(I : K J ) : J ∈ Ireg (R), J  = R}, I =  = {(I : K J ) : J ∈ Ireg (R) is f.g. and J  = R}.

I t =

I w

We note the following equivalent characterizations of stability. Proposition 2.4.23. Let R be a ring with total quotient ring K , and let  be a semistar operation on R. The following conditions are equivalent. (1)  is stable. (2) (I : K J ) = (I  : K J ) and (I ∩ R) = I  ∩ R for all I, J ∈ K(R) with J finitely generated. (3) (I : R J ) = (I  : R J ) for all I, J ∈ K(R) with J finitely generated. (4) (I : R x) = (I  : R x) for all I ∈ K(R) and all x ∈ K . (5)  = . Proof. The implications (1) ⇒ (2) ⇒ (3) ⇒ (4) are clear. Suppose that (4) holds. Let I ∈ K(R) and x ∈ I  . Let J = (I : R x). Then J is an ideal of R with J  = (I  : R x) = R. One therefore has x ∈ (I : K (I : R x)) = (I : K J ) ⊆ I  . It follows that I  = I  . Thus (4) ⇒ (5). Finally, one easily checks that  is a stable semistar operation on R, and therefore (5) ⇒ (1).  See Theorem 6.6.7 for a generalization of the proposition to nuclei. By the proposition above,  is stable if and only if  = . It is also clear that  is of finite type if and only if  = t , and  is stable and of finite type if and only if  = w . From these facts one may readily deduce the following. Proposition 2.4.24. Let  be a semistar operation on a ring R. (1) t is the largest finite type semistar operation less than or equal to . (2)  = s is largest stable semistar operation less than or equal to . (3) w is the largest stable finite type semistar operation less than or equal to , and one has w = t  ()t . In particular, one has w  t   and w    .

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See Theorems 6.5.7, 6.6.7, and 6.6.9 for generalizations of the proposition above to nuclei. Note that t and  are not necessarily comparable. Corollary 2.4.25. Let R be a ring. Then v is the largest semistar operation, t is the largest finite type semistar operation, s is the largest stable semistar operation, and w is the largest stable finite type semistar operation, on R. In particular, one has w  t  v and w  s  v. For any semistar operation , the semistar operations , t , , w , v, t, s, and w are related according to the following Hasse diagram.

v

t



s

t

w



w

The semistar operations t, s, and w repair certain deficiencies in the operation v in an optimal way. The four semistar operations v, t, w, and d are by many accounts the most important and useful of all semistar operations, with the possible exception of those studied in Chapter 4. Proposition 2.4.24 and Corollary 2.4.25 are generalized as follows. Definition 2.4.26. Let R be a ring with total quotient ring K . For any I, J ∈ K(R), we set I v(J ) = I v ∩ (J : K (J : K I )). The operation v(J ) : I −→ I v(J ) on K(R) for any fixed J ∈ K(R) is a semistar operation on R called divisorial closure on R with respect to J. We set t (J ) = v(J )t , s(J ) = v(J )s , and w(J ) = v(J )w . Proposition 2.4.27. Let R be a ring and J ∈ K(R). (1) (2) (3) (4)

v(J ) is the largest semistar operation  on R such that J  = J . t (J ) is the largest finite type semistar operation  on R such that J  = J . s(J ) is the largest stable semistar operation  on R such that J  = J . w(J ) is the largest stable finite type semistar operation  on R such that J  = J .

Proof. See Theorems 6.3.11 and 6.6.11.



Note that v = v(R), t = t (R), s = s(R), and w = w(R). Remark 2.4.28. For semistar operations that are not necessarily unital, one may define I v(J ) = (J : K (J : K I )) instead of I v(J ) = I v ∩ (J : K (J : K I )).

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101

Given an element J of K(R), the semistar operation v(J ) is the largest semistar operation  on R such that J is -closed. But suppose more generally we take an arbitrary subset S of K(R) and ask, is there a largest semistar operation  on R such that every element of S is -closed? The answer is in the affirmative, essentially because the poset of all semistar operations is complete. Definition 2.4.29. Let R be a ring and S ⊆ K(R). We define v(S) = inf{v(J ) : J ∈ S}, which we call divisorial closure on R with respect to S. We set t (S) = v(S)t , s(S) = v(S)s , and w(S) = v(S)w . Proposition 2.4.30. Let R be a ring and S ⊆ K(R). v(S) is the largest semistar operation  on R such that S ⊆ K(R) . t (S) is the largest finite type semistar operation  on R such that S ⊆ K(R) . s(S) is the largest stable semistar operation  on R such that S ⊆ K(R) . w(S) is the largest stable finite type semistar operation  on R such that S ⊆ K(R) . (5) v(S) = inf{v(J ) : J ∈ S} and s(S) = inf{s(J ) : J ∈ S}. (6) t (S) = inf{t (J ) : J ∈ S} and w(S) = inf{w(J ) : J ∈ S} if S is finite. (1) (2) (3) (4)

Moreover, for any semistar operation  on R, one has the following. (7) (8) (9) (10)

 = v(K(R) ) = inf{v(J ) : J ∈ K(R) }. t = t (K(R)t ) = inf{t (J ) : J ∈ K(R)t }. s = s(K(R)s ) = inf{s(J ) : J ∈ K(R)s }. w = w(K(R)w ) = inf{w(J ) : J ∈ K(R)w }.

Proof. See Theorems 6.3.11 and 6.6.11.



In particular, every semistar operation on a ring is an infimum of divisorial closure semistar operations v(J ). Moreover, we have the following. Corollary 2.4.31. Let  be a semistar operation on a ring R. (1) (2) (3) (4)

 = v(S) for some S ⊆ K(R).  is of finite type if and only if  = t (S) for some S ⊆ K(R).  is stable if and only if  = s(S) for some S ⊆ K(R).  is stable of finite type if and only if  = w(S) for some S ⊆ K(R).

Semistar operations are only one of several types of closure operation associated with a commutative ring. The following definition of a star operation on a commutative ring provides a generalization of the notion of a star operation, or  -operation, on an integral domain, originally due to Krull [124, p. 118]. Definition 2.4.32. Let R be a ring. A star operation on R is a unital nucleus on the ordered monoid Freg (R), that is, it is a self-map ∗ : I −→ I ∗ of the set Freg (R) such that ∗ is: (1) order-preserving: I ⊆ J implies I ∗ ⊆ J ∗ , (2) expansive: I ⊆ I ∗ ,

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(3) idempotent: (I ∗ )∗ = I ∗ , (4) sub-multiplicative: I ∗ J ∗ ⊆ (I J )∗ , (5) unital: R ∗ = R, where each of these axioms holds for all I, J ∈ Freg (R). The set of all star operations  on a ring R is a partially ordered set, where one defines ∗  ∗ if I ∗ ⊆ I ∗ for all I ∈ Freg (R), or equivalently if every ∗-closed regular fractional ideal of R is ∗ -closed. It is clear that every semistar operation on R restricts to a star operation on R. It is not so obvious that, conversely, every star operation ∗ on R is the restriction to Freg (R) of some semistar operation on R. Definition 2.4.33. For any star operation ∗ on a ring R, let e∗ = v(Freg (R)∗ ) = inf{v(I ) : I ∈ Freg (R)∗ } and d∗ = inf{ ∈ Semistar(R) : |Freg (R) = ∗}. Proposition 2.4.34. Let ∗ be a star operation on a ring R. (1) e∗ is the largest semistar operation on R that restricts to ∗. (2) d∗ is the smallest semistar operation on R that restricts to ∗. (3) A semistar operation  on R restricts to ∗ if and only if d∗    e∗ . Star operations are discussed further in Section 4.1. Definition 2.4.35. Let  and  be semistar operations on a ring R. We say that  and  are star equivalent, written    , if  and  restrict to the same star operation  on R, that is, if I  = I  for all regular (fractional) ideals I of R. We write    if  I  ⊆ I  for all regular (fractional) ideals I of R. Clearly one has    if and only if    and   . The relation  is a preorder refined by the partial ordering , and the relation  is an equivalence relation refined by the equality relation, on the set of all semistar operations on R. A star operation can be thought of as a -equivalence class of semistar operations, and by Proposition 2.4.34 every such -equivalence class is a -interval. Semistar operations have many advantages over star operations (besides being more general), so we focus in Chapters 2 and 3 on the former. With semistar operations, one can consider the closure of Kaplansky fractional ideals that are not regular or not fractional, which is useful for many purposes. While it would be possible to reframe Chapters 2 and 3 around star operations, as was done in Chapter 1, much information would be lost in doing so. Nevertheless, any properties of semistar operations involving only regular fractional ideals—and this includes many of those discussed in Chapters 2 and 3—are invariant under star equivalence. This is not the case, however, for the finite type and stable properties. To rectify this we make the following definitions.

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103

Definition 2.4.36. A semistar operation  on a ring R is of r-finite type if I =

 {J  : J ∈ Freg (R) is f.g. and J ⊆ I }

for all I ∈ Freg (R). Definition 2.4.37. A semistar operation  on a ring R with total quotient ring K is r-stable if (I ∩ J ) = I  ∩ J  for all I, J ∈ Freg (R) and (I : K J ) = (I  : K J ) for all I, J ∈ Freg (R) with J finitely generated. Clearly, if  is of finite type then  is of r-finite type, and if  is stable then  is r-stable. The “r-” prefix stands for “regularly.” Proposition 2.4.38. Let  be a semistar operation on a ring R. (1)  is of r-finite type if and only if   t . (2)  is r-stable if and only if   . (3)  is r-stable and of r-finite type if and only if   w . Proposition 2.4.39. Let R be a ring and  a set of semistar operations on R. (1) If  is of r-finite type for all  ∈  and  is finite, then inf  is of r-finite type. (2) If  is r-stable for all  ∈ , then inf  is r-stable. Proposition 2.4.40. Let R be a ring with total quotient ring K , and let  be a semistar operation on R. The following conditions are equivalent. (1)  is r-stable. (2) (I ∩ R) = I  ∩ R and (I : K J ) = (I  : K J ) for all I, J ∈ Freg (R) with J finitely generated. (3) (I : R J ) = (I  : R J ) for all I, J ∈ Freg (R) with J finitely generated. (4) (I : R J ) = (I  : R J ) for all regular ideals I and J of R with J finitely generated. (5)   . Proof. See Theorem 6.6.7.



Proposition 2.4.42 below shows that t , s , and w have analogues for the r-finite and r-stable properties. Definition 2.4.41. Let  be a semistar operation on a ring R. We let r t = inf{v(I ) : I ∈ Freg (R) and I t = I }, r s = inf{v(I ) : I ∈ Freg (R) and I s = I }, r w = inf{v(I ) : I ∈ Freg (R) and I w = I }.

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Note that r t = e∗ , where ∗ is the restriction of t to a star operation on R. Thus, r t is the largest semistar operation on R that has the same restriction to Freg (R) as t . Similar comments hold for r s and r w . Proposition 2.4.42. Let  be a semistar operation on a ring R. (1) r t is the largest r-finite type semistar operation less than or equal to . (2) r s is largest r-stable semistar operation less than or equal to . (3) r w = (r t )r s is the largest r-stable r-finite type semistar operation less than or equal to . (4) (r t )t = (t )r t = t  r t  t . (5) (r s )s = (s )r s = s  r s  s . (6) (r w )w = (w )r w = w  r w  w . The semistar operations r t , r t , r w are in a sense more general, respectively, than the “simpler” semistar operations t , s , w , since the latter can be recovered from the former, but not vice versa. In the interest of generality, it would better to use the former operations over the latter whenever possible. However, since most of the results on semistar operations we discuss in Chapters 2 and 3 are invariant under star invariance, in the interest of simplicity we use the operations t , s , w instead.

2.5 -Dedekind Rings and -Prüfer Rings In this section, we introduce natural generalizations of the Dedekind and Prüfer rings to classes of rings defined relative to semistar operations. These so-called -Dedekind and -Prüfer rings are the main subjects of Chapters 2 and 3. The rather extensive theories of Dedekind domains and Prüfer domains bear witness to the fact that rings with lots of invertible ideals have nice properties. Rings whose invertible ideals are all principal (such as all semilocal rings and all UFDs), that is, rings with trivial ideal class group, can be judged as well behaved only if they are known to have lots of invertible ideals, as do, for example, Dedekind rings and Prüfer rings. One of the many advantages of using semistar operations is that many of the benefits of invertibility and class groups can be made available to much more general classes of rings via the notions of -invertibility and -class groups. We saw hints of this in Section 2.3, where we showed, for example, that a ring is completely integrally closed if and only if every regular fractional ideal is v-invertible. Definition 2.5.1. Let  be a semistar operation on a ring R. A Kaplansky fractional ideal I of R is  -invertible if (I J ) = R for some Kaplansky fractional ideal J of R. Example 2.5.2. A Kaplansky fractional ideal I of R is invertible if and only if it is d-invertible. A Kaplansky fractional ideal I of R is v-invertible as defined in Definition 2.3.26 if and only if it is -invertible, where  = v. We note the following.

2.5 -Dedekind Rings and -Prüfer Rings

105

Proposition 2.5.3. Let R be a ring and I, J ∈ K(R), let c ∈ T (R), and let  be a semistar operation on R. (1) If (I J ) = R, then J  = J v = I −1 and therefore R = (I J ) = (I J  ) = (I I −1 ) . (2) I is -invertible if and only if (I I −1 ) = R, if and only if I  is -invertible. (3) I J is -invertible if and only if both I and J are -invertible. (4) cI is -invertible if I is -invertible and c is regular. Proposition 2.5.4. Let R be a ring, and let  and  be semistar operations on R. (1) If   , then one has the following. (a) Every  -invertible regular fractional ideal of R is -invertible. (b) If I ∈ Freg (R) is  -closed and  -invertible, then I is -closed and -invertible. (2) If   , then one has the following. (a) Every  -invertible Kaplansky fractional ideal of R is -invertible. (b) If I ∈ K(R) is  -closed and  -invertible, then I is -closed and -invertible. Semistar operations and the notion of -invertibility allow us to define the following natural generalizations of the Dedekind and Prüfer rings. Definition 2.5.5. Let  be a semistar operation on a ring R. (1) R is  -Dedekind if every regular (fractional) ideal of R is -invertible. (2) R is -Prüfer if every finitely generated regular (fractional) ideal of R is -invertible. The -Dedekind rings and -Prüfer rings are the primary focus of Chapters 2 and 3. The case  = d was mentioned in Section 2.2, while the case  = v was mentioned in Section 2.3. Indeed, the following result follows immediately from Proposition 2.3.28 and the definitions of the Dedekind, Prüfer, and v-Prüfer rings provided in Sections 2.2 and 2.3. Proposition 2.5.6. One has the following. (1) (2) (3) (4)

A d-Dedekind ring is equivalently a Dedekind ring. A d-Prüfer ring is equivalently a Prüfer ring. A v-Dedekind ring is equivalently a completely integrally closed ring. A v-Prüfer ring as defined in Definition 2.3.30 is equivalently a -Prüfer ring, where  = v.

Remark 2.5.7. In the literature, a t -Prüfer domain is called a PMD, or Prüfer -multiplication domain, and a t -Prüfer ring is called a PMR, or Prüfer -multiplication ring. The notion of a -Prüfer ring is more general than that of a PMR.

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Krull domains are a well-known and important class of integral domains that generalize the Dedekind domains, the integrally closed Noetherian domains, and the unique factorization domains. We have seen that a Dedekind domain is equivalently a Noetherian Prüfer domain. We show in Section 3.4 that a Dedekind domain is also equivalently a Krull domain of dimension at most one. Thus, the Prüfer domains are a non-Noetherian analogue of the Dedekind domains, while the Krull domains are a higher dimensional analogue of the Dedekind domains.  A Krull domain is typically defined as an intersection D = λ∈ Dλ of DVR overrings Dλ of D such that every nonzero element of D is a unit in Dλ for all but finitely many λ. Here we part from tradition and define a Krull domain to be a t-Dedekind domain, and we show in Section 3.4, using local methods, that this agrees with the more standard definitions. With this approach, the Krull domains can be studied in a much broader context, since they can then be generalized much more easily to rings with zerodivisors, as follows. Definition 2.5.8. A Krull ring is a t-Dedekind ring, that is, a ring in which every regular (fractional) ideal is t-invertible. Thus, a Krull domain is a t-Dedekind domain, that is, a domain in which every nonzero (fractional) ideal is t-invertible. PVMDs, or Prüfer v -multiplication domains, are an important class of domains that generalize both the Krull domains and the Prüfer domains. A useful analogy is that PVMDs are to Krull domains as Prüfer domains are to Dedekind domains, or, equivalently, PVMDs are to Prüfer domains as Krull domains are to Dedekind domains. In Bourbaki, PVMDs are called pseudoPrüferian domains [23, Exercise VII.2.19]. Unfortunately, and undeservedly, the PVMDs are not as widely known as are the Krull domains and the Prüfer domains. Definition 2.5.9. (1) A PVMR, or Prüfer v-multiplication ring, is a t-Prüfer ring, that is, a ring in which every finitely generated regular (fractional) ideal is t-invertible. (2) A PVMD, or Prüfer v-multiplication domain, is a t-Prüfer domain, that is, a domain in which every nonzero finitely generated (fractional) ideal is t-invertible. The following result is a consequence of Proposition 2.5.4. Proposition 2.5.10. Let  and  be semistar operations on a ring R. If   , then every  -Dedekind ring is -Dedekind and every  -Prüfer ring is -Prüfer. In particular, since d  t  t  v and t    v, we have the following. Corollary 2.5.11. One has the following implications for any semistar operation  on a ring R.

2.5 -Dedekind Rings and -Prüfer Rings

107

Dedekind

d-Dedekind

t -Dedekind

t-Dedekind

Krull

Prüfer

d-Prüfer

t -Prüfer

t-Prüfer

PVMR

t -Dedekind

-Dedekind

v-Dedekind

CIC

t -Prüfer

-Prüfer

v-Prüfer

integrally closed

The following result implies that a ring is -Dedekind (resp., t -Dedekind, t -Prüfer) if and only if it is CIC (resp., Krull, a PVMR) and   v (resp, t  t, t  t). Thus, for example, a ring is Dedekind if and only if it is CIC and d  v, as we saw in Proposition 2.3.34, and likewise a ring is Prüfer if and only if it is a PVMR and d  t. However, it is important to note that, if a ring is -Prüfer, then one need not have   v. For example, a t-Prüfer domain, or PVMD, need not satisfy t  v. Proposition 2.5.12. Let R be a ring and  a semistar operation on R. (1) (2) (3) (4)

R is -Dedekind if and only if R is CIC and   v. R is t -Dedekind if and only if R is Krull and t  t. R is t -Prüfer if and only if R is a PVMR and t  t. R is -Prüfer if R is v-Prüfer and   v.

Proof. Clearly, if R is CIC, or v-Dedekind, and  = v on Freg (R) then R is -Dedekind. Conversely, suppose that R is -Dedekind. Then R is v-Dedekind since   v. Let I be a regular ideal of R. Then I  is -closed and -invertible and therefore is also v-closed and v-invertible, by Proposition 2.5.4. Therefore I  = I v . This proves (1), and the proofs of (2)–(4) are similar.  The implication in statement (4) of the proposition is not reversible. For example, there exist (d-)Prüfer domains for which d  v, e.g., the domain D + X K [X ] for any non-local semilocal PID D with quotient field K . (See Theorem 1.2.6.) Corollary 2.5.13. Let R be a ring and    semistar operations on R. (1) R is  -Dedekind if and only if R is -Dedekind and    . (2) R is t -Prüfer if and only if R is t -Prüfer and t  t . Corollary 2.5.14. One has the following. (1) A ring is Dedekind if and only if it is completely integrally closed and v  d. (2) A ring is Krull if and only if it is completely integrally closed and t  v. (3) A ring is Prüfer if and only if it is a PVMR and t  d.

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Consequently, a ring is Dedekind if and only if it is Krull and Prüfer. Next, recall from Section 2.4 that  (resp., w ) denotes the largest stable (resp., finite type stable) semistar operation less than or equal to . By the following lemma and proposition, the -Prüfer rings and the w -Prüfer rings coincide, respectively, with the -Prüfer rings and the t -Prüfer rings (and also, respectively, with the r s -Prüfer rings and the r w -Prüfer rings). Lemma 2.5.15. Let  be a semistar operation on a ring R, and let I ∈ K(R). (1) I  = R if and only if I  = R. (2) I is -invertible if and only if I is -invertible. (3) I is t -invertible if and only if I is w -invertible. Proposition 2.5.16. Let  be a semistar operation on a ring R. (1) (2) (3) (4)

A -Prüfer ring is equivalently a -Prüfer ring. A t -Prüfer ring is equivalently a w -Prüfer ring. A -Dedekind ring is equivalently a -Dedekind ring. A t -Dedekind ring is equivalently a w -Dedekind ring.

Corollary 2.5.17. One has the following. (1) (2) (3) (4)

A v-Prüfer ring is equivalently a v-Prüfer ring. A PVMR is equivalently a w-Prüfer ring. A completely integrally closed ring is equivalently a v-Dedekind ring. A Krull ring is equivalently a w-Dedekind ring.

It is well-known that the so-called “ideal class group” of a Dedekind domain is an important measure of how far it is from being a PID. Definition 2.5.18. Let R be a ring. (1) Prin(R) denotes the group of all principal regular fractional ideals of R. (2) Inv(R) denotes the group of all invertible (Kaplansky) fractional ideals of R. (3) Cl(R) = Inv(R)/ Prin(R) denotes the quotient of the group Inv(R) of all invertible fractional ideals of R by its subgroup Prin(R) of all principal regular fractional ideals of R. It is called the (ideal) class group of R. Note that a ring R has trivial ideal class group (written Cl(R) = 0) if and only if Inv(R) = Prin(R), if and only if every invertible ideal of R is principal. Proposition 2.5.19. One has the following. (1) An r-PIR is equivalently a Dedekind ring with trivial ideal class group. (2) An r-Bézout ring is equivalently a Prüfer ring with trivial ideal class group. Proof. A ring R is Dedekind with trivial ideal class group if and only if every regular ideal of R is invertible and every invertible ideal of R is principal, if and only if every regular ideal of R is principal, if and only if R is an r-PIR. Similarly, a ring R is Prüfer with trivial ideal class group if and only if every finitely generated regular ideal of R is invertible and every invertible ideal of R is principal, if and only if every finitely generated regular ideal of R is principal, if and only if R is r-Bézout. 

2.5 -Dedekind Rings and -Prüfer Rings

109

For more general classes of rings, like the Krull rings and PVMRs, it is the “t-class group,” rather than the ideal class group, that carries the most relevant information (and for Dedekind rings and Prüfer rings the two groups are the same). Definition 2.5.20. Let  be a semistar operation on a ring R. (1) Inv (R) denotes the group of all -closed -invertible regular fractional ideals of R under -multiplication. (2) Cl (R) = Inv (R)/ Prin(R) denotes the quotient of the group Inv (R) of all -closed -invertible regular fractional ideals of R under -multiplication by its subgroup Prin(R) of all principal regular fractional ideals of R. It is called the -class group, or -divisor class group, of R. Definition 2.5.21. Let  be a semistar operation on a ring R. (1) Inv (R) denotes the group of all -closed -invertible Kaplansky fractional ideals of R under -multiplication. (2) Cl (R) = Inv (R)/ Prin(R) denotes the quotient of the group Inv (R) of all -closed -invertible Kaplansky fractional ideals of R under -multiplication by its subgroup Prin(R) of all principal regular fractional ideals of R. It is called the semistar -class group of R. We note the following. Proposition 2.5.22. Let R be a ring, and let  and  be semistar operations on R. Inv (R) is the group of units of the ordered monoid Freg (R) . Inv (R) is the group of units of the ordered monoid K(R) . Inv (R) is a subgroup of Inv (R).  If   , then the group Inv (R) is a subgroup of Inv (R) (and  -multiplication on the former group coincides with -multiplication).  (5) If   , then the group Inv (R) is a subgroup of Inv (R) (and  -multiplication on the former group coincides with -multiplication).

(1) (2) (3) (4)

Corollary 2.5.23. Let R be a ring, and let  and  be semistar operations on R with   . (1) (2) (3) (4) (5)

Cl(R) = Cld (R) is a subgroup of Cl (R).  Cl (R) is a subgroup of Cl (R). Cl (R) is a subgroup of Clv (R). If  is of r-finite type, then Cl (R) is a subgroup of Clt (R).  If   , then Cl (R) = Cl (R).

Corollary 2.5.24. Let R be a ring, and let  and  be semistar operations on R with   . (1) (2) (3) (4)

Cl(R) = Cld (R) is a subgroup of Cl (R).  Cl (R) is a subgroup of Cl (R). Cl (R) is a subgroup of Clv (R). If  is of finite type, then Cl (R) is a subgroup of Clt (R).

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Corollary 2.5.25. For any Prüfer ring R one has Clt (R) = Cl(R). Remark 2.5.26. Let  be a semistar operation on a ring R. (1) For any I, J ∈ K(R), write I ∼ J if I  = J  . The relation ∼ is an equivalence relation on K(R). Any ∼ -equivalence class contains a unique -closed Kaplansky fractional ideal, which is the largest element in that class. A -divisor of R is a ∼ -equivalence class div I of some regular fractional ideal I of R. A -divisor div I of R may then be identified with the -closed regular fractional ideal I  . One may give an alternative description of the -divisor class group Cl (R) as in Remark 2.3.35, by replacing the v-operation in the remark with . (2) The semistar -class group Cl (R) of R may be strictly larger than the -class group Cl (R) of R, but they are the same if every -invertible Kaplansky fractional ideal of R is regular, which holds if and only if I  = R implies that I is  regular for any ideal I of R. If   , then one need not have Cl (R) = Cl (R). Since we are mostly interested in semistar operations only up to star equivalence, we are more concerned here with the group Cl (R) rather than the larger group Cl (R). Definition 2.5.27. (1) An r-UFR, or unique r-factorization ring, is a Krull ring with trivial t-class group. (2) An r-GCD ring is a PVMR with trivial t-class group. It is clear that one has the following diagram of implications.

r-PIR

r-UFR

Dedekind

r-Bézout

Krull

r-GCD

Prüfer

PVMR In fact, by Proposition 2.5.19 and Corollaries 2.5.14 and 2.5.25, the diagram above is a full implication lattice. Note also that none of the implications are reversible. The full and complete implication lattice above can be analyzed further, as follows. By Proposition 2.5.19 and Corollary 2.5.25, the classes on the upper face of the cube (r-PIR, r-UFR, r-Bézout, r-GCD) are the corresponding classes on the lower face (Dedekind, Krull, Prüfer, PVMR) with trivial t-class group. By Corollary 2.5.14, the classes on the upper right face of the cube (r-PIR, Dedekind, r-Bézout, Prüfer) are the corresponding classes on the lower left face (r-UFR, Krull, r-GCD, PVMR) with t  d. Finally, by results in the next section, the classes on the upper left face

2.5 -Dedekind Rings and -Prüfer Rings

111

(r-PIR, Dedekind, r-UFR, Krull) are the corresponding classes on the lower right face (r-Bézout, Prüfer, r-GCD, PVMR) that are t-Noetherian, that is, which satisfy the ascending chain condition on the regular t-closed ideals. Thus it makes sense to think of the upward direction in the lattice as the “impose trivial t-class-group” axis, the upward-right direction as the “impose t  d” axis, and the upward-left direction as the “impose t-Noetherian” axis. We illustrate this as follows.

r-PIR

r-UFR

Dedekind

r-Bézout

Krull

r-GCD

Prüfer

PVMR t -Noetherian

td

Clt (R)=0

A GCD domain is a domain in which every finite set of elements has a gcd. It follows that the full lattice of implications above yields exactly the full lattice of implications below, once we have shown the following: (1) A UFD is equivalently a Krull domain with trivial t-class group. (2) A GCD domain is equivalently a PVMD with trivial t-class group. These two facts will be proved in Sections 2.9 and 2.8, respectively.

PID

UFD

Dedekind

Bézout

Krull

GCD

Prüfer

PVMD

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In Theorem 3.8.8 we show that a ring R is a PMVR if and only if R is integrally closed and t  w. From that result and the results of this section, we deduce the following characterization table. Ring R

Characterization 1

Characterization 2

r-PIR Dedekind r-Bézout Prüfer r-UFR Krull r-GCD PVMR

d-Dedekind, Cl(R) = 0 d-Dedekind d-Prüfer, Cl(R) = 0 d-Prüfer t-Dedekind, Clt (R) = 0 t-Dedekind t-Prüfer, Clt (R) = 0 t-Prüfer

t t t t t t t t

 d, r-UFR  d, Krull  d, r-GCD  d, PVMR  v, CIC, Clt (R) = 0  v, CIC  w, integrally closed, Clt (R) = 0  w, integrally closed

Remark 2.5.28. In algebraic number theory, one studies the ring O K of algebraic integers in an arbitrary number field K . The ring O K is known to be a Dedekind domain with finite class group. The class group of O K is in some sense a measure of the non-uniqueness of factorizations in O K . The order of the class group of O K is called the class number of K and is denoted h K . If e K is the exponent of the class group, then e K divides h K , and ae K and ah K are principal for every ideal a of O K . Most graduate texts in algebraic number theory provide techniques for computing the class group of a number field.√Using such methods, one can show, for √ example, that the number field Q( −65), class group of O K = Z[ −65], where K is the√ √ is generated by the equivalence classes of the ideals (2, 1 + −65) and (3, 1 + −65), of orders 2 and 4, respectively, and is isomorphic to Z/2Z × Z/4Z. √ One can deduce from this, for example, that a4 is principal for every ideal a of Z[ −65]. The text [69] provides techniques for computing the t-class group (or v-class group) of a Krull domain. For example, if k is any field of characteristic not equal to 2, and if F is any nondegenerate quadratic form in k[X 1 , X 2 , . . . , X n ], where n  3, then the ring k[X 1 , X 2 , . . . , X n ]/(F) is a Noetherian integrally closed domain, hence a Krull domain, and its t-class group is trivial or else is isomorphic either to Z/2Z or to Z; moreover, if n  5, then the t-class group is trivial [69, Propositions 11.2, 11.3, and 11.4]. The t-class group of k[X, Y, Z ]/(X Y − Z 2 ), for example, is cyclic of order two, and the t-class group of k[X, Y, Z , W ]/(X Y − Z W ) is infinite cyclic [69, Proposition 11.4]. (See Exercises 3.5.14 and 3.5.15.) Note that neither of these domains is a UFD because the image of X Y factors as a product of irreducible elements in two distinct ways. Remark 2.5.29. Let R be a ring. The following equivalences are known. (1) R[X ] is a PVMR if and only if R[X] is a PVMR for any nonempty set X of indeterminates, if and only if R is a reduced PVMR such that every non-regular prime of R is minimal, if and only if R is a PVMR and T (R) is von Neumann regular. (2) R[X ] is a Krull ring if and only if R[X] is a Krull ring for any nonempty set X of indeterminates, if and only if R is a reduced Krull ring with finitely many minimal primes, if and only if R is isomorphic to a finite direct product of Krull domains.

2.5 -Dedekind Rings and -Prüfer Rings

113

(3) R[X ] is a Prüfer ring if and only if R is von Neumann regular. (4) R[X ] is a Dedekind ring if and only if R is a von Neumann regular ring with finitely many primes, if and only if R is isomorphic to a finite direct product of fields. (5) R[X, Y ] is a Prüfer ring if and only if R is trivial. Remark 2.5.30. A Kaplansky fractional ideal I of a ring R is said to be dense if I has trivial annihilator (that is, (0 : R I ) = 0), and I is said to be semiregular if I contains a finitely generated dense J ∈ K(R). One has the following irreversible implications (equivalent for Noetherian rings): generated by regular elements ⇒ regular ⇒ semiregular ⇒ dense. Let  be a reduced semistar operation on a ring R. Note that if I ∈ K(R) is -invertible, then I and I −1 are dense. Indeed, if (I J ) = R and cI = 0, where c ∈ R, then c R = c(I J ) ⊆ (cI J ) = 0 and therefore c = 0. Similarly, if I ∈ K(R) is t -invertible, then I and I −1 are semiregular. By contrast, if I ∈ K(R) is -invertible or t -invertible, then I need not be regular or fractional. One might consider, then, the following natural analogues of the -Dedekind and -Prüfer conditions. (1) Every I ∈ K(R) such that I and I −1 are dense (resp., semiregular) is -invertible. (2) Every finitely generated I ∈ K(R) such that I (and I −1 ) is semiregular (resp., dense) is -invertible. In fact, one can improve on the conditions above by replacing K(R) with the ordered monoid of all R-submodules of Q(R) (resp., Q 0 (R)), where Q(R) (resp., Q 0 (R)) is the localization of R with respect to the Gabriel filter of all dense ideals (resp., all semiregular ideals) of R. (The ring Q(R) is called the complete ring of quotients of R, and the ring Q 0 (R) is called the ring of finite fractions of R.) Since the total quotient ring T (R) is the localization of R with respect to the Gabriel filter of all regular ideals of R, such a modification of the definitions would provide the most natural analogues of the -Dedekind and -Prüfer conditions. This suggestion is witnessed, for example, by the work of Lucas [134, 135, 137]. All of this indicates that it is most natural to study the regular ideals, rather than dense ideals or the semiregular ideals, with respect to semistar operations. The natural alternatives employing Q(R) are discussed in Section 5.7. Remark 2.5.31. If  is a semistar operation on a ring R, then, somewhat awkwardly, the class of -Dedekind rings, or, alternatively, of -Prüfer rings, is either ∅ or {R}. To make sense of these properties as classes, we say that a universal semistar operation  is an assignment  : R −→  R of a semistar operation  R on R to every ring R (or, alternatively, to every domain R [47]). The semistar operations d, t, w, v are universal semistar operations. If  is a universal semistar operation, then the properties of -Dedekind and -Prüfer are genuine properties of rings rather than of a fixed semistar operation  R on a fixed ring R. Thus, for example, it makes sense to say “every -Dedekind ring is a -Prüfer ring” rather than merely “if R is -Dedekind, then R is -Prüfer.” More importantly, it then also becomes quite natural to compare

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the semistar operations  R and  S for any homomorphism R −→ S of rings. This is important even for the operations v and t, because, for example, they do not in general commute with localization and in general Clv (R) and Clt (R) are not functorial in R. For a study of the functoriality of -class groups and universal semistar operations, see [47, 52]. The reader is invited to translate the language of semistar operations in this book to that of universal semistar operations. There is no harm (and much benefit) in doing so, because every semistar operation  on a ring R can be thought of as the unique universal semistar operation  such that  R =  and  S = d S for all S = R.

2.6 TV, TW, -Noetherian, and Related Conditions For an arbitrary commutative ring, one expects the semistar operations d  w  t  v to be star inequivalent, and indeed there exist integrally closed domains on which d  w  t v, such as Example 1.2.17(3). The following definition provides names for the 6 = 24 distinguished classes of rings defined by setting any two of these four semistar operations star equivalent to one another. Definition 2.6.1. Let R be a ring. (1) (2) (3) (4) (5) (6)

R is divisorial if v  d. R is w-divisorial if v  w. R is TD if t  d. R is TV if t  v. R is TW if t  w. R is DW if d  w.

In the case of domains, each of these classes has received attention in the literature [45, 98, 104, 147, 148, 186]. TD domains are known as fgv domains since they are characterized as domains in which every finitely generated ideal is v-closed. The above classes of rings are related by the following full and complete implication lattice. divisorial (v  d)

w-divisorial (v  w)

TV DW

TD (t  d)

TV (v  t)

TW (t  w)

DW (w  d)

TV or TW or DW

2.6 TV, TW, -Noetherian, and Related Conditions

115

Two basic classes of examples are provided by Corollary 2.5.14, which may be restated as follows. Proposition 2.6.2. One has the following. (1) A Dedekind ring is equivalently a divisorial completely integrally closed ring. (2) A Prüfer ring is equivalently a TD PVMR. Note, by contrast, that a divisorial integrally closed domain need not be Dedekind, and a Prüfer domain need not be divisorial or completely integrally closed. Another basic class of examples is provided by the Noetherian rings: clearly, any Noetherian ring is a TV ring. More generally, one has the following. Proposition 2.6.3. Every r-Noetherian ring is TV. Proof. If R is an r-Noetherian ring, then every I ∈ Freg (R) is finitely generated and  therefore satisfies I t = I v , and thus t  v. It follows that the TV rings can be thought of a generalization of the r-Noetherian rings. Moreover, by Example 2.6.4(8), a Noetherian domain need not satisfy any of the six conditions in Definition 2.6.1 besides the TV condition. Nevertheless, all six of these conditions arise in very natural settings, including the following. Example 2.6.4. (1) By Corollary 2.5.14(2), a Krull ring is equivalently a completely integrally closed TV ring. However, an integrally closed TV domain (and even a TV PVMD) need not be Krull. (2) In Theorem 3.8.8 we show that a PVMR is equivalently an integrally closed TW ring. It follows from this that a Prüfer ring is equivalently an integrally closed TD ring. (3) By [19, Theorem 6.3] and [98, Corollary 4.3], a Noetherian divisorial domain is equivalently a Gorenstein domain of dimension at most one. Moreover, by [19, Theorems 6.2 and 6.3], a one dimensional Noetherian ring is Gorenstein if and only if it is divisorial and its total quotient ring is Gorenstein. (4) An order of a number field K is a subring O of K that is finitely generated as an abelian group and satisfies K = QO. Every order of a number field K is contained in the ring of integers O K of K , which is the unique maximal order of K . Any order of a number field is Noetherian and one dimensional. An order O is divisorial if and only if it is Gorenstein, if and only if the O-module HomZ (O, Z) is invertible [113, Proposition 3.5]. Any number field of degree greater than two has infinitely many orders that are divisorial and infinitely many orders that are not divisorial [113]. Moreover, if ξ1 , ξ2 , . . . , ξr are algebraic integers and the number fields Q(ξ1 ), Q(ξ2 ), . . . , Q(ξr ) are linearly . , ξr ] √ is divisorial [113, Proposition disjoint over Q, then the √ order√Z[ξ1 , ξ2 , . . √ 3.6]. The orders Z[2i, 2 2, 2i 2] and Z[2 3 2, 2 3 4], on the other hand, are not divisorial [113, Examples 7.3 and 7.7].

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(5) A domain is DW if and only if all of its maximal ideals are t-closed [148, Proposition 2.2]. Therefore, by Lemma 3.4.2 of Chapter 3, since every height one prime is t-closed, every one dimensional domain is DW. (6) If D is a Noetherian DW domain of dimension greater than one, e.g., [148, Example 2.10], then D is TV and DW but not divisorial, hence not TW. (7) If D is a domain that is not TW, then D[X ] is a domain that is neither TW nor DW (that is, t  w  d). (8) Let L be an extension of a field K , and let D = K + X L[X ]. By [148, Corollaries 3.10 and 3.13], D is Noetherian if and only if [L : K ] is finite, but D is TW if and only if [L : K ]  2. Thus, D is a (one dimensional) Noetherian DW domain that is not TW if and only if 2 < [L : K ] < ∞, and in that case D[Y ] is a Noetherian domain on which t  w  d. Next we prove the following. Proposition 2.6.5. Every PVMR is TW, and every Krull ring is w-divisorial. Proof. Let R be a PVMR, and let I, J ∈ Freg (R) with J ⊆ I and J finitely generated. Then J is t-invertible, so we have (J J −1 )t = R and therefore H v = R for some finitely generated regular subideal H of J J −1 . Now J v J −1 ⊆ R v = R = (J : K J ), so J v H ⊆ J v J −1 J ⊆ (J : K J )J ⊆ J ⊆ I . Therefore J v ⊆ (I : K H ). But H is finitely generated and H v = R, so (I : K H ) ⊆ I w . Therefore J v ⊆ I w for all finitely generated J ∈ Freg (R) with J ⊆ I , so I t ⊆ I w and therefore I t = I w . Therefore R is a TW ring. A similar argument shows that, if R is a Krull ring, then for all I ∈ Freg (R) one has I v = I w , and therefore R is w-divisorial. (Apply the preceding argument to I rather than J .)  By Propositions 2.6.2 and 2.6.5, one has the following implication diagram.

Dedekind

Krull

divisorial

Prüfer

w-divisorial

PVMR

TD

TV

TW

DW

However, the implication diagram above is not a full implication lattice. For example, it turns out that a divisorial Prüfer ring need not be Dedekind and a w-divisorial PVMR need not be Krull. Below is the smallest full and complete implication lattice that includes the r-PIR, Dedekind, r-Bézout, Prüfer, r-UFR, Krull, r-GCD, and PVMR conditions along with the divisorial, w-divisorial, TD, TV, DW, and TW conditions.

2.6 TV, TW, -Noetherian, and Related Conditions

117

r-PIR

r-UFR

Dedekind

div. r-Bézout

Krull

TV r-GCD

div. Prüfer

r-Bézout

TV PVMR

r-GCD

Prüfer

PVMR

divisorial

w-divisorial

TV DW

TD

TV

TW

DW

ring

Among the classes of rings in the diagram above, only the r-PIRs and Dedekind rings need be r-Noetherian. The r-UFRs and Krull rings, however, must be v-Noetherian, or Mori, which is an important condition defined as follows. Definition 2.6.6. Let R be a ring and I a set of ideals of R. The ring R satisfies the ascending chain condition on I if there are no infinite ascending chains I1  I2  I3  · · · of ideals Ii ∈ I. Definition 2.6.7. Let  be a semistar operation on a ring R. The ring R is -Noetherian if R satisfies the ascending chain condition on the regular -closed ideals of R. A Mori ring is a v-Noetherian ring. Thus, an r-Noetherian ring is equivalently a d-Noetherian ring, where d is the trivial semistar operation. A Mori ring is equivalently a ring R that satisfies the ascending chain condition on the regular divisorial ideals of R. Clearly, every r-Noetherian ring is Mori. More generally, if   , then every  -Noetherian ring is -Noetherian. In particular, one has the implications Noetherian ⇒ r-Noetherian ⇔ d-Noetherian ⇒ -Noetherian ⇒ Mori.

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Remark 2.6.8. A ring R is semistar -Noetherian if R satisfies the ascending chain condition on the -closed ideals of R. (For example, a ring is semistar d-Noetherian if and only if it is Noetherian.) This is a stronger condition than the -Noetherian condition, but unlike the latter it is not invariant under star equivalence. The notion of a finitely generated regular ideal generalizes as follows. Definition 2.6.9. Let  be a semistar operation on a ring R and I a Kaplansky fractional ideal of R. (1) I is -finite if I  = J  for some finitely generated Kaplansky fractional ideal J of R. (2) I is strictly -finite I  = J  for some finitely generated Kaplansky fractional ideal J of R contained in I . Note specifically that I ∈ K(R) is d-finite if and only if I is strictly d-finite, if and only if I is finitely generated. Any -finite Kaplansky fractional ideal is necessarily fractional but not necessarily regular. If I is a regular Kaplansky fractional ideal, then I is strictly -finite if and only if I  = J  for some finitely generated regular fractional ideal J of R contained in I . Lemma 2.6.10. Let  be a semistar operation on a ring R and let I ∈ K(R). The following conditions are equivalent. (1) I is t -finite. (2) I is strictly -finite. (3) I is strictly t -finite. Moreover, if the conditions above hold, then I  = I t . Proof. Suppose that (2) holds, so that I  = J  for some finitely generated fractional subideal J of I . Since J is finitely generated, one has J  = J t , so that I  = J t ⊆ I t and therefore I  = I t = J t . Therefore (3) holds. Conversely, if (3) holds, then I t = J t for some finitely generated subideal J of I , so that, applying , we see that I  = J  , and therefore (2) holds. Therefore (2) and (3) are equivalent. Since (3) implies (1), it remains only to show that (1) implies (3). Suppose that (1) holds, so that I t = J t for some finitely generated fractional ideal J = (a1 , a2 , . . . , an ) of R. Since ai ∈ I t , one has ai ∈ Ii for some finitely generated fractional ideal Ii ⊆ I of R, for each i. Therefore H = I1 + I2 + · · · + In is finitely generated and contained in I , and one has J ⊆ H t ⊆ I t and therefore J t = H t = I t . Therefore (3) holds.  Proposition 2.6.11. Let  be a semistar operation on a ring R. The following conditions are equivalent. (1) (2) (3) (4) (5)

R is -Noetherian. R is t -Noetherian. Every I ∈ Freg (R) is strictly -finite. Every I ∈ Freg (R) is strictly t -finite. Every I ∈ Freg (R) is t -finite.

2.6 TV, TW, -Noetherian, and Related Conditions

119

Moreover, if any of the conditions above holds, then   t is of r-finite type. Proof. Suppose (3) does not hold, so there exists an I ∈ Freg (R) such that I   J  for all finitely generated fractional subideals J of I . We may suppose without loss of generality that I ⊆ R. Let a1 ∈ I be regular. Then there exists a2 ∈ I − (a1 ) , so that (a1 )  (a1 , a2 )  I  . Then there exists a3 ∈ I − (a1 , a2 ) , so that (a1 , a2 )  (a1 , a2 , a3 )  I  . Continuing in this manner, we see that (1) does not hold. Therefore (1) ⇒ (3) and (2) ⇒ (4). By Lemma 2.6.10 we have (3) ⇔ (4) ⇔ (5), and since t   we also have (2) ⇒ (1). Suppose that (4) holds,  and let I1 ⊆ I2 ⊆ · · · be a chain oftregular t t -closed ideals of R. Then I = ∞ n=1 In is a regular ideal of R. Therefore I = J for some finitely generated regular subideal J of I . Moreover, since J is finitely generated, one has J ⊆ I N for some N , so I t = J t ⊆ In ⊆ I ⊆ I t , whence In = I t , for all n  N . Therefore (4) ⇒ (2). Finally, the last statement of the proposition follows from Lemma 2.6.10.  Corollary 2.6.12. One has the implications -Noetherian ⇔ t -Noetherian ⇒ t-Noetherian ⇔ v-Noetherian ⇔ Mori. Corollary 2.6.13. Every Mori ring is a TV ring. Proof. If R is Mori, then every I ∈ Freg (R) is strictly v-finite and therefore satisfies  I v = I t by Lemma 2.6.10. The following lemma generalizes the well-known fact, proved in Section 2.2, that every invertible ideal is finitely generated. Lemma 2.6.14. Let  be a semistar operation on a ring R, and let I ∈ K(R). Then I is t -invertible if and only if I is -invertible and both I and I −1 are strictly -finite. Proof. Suppose that I is t -invertible, so that (I I −1 )t = R. Then I is -invertible, and 1 ∈ (I I −1 )t , so 1 ∈ J  for some finitely generated fractional subideal J of I I −1 . Since J is finitely generated, one has J ⊆ G H for finitely generated fractional subideals G and H of I and I −1 , respectively. Then one has R = J  ⊆ (G H ) ⊆ (I I −1 )t = R, so (G H ) = R. Thus G  ⊇ (G H I ) = ((G H ) I ) = I  , so I  = G  . Therefore I is strictly -finite. Since I −1 is also t -invertible, I −1 is strictly -finite as well. Conversely, suppose that I is -invertible and both I and I −1 are strictly -finite. Then (I I −1 ) = R and I  = J  = I t = J t and I −1 = H  = H t for some finitely generated fractional subideals J of I and H of I −1 . Then (I I −1 )t = (I t I −1 )t = (J t H t )t = (J H )t = (J H ) = (J  H  ) = (I  I −1 ) = (I I −1 ) = R. Therefore  I is t -invertible. Definition 2.6.15. Let  be a semistar operation on a ring R. The ring R is -coherent if I −1 is strictly -finite for every finitely generated regular (fractional) ideal I of R.

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Example 2.6.16. (1) Trivially, any -Noetherian ring is -coherent. (2) A d-coherent domain is said to be quasi-coherent. A domain D is quasi-coherent if and only if I −1 is finitely generated for all finitely generated I ∈ Freg (D), if and only if every finite intersection of principal ideals is finitely generated. See [84] for a study of quasi-coherent domains and related rings. By Lemma 2.6.10, a ring is -coherent if and only if it is t -coherent. Moreover, if   , then I ∈ Freg (R) strictly  -finite implies that I is strictly -finite, and therefore every  -coherent ring is -coherent. In particular, one has the implications d-coherent ⇒ -coherent ⇔ t -coherent ⇒ t-coherent ⇔ v-coherent. Lemma 2.6.14 immediately implies the following. Proposition 2.6.17. Let  be a semistar operation. A t -Prüfer ring is equivalently a -coherent -Prüfer ring. In particular, a PVMR is equivalently a v-coherent vPrüfer ring, and a PVMD is equivalently a v-coherent v-domain. Remark 2.6.18. Let  be a semistar operation on a ring R. Let us say that R is quasi--Noetherian if every I ∈ Freg (R) is -finite. By Proposition 2.6.11, R is Noetherian if and only if R is quasi--Noetherian and  is of r-finite type. Thus, for example, R is Mori if and only if R is quasi-v-Noetherian and TV. We do not know an example of a quasi-v-Noetherian domain that is not v-Noetherian. Next, we apply the various finiteness conditions defined above to the study of the t -Dedekind rings. Proposition 2.6.19. Let  be a semistar operation on a ring R. The following conditions are equivalent. (1) (2) (3) (4) (5) (6) (7)

R is t -Dedekind. R is -Dedekind and   t . R is t -Prüfer and t -Noetherian. R is -Prüfer and -Noetherian. R is -Dedekind and -Noetherian. R is Krull and t  t. R is Krull and t  v.

Proof. Statements (1) and (2) are equivalent by Corollary 2.5.13, statements (1) and (6) are equivalent by Proposition 2.5.12(2), and statements (6) and (7) are equivalent because every Krull ring is TV. One clearly has (1) ⇒ (3) ⇒ (4). Moreover, by Proposition 2.6.11 and Lemma 2.6.14 one has (4) ⇒ (5) ⇒ (1).  As immediate corollaries of Proposition 2.6.19, we have the following. Corollary 2.6.20. Let R be a ring. The following conditions are equivalent.

2.6 TV, TW, -Noetherian, and Related Conditions

(1) (2) (3) (4)

121

R is a Krull ring. R is a Mori PVMR. R is a Mori v-Prüfer ring. R is a completely integrally closed Mori ring.

Corollary 2.6.21. Let R be a ring. The following conditions are equivalent. (1) (2) (3) (4) (5) (6) (7)

R is a Dedekind ring. R is an r-Noetherian Prüfer ring. R is a Mori Prüfer ring. R is Krull and Prüfer. R is Krull and TD. R is Krull and divisorial. R is completely integrally closed and divisorial.

Remark 2.6.22. (1) The reader familiar with Krull domains likely knows that a Krull domain is equivalently a completely integrally closed Mori domain and will now agree that our definition of a Krull ring as a t-Dedekind ring is consistent with the more standard definitions of a Krull domain. Corollary 2.6.20 also implies that our definition of Krull rings for general rings agrees with the standard definitions, as in [10, 118, 135]. (2) Let us say that a -Dedekind domain in the sense of [44] is a -Noetherian PMD. By Proposition 2.6.19, a -Dedekind domain in the sense of [44] is equivalently a t -Dedekind domain in the sense that we have defined. Our notion of a -Dedekind domain (or ring) is more general, and we feel that the slight drift in terminology is justified. In [44], for example, the v-Dedekind domains, t-Dedekind domains, and Krull domains are all the same, while for us a t-Dedekind domain is equivalently a Krull domain and a v-Dedekind domain is equivalently a completely integrally closed domain. Likewise, a PMD is equivalently a Pt MD, while a -Prüfer ring need not be t -Prüfer. Our slight departure from the previous terminology allows for greater flexibility. It is shown in [58, Theorem 2.8] that a Krull domain is equivalently an integrally closed w-Noetherian domain. A proof is outlined in Exercise 3.4.12. It is shown in Theorem 3.8.8 that an PVMR is equivalently an integrally closed TW ring. It follows that any Krull ring is an integrally closed w-Noetherian ring. In fact, the converse also holds: a Krull ring is equivalently an integrally closed w-Noetherian ring [53]. In general w-Noetherian implies t-Noetherian (i.e., Mori), but not conversely. In fact, there exist integrally closed Mori domains that are not Krull, hence not w-Noetherian [68, p. 161]. The w-Noetherian domains, first introduced and studied by Fanggui and McCasland in [57, 58], are called strong Mori domains. In those two papers, they show that many results, including analogues of the principal ideal theorem, the Hilbert basis theorem, the Krull intersection theorem, and the existence of primary decompositions, that one would want to hold for Mori domains, but don’t, actually do hold for the strong Mori domains. The w -Noetherian domains are studied, for example, in [110].

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Open Problem 2.6.23. Generalize results in [57, 58, 110, 147] on strong Mori domains and w -Noetherian domains to w-Noetherian rings and w -Noetherian rings. Next we define an important class of rings that generalize the TV rings. Definition 2.6.24. A ring R is an H ring if R if every regular v-invertible (fractional) ideal of R is t-invertible. Clearly, every TV ring is an H ring, so we have the (generally irreversible) implications Noetherian ⇒ r-Noetherian ⇒ Mori ⇒ TV ⇒ H. Proposition 2.6.25. Let R be a ring. The following conditions are equivalent. (1) (2) (3) (4)

R is Krull. R is a completely integrally closed Mori ring. R is a completely integrally closed TV ring. R is a completely integrally closed H ring.

Proof. We proved in Corollary 2.6.20 that a Krull ring is equivalently a completely integrally closed Mori ring. Therefore we have (1) ⇔ (2) ⇒ (3) ⇒ (4). Suppose that (4) holds. Since a completely integrally closed ring is equivalently a v-Dedekind ring, every I ∈ Freg (R) is v-invertible, hence t-invertible, since R is an H ring. Therefore R is t-Dedekind, that is, R is Krull. Thus (4) implies (1).  It is well-known that an integrally closed Noetherian domain is a Krull domain. This generalizes as follows. Corollary 2.6.26. Let R be an r-Noetherian ring. Then the integral closure of R is equal to the complete integral closure of R. Moreover, if R is r-Noetherian and integrally closed, then R is Krull. Proof. Let a ∈ T (R) be almost integral over R. Then R[a] is a regular fractional ideal of R, whence R[a] is finitely generated, since R is r-Noetherian. Therefore a is integral over R. It follows that the complete integral closure of R is equal to the integral closure of R. Suppose that R is integrally closed. Then R is completely integrally closed and r-Noetherian and therefore Mori, whence by Proposition 2.6.25 R is a Krull ring.  Although we do not prove the result here, the famous Mori–Nagata theorem (1953, 1955) states that the integral closure of any Noetherian domain is a Krull domain [149, 154]. The theorem is often stated in a slightly more general form. Theorem 2.6.27 (Mori–Nagata theorem). The integral closure of any reduced Noetherian ring R is isomorphic to a direct product of k Krull domains, where k is the number of minimal primes of R. In 1976, Huckaba proved that the integral closure of any Noetherian ring is a Krull ring [106, Corollary 2.3]. In 2000, Chang and Kang generalized this as follows.

2.6 TV, TW, -Noetherian, and Related Conditions

123

Theorem 2.6.28 ([32, Theorem 13]). The integral closure of an r-Noetherian ring is a Krull ring. Also, analogous to Corollary 2.3.37, one has the following theorem, proved by combining [23, Proposition VII.1.12] with the Mori–Nagata theorem. Theorem 2.6.29. Let D be a Noetherian or Krull domain with quotient field K , let L be a finite extension of K , and let D  be the integral closure of D in L. Then D  is a Krull domain. Naturally, one might conjecture the following. Conjecture 2.6.30. Let R be an (r-Noetherian or) Krull ring with total quotient ring K , let L be any total quotient ring that is a finite extension of K , and let S be the integral closure of R in L. Then S is a Krull ring. Conjecture 2.6.31. Let R be a PVMR with total quotient ring K , let L be any total quotient ring that is an integral extension of K , and let S be the integral closure of R in L. Then S is a PVMR.

2.7 -Marot Rings Before studying the -Dedekind and -Prüfer rings any further, we need to take a slight detour into the study of the Marot rings and various generalizations thereof. Definition 2.7.1. Let R be a ring with total quotient ring K . A Kaplansky fractional ideal of R is regularly generated if it is generated by regular elements of K . Note that a Kaplansky fractional ideal of a ring is regularly generated if and only if it is generated by its regular elements. Also, a Kaplansky fractional ideal is generated by finitely many regular elements if and only if it is both finitely generated and regularly generated. Definition 2.7.2. A ring R is Marot if every regular ideal of R is regularly generated. A Marot ring, loosely speaking, is a ring with “many” non-zerodivisors. Definition 2.7.3. Let R be a ring. For any I ∈ K(R), let I rg = R I reg denote the element of K(R) generated by the set I reg of all regular elements of I . Equivalently, I rg is the largest regularly generated Kaplansky fractional ideal of R contained in I . Clearly, I ∈ K(R) is regularly generated if and only if I = I rg . In particular, a ring R is Marot if and only if I = I rg for every regular ideal I of R. The following proposition is also clear. Proposition 2.7.4. Let R be a ring with total quotient ring K . The following conditions are equivalent.

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(1) R is Marot. (2) Every regular Kaplansky fractional ideal of R is regularly generated. (3) Every finitely generated regular ideal of R is generated by finitely many regular elements of R. (4) Every finitely generated regular fractional ideal of R is generated by finitely many regular elements of K . (5) (a, b) is regularly generated for all a, b ∈ R with a regular. Corollary 2.7.5. Every overring of a Marot ring is Marot. Corollary 2.7.6. Every r-Bézout ring (hence every total quotient ring) is Marot. Trivially, any integral domain is Marot, as is, not so trivially, any Noetherian ring. The Marot rings are therefore a common generalization of integral domains and Noetherian rings, sharing with them both an important property with respect to regular ideals. A ring R has few zerodivisors if the set of all zerodivisors of R is a finite union of prime ideals, or equivalently if the total quotient ring of R has only finitely many maximal ideals. For example, any Noetherian ring has few zerodivisors, as does any reduced ring with only finitely many minimal primes, since in any reduced ring the set of all zerodivisors is equal to the union of the minimal primes. A ring R is said to be additively regular if for every x ∈ T (R) there exists a y ∈ R such that x + y is a unit in T (R). Any total quotient ring, for example, is additively regular. One has the following. Theorem 2.7.7 ([107, Theorems 7.2 and 7.4]). Let R be a ring. Each of the following conditions implies the next. (1) (2) (3) (4) (5)

R is Noetherian. R has few zerodivisors. The total quotient ring of R is zero dimensional. R is additively regular. R is Marot.

Proof. One has (1) ⇒ (2) by Exercise 2.1.29 and (2) ⇒ (3) by Exercise 2.1.31. One also has (3) ⇒ (4) by Exercise 2.7.17 (or [16, Theorem 10], or [107, Theorem 7.4]), and one has (4) ⇒ (5) by Exercise 2.7.19.  The theorem demonstrates that the class of Marot rings is substantial. However, as we show in Example 2.7.13 below, there exist Dedekind rings that are not Marot. The Marot property can be generalized in many ways, including the following. Definition 2.7.8. A ring R is quasi-Marot if every regular ideal of R is a sum of invertible ideals of R. Analogous to Proposition 2.7.4, we have the following. Proposition 2.7.9. Let R be a ring. The following conditions are equivalent.

2.7 -Marot Rings

125

(1) R is quasi-Marot. (2) Every regular Kaplansky fractional ideal of R is a sum of invertible fractional ideals of R. (3) Every finitely generated regular ideal of R is a finite sum of invertible ideals of R. (4) Every finitely generated regular fractional ideal of R is a finite sum of invertible fractional ideals of R. (5) (a, b) is a sum of invertible ideals for all a, b ∈ R with a regular. Corollary 2.7.10. Every overring of a quasi-Marot ring is quasi-Marot. It is easy to see that every Prüfer ring is quasi-Marot (even if not Marot). Indeed, every finitely generated regular ideal of a Prüfer ring is invertible, hence trivially is a finite sum of invertible ideals. In particular, the quasi-Marot rings are a common generalization of the Prüfer rings and the Marot rings. Since any invertible ideal of a local ring is regular and principal, it is clear that a local ring is quasi-Marot if and only if it is Marot. Thus a local ring that is not Marot is not quasi-Marot either. D. D. Anderson and J. Pascual introduced the notion of a quasi-Marot ring in 1985, although the authors did not provide a name for them [12]. They gave an example of a Dedekind ring that is not Marot [12, Example 3.6] and an example of a local ring that is not quasi-Marot [12, Example 3.4], both of which are described below. To construct these examples, they used the “idealization” construction, which was introduced by Nagata in [155] and is extremely useful for reducing questions about modules to questions about ideals and for producing examples of rings with zerodivisors having various pathological properties. Idealization is defined as follows. Definition 2.7.11. Let R be a ring and M an R-module. The R-module R ⊕ M is a ring R(+)M, called the idealization of M over R, or the trivial extension of R by M, with respect to the multiplication (r, m)(s, n) = (r s, r n + sm) and multiplicative identity (1, 0). The map R −→ R(+)M given by r −→ (r, 0) is a ring embedding of R in R(+)M, so in particular R(+)M is an R-algebra, where r (s, n) = (r, 0)(s, n) = (r s, r n) for all r ∈ R and all (s, n) ∈ R(+)M. If N is a submodule of M, then 0(+)N = 0 ⊕ N is an ideal of R(+)M with (0(+)N )2 = 0 and (R(+)M)/(0(+)N ) ∼ = R(+)(M/N ). Moreover, every ideal of R(+)M containing 0(+)M is of the form I (+)M for some ideal I of R, and one has (R(+)M)/(I (+)M) ∼ = R/I . Moreover, since (0(+)M)2 = 0, every prime ideal of R(+)M contains 0(+)M, and therefore the prime ideals of R(+)M are precisely the ideals P(+)M, where P is an arbitrary prime ideal of R. The reference [13] provides an excellent survey of the idealization construction; also see Exercise 2.7.14. Remark 2.7.12. The idealization of M over isomorphic to the quotient

∞R is naturally n (M) = Sym (M) of M over R by the ideal of the symmetric algebra Sym R R n=0

∞ n Sym (M). R n=2

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The following provides an example of a Dedekind ring that is not Marot. Example 2.7.13 ([12, Example 3.6]). Let D be a Dedekind domain with a maximal ideal p that is not principal but has a principal power. Let m > 1 be the least positive integer with pm = π D principal. Let M denote the D-module {D/q : q ∈ Max(D), q = p}. Let R denote the idealization D(+)M, and let P = p(+)M. Then P is a maximal ideal of R, and P m = π R, so P is invertible. Moreover, as shown in [12], the set {R, P, P 2 , . . .} is the set of all regular ideals of R, and therefore Freg (R) = {. . . , P −2 , P −1 , R, P 1 , P 2 , . . .} is the set of all regular fractional ideals of R, which, as an ordered monoid, is anti-isomorphic to the ordered group Z. Moreover, {R, π R, π 2 R, . . .} = {R, P m , P 2m , . . .} is the set of all regular principal ideals of R. In particular, R[π −1 ] is the total quotient ring of R, and every regular ideal of R is invertible, so R is a Dedekind ring. However, since π R  P, the ideal P is not generated by regular elements of R, and therefore R is not Marot. Note also that the class group Cl(R) = Inv(R)/ Prin(R) of R is cyclic of order m since the group Inv(R) = Freg (R) is infinite cyclic and Prin(R) = {π k R : k ∈ Z} has index m in Inv(R). The following provides an example of a local ring that is not quasi-Marot. Example 2.7.14 ([12, Example 3.4]). Let k be a field and D = k[[X, Y ]], and let

M denote the D-module {D/p : p ∈ X 1 (D), p = (X )}, where X 1 (S) for any ring S denotes the set of all height one primes of S. Let R be the idealization D(+)M. Then the regular elements of R have the form (u X n , m), where n is a nonnegative integer, u is a unit of D, and m ∈ M. Moreover, R is local with regular maximal ideal (X, Y )(+)M that is not generated by regular elements. In particular, R is not Marot. Neither, then, is it R quasi-Marot, since any local quasi-Marot ring is Marot. Proofs of the unsubstantiated claims in the two examples above are contained in [12] and are also outlined in Exercises 2.7.14 and 2.7.15. The following result follows from Proposition 6.6.2 of Chapter 6. Proposition 2.7.15. Let R be a quasi-Marot ring. A semistar operation  on R is r-stable if and only if (I ∩ J ) = I  ∩ J  for all I, J ∈ Freg (R). It is well-known  that, if R is an integral domain, then I v for any I ∈ K(R) is equal to the intersection {x R : x ∈ K , x R ⊇ I } of all principal fractional ideals of R containing I . This equality, however, does not necessarily hold if R is a ring with zerodivisors. Famously, the papers [118] and [10] falsely assumed otherwise, but the error was corrected in [11]. Nevertheless, it was shown in [11] and is proved below that the equality does hold for all I ∈ Freg (R) if R is Marot, or, more generally, if and only if R is “v-Marot.” Lemma  2.7.16. Let R be a ring with total quotient ring K . For all I ∈ K(R), let I p = {x R : x ∈ K reg , x R ⊇ I }. Let I ∈ K(R). (1) I p is the smallest Kaplansky fractional ideal of R containing I that is an intersection of regular principal fractional ideals of R.

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 (2) If I is regular, then I p = {x R : x ∈ K , x R ⊇ I }. (3) The map p : I −→ I p is a closure operation on K(R) with v  p. It is the largest closure operation  on R such that R  = R and a I  = (a I ) for all I ∈ K(R) and all a ∈ K reg . (4) p is a semistar operation on R if and only if p = v. (5) p restricts to a star operation on R if and only if p = v on Freg (R). (6) I p = ((I −1 )rg )−1 . (7) If I is regularly generated, then (I −1 ) p = I −1 . Proof. Statements (1)–(5) are easy to check. To prove (6), observe  that for all x ∈ K reg one has x R ⊇ I if and only if x −1 ∈ I −1 and therefore {x R : x ∈  −1 −1 R = ((I −1 )rg )−1 . Finally, (7) follows from the K reg , x R ⊇ I } = x −1 ∈I −1 x     −1 −1 = x∈I reg x −1 R is an intersection of regular princifact that I = x∈I reg x R pal fractional ideals if I is regularly generated.  Corollary 2.7.17. Let R be a ring with total quotient ring K and let I ∈ K(R).  Then I v ⊆ {x R : x ∈ K reg , x R ⊇ I }, and the containment is an equality if and only if (I −1 )−1 = ((I −1 )rg )−1 , if and only if I v is an intersection of principal regular fractional ideals of R. Definition 2.7.18. A ring R is v-Marot if for every I ∈ Freg (R) there exists a regularly generated J ∈ Freg (R) such that I v = J v . The v-Marot rings were first introduced in [11], as rings satisfying “Property (D).” Every Marot ring, for example, is v-Marot. From the lemma one may easily deduce the following. Proposition 2.7.19. Let R be a ring. The following conditions are equivalent. (1) R is v-Marot. (2) For every I ∈ Freg (R) there exists a regularly generated J ∈ Freg (R) such that I −1 = J v . −1 rg −1 reg (3) (I −1 )−1  = ((I ) ) for all I ∈ F (R). reg v (4) I = {x R : x ∈ K , x R ⊇ I } for all I ∈ F (R). (5) I ∈ Freg (R) is v-closed if and only if I is an intersection of principal fractional ideals of R.  (6) The closure operation I −→ ((I −1 )rg )−1 = {x R : x ∈ K , x R ⊇ I } on Freg (R) is a star operation on R. Example 2.7.20. The maximal ideal P of the ring R of Example 2.7.13 is invertible, hence v-closed, yet P is not an intersection of (regular) principal fractional ideals of R, since all principal  fractional ideals of R containing P contain R as well. Thus P v = P  R = {x R : x ∈ K , x R ⊇ P}. Therefore R is not v-Marot. Alternatively, since R is not Marot and v  d, the ring R is not v-Marot either. The Marot rings and the v-Marot rings generalize as follows.

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Definition 2.7.21. Let  be a semistar operation on a ring R. We say that R is -Marot if for every regular ideal I of R there exists a regularly generated ideal J of R such that I  = J  . Note that a Marot ring is equivalently a d-Marot ring; if   , then every  -Marot ring is -Marot; every Marot ring is -Marot; and every -Marot ring is v-Marot. Thus one has the implications Marot ⇔ d-Marot ⇒ t -Marot ⇒ -Marot ⇒ v-Marot. The following two propositions suggest that a v-Marot ring need not be t-Marot. Proposition 2.7.22. Let  be a semistar operation on a ring R. The following conditions are equivalent. (1) R is -Marot. (2) For every regular I ∈ K(R) there exists a regularly generated J ∈ K(R) such that I  = J  . (3) For every finitely generated regular ideal I of R there exists a regularly generated ideal J of R such that I  = J  . (4) For all a, b ∈ R with a regular one has (a, b) = J  for some regularly generated ideal J of R. Proposition 2.7.23. Let  be a semistar operation on a ring R. The following conditions are equivalent. (1) R is t -Marot. (2) For every regular I ∈ K(R) there exists a regularly generated J ∈ K(R) such that I t = J t . (3) For every finitely generated regular ideal I of R there exists an ideal J of R generated by finitely many regular elements of R such that I  = J  . (4) For all a, b ∈ R with a regular one has (a, b) = J  for some ideal J of R generated by finitely many regular elements of R. Open Problem 2.7.24. Must a v-Marot ring be t-Marot? The -Marot rings are employed in the next two sections.

2.8 -PIRs and -Bézout Rings If    are r-finite type semistar operations on a ring R, then our results in Section 2.5 yield the full lattice of implications below. Trivially, the upward direction in the lattice is the “impose Cl (R) = 0” axis, by Corollary 2.5.13 the upward-right direction is the “impose    ” axis, and by Proposition 2.6.19 the upward-left direction is the “impose -Noetherian” axis.

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 -Dedekind, Cl = 0



-Dedekind, Cl = 0

 -Dedekind

 -Prüfer, Cl = 0

-Dedekind

-Prüfer, Cl = 0

 -Prüfer

-Prüfer In this section and the next, we carry out a study of the top face of the cube, that is, the -Dedekind and -Prüfer rings rings with trivial -class group, where  is an r-finite type semistar operation. (As a corollary, this will yield several equivalent characterizations of the r-UFRs and r-GCD rings.) For this purpose, we introduce the following definitions. Definition 2.8.1. Let  be a semistar operation on a ring R. (1) R is a -PIR, or -principal ideal ring, if every -closed regular (fractional) ideal of R is principal. (2) R is a -PID, or -principal ideal domain, if R is a -PIR and an integral domain. (3) R is -Bézout if the -closure of every finitely generated regular ideal of R (or equivalently if every -finite -closed ideal of R) is principal. Example 2.8.2. Let R be a ring. (1) R is a d-PIR if and only if it is an r-PIR, if and only if every regular ideal of R is principal. (2) R is d-Bézout if and only if R is r-Bézout, if and only if every finitely generated regular ideal of R is principal. (3) A domain is a d-PID (or a d-PIR) if and only if it is a PID. (4) A domain is d-Bézout if and only if it is a Bézout domain. (5) A principal ideal ring, or PIR, is a ring in which every ideal is principal, and a Bézout ring is a ring in which every finitely generated ideal is principal. A d-PIR need not be a PIR, and a d-Bézout ring need not be Bézout. The following proposition generalizes Proposition 2.5.19. Proposition 2.8.3. Let  be a semistar operation on a ring R. (1) R is a -PIR if and only if R is a -Dedekind ring with trivial -class group. (2) R is a -Bézout ring if and only if R is a t -Prüfer ring with trivial t -class group. Corollary 2.8.4. Let R be a ring.

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(1) R is a t-PIR if and only if R is an r-UFR. (2) R is a t-Bézout ring if and only if R is an r-GCD ring. For any r-finite type semistar operations    on a ring R, one has the following full lattice of implications. The upward direction in the lattice is the “impose Cl (R) = 0” axis, the upward-right direction is the “impose    ” axis, and the upward-left direction is the “impose -Noetherian” axis.  -PIR -PIR

 -Dedekind

 -Bézout

-Dedekind

-Bézout

 -Prüfer

-Prüfer Consequently, if  is an r-finite type semistar operation on a ring R, then the implication diagram below is full. The first axis to the upper right (e.g., from PVMR to -Prüfer) is the “impose   t” axis, and the second axis to the upper right (e.g., from -Prüfer to Prüfer) is the “impose   d” axis; the upward direction is the “impose Clt (R) = 0” axis; and the upward-left direction is the “impose Mori” axis.

r-PIR -PIR

Dedekind

r-Bézout

r-UFR

-Dedekind

-Bézout

Prüfer

Krull

r-GCD

-Prüfer

PVMR The following proposition is an immediate consequence of Proposition 2.7.23.

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Proposition 2.8.5. Let  be a semistar operation on a ring R. If R is -Bézout, then R is t -Marot. In order to study the -PIRs and -Bézout rings in greater depth, we need a general theory of gcds and lcms in any commutative ring. Definition 2.8.6. Let R be a ring and S a subset of R. (1) Let a, b ∈ R. We say that a divides b, or a is a divisor of b, or b is a multiple of a, if there exists a c ∈ R such that b = ac. (2) A common divisor of S is an element of R that divides every element of S. (3) A greatest common divisor, or gcd, of S, is a common divisor of S that is a multiple of every common divisor of S. If g is a gcd of S, then we write g = gcd(S), even though g may not be unique. (4) A common multiple of S is an element of R that is a multiple of every element of S. (5) A least common multiple, or lcm, of S, is an element of R that is a divisor of every common multiple of S. If l is an lcm of S, then we write l = lcm(S), even though l may not be unique. Thus gcds are infima, and lcms are suprema, under the divides relation (which is a preorder rather than a partial ordering). Note that every element of a ring R is a divisor of 0, but 0 is a divisor only of 0. Dually, every element of a ring R is a multiple of 1, but 1 is a multiple only of the units of R. The following proposition describes some fundamental properties of gcds and lcms and generalizes [1, p. 2–4] to rings with zerodivisors. Statement (7) of the proposition is important for characterizing the r-GCD rings. Proposition 2.8.7 (cf. [1, p. 2–4]). Let R be a ring, let S be a subset of R, and let I = S R be the ideal generated by S. (1) If a gcd (resp., lcm) of S in R exists, then the gcds (resp., lcms) of S are precisely the generators of a unique principal  ideal. (2) l = lcm(S) exists if and only if {s R : s ∈ S} = l R is principal. (3) If lcm(S) exists, then lcm(x S) =x lcm(S) exists for all regular x ∈ I −1 . (4) g = gcd(S) exists if and only if {d R : d ∈ R, d R ⊇ S} = g R is principal, if and only if g R is the smallest principal ideal of R containing S. (5) If gcd(S) exists, then gcd(I ) = gcd(S) and gcd(c−1 S) = c−1 gcd(S) exist for all regular c ∈ R dividinggcd(S). (6) If I is regular, then I p = {x R : x ∈ T (R)reg , x R ⊇ I } is principal if and only if gcd(cS) exists for all regular c ∈ R. If these conditions hold, then one has I p = (gcd(S)) and gcd(cS) = c gcd(S) for all regular c ∈ R. (7) If I is regular, then I v is principal if and only if I v = I p and I p is principal, if and only if I v = I p and gcd(d S) exists for all regular d ∈ R. Proof. Statement (1) is clear. An element m of R is a common multiple of S if and only if m R ⊆ s R for  all s ∈ R, if and only if m R ⊆ {s R : s ∈ S}. Therefore, l ∈ R a least common

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 multiple of  S if and only if l R ⊆ {s R : s ∈ S} and l R ⊇ m R for all m ∈ R such that m R ⊆ {s R : s ∈ S}, if and only if l R = {s R : s ∈ S}. This proves (2). Next, we prove (3). Suppose x ∈ I −1 be regu that l = lcm(S) exists. Let  lar,  so x S ⊆ R. Since l R = {s R : s ∈ S}, one has xl R = {xs R : s ∈ S} = {t R : t ∈ x S} = lcm(x S). This proves (3). Next, an element d of R is a common divisor of S if and only if d R ⊇ s R for all s ∈ R, if and only if d R ⊇ {s R : s ∈ S} = S R = I . Therefore, g ∈ R is a greatest common divisor of S if and only  if g R ⊇ I and g R ⊆ d R for all d ∈ R such that d R ⊇ I , if and only if g R = {d R : d ∈ R, d R ⊇ S}. This proves (4) (and (2) and (4) together imply (1)). Next, we prove (5). Suppose that g = gcd(S) exists, so that (g) is the smallest principal ideal of R containing S. It is clear that gcd(I ) = gcd(S) also exists. Let c ∈ R be any regular element of R dividing gcd(S). Then c divides s for all s ∈ S, so that c−1 S ⊆ R. We claim that (c−1 gcd(S)) ⊇ c−1 S is the smallest principal ideal of R containing c−1 S. Indeed, if (a) ⊇ c−1 S, then (ca) ⊇ S, whence (ca) ⊇ (gcd(S)) and therefore (a) ⊇ (c−1 gcd(S)). Therefore, by statement (4), gcd(c−1 S) = c−1 gcd(S) exists. This proves (5). Next, we prove (6). Suppose that I is regular. Suppose first that I p = (g) is principal. Then I ⊆ (g), and so g is regular since I is regular. Moreover, if I ⊆ (a) for some a ∈ R, then a is regular, whence (g) = I p ⊆ (a) p = (a). Therefore (g) is the smallest principal ideal of R containing I , whence g = gcd(I ) exists. Furthermore, if c ∈ R is regular, then (cg) = c(g) = cI p = (cI ) p , so that cg = gcd(cI ) exists. Conversely, suppose that gcd(cI ) exists for all regular c ∈ R. We claim that I p = (gcd(I )). Since I ⊆ (gcd(I )), one has I p ⊆ (gcd(I )) p = gcd(I ). We prove the reverse containment as follows. Let c ∈ R be regular. Since c divides gcd(cI ), by statement (5) one has gcd(c−1 cI ) = c−1 gcd(cI ) and therefore c gcd(I ) = gcd(cI ). Now, let x = r/s ∈ K be regular, where r, s ∈ R are regular. If x R ⊇ I , then r R ⊇ s I , whence r R ⊇ (gcd(s I )) = (s gcd(I )), so that x R ⊇ (gcd(I )). Taking the intersection over all such x ∈ K , we see that I p ⊇ (gcd(I )), and therefore equality holds. Finally, we prove (7). If I p is principal and I v = I p , then I v is principal. Conversely, if I v = (g) is principal, then I ⊆ (g) = I v ⊆ I p , whence I p ⊆ (g) p =  (g) ⊆ I p , so that I p = (g) = I v is principal. Corollary 2.8.8. Let I be a regular ideal of a ring R. If I v is principal, then gcd(cI ) exists for all regular c ∈ R, and the converse holds if R is v-Marot. Definition 2.8.9. Let R be a ring. (1) R is GCD if gcd(S) exists for every finite subset S of R containing a regular element of R. (2) R is strong GCD if gcd(S) exists for every subset S of R containing a regular element of R. (3) R is weak GCD if gcd(S) exists for every finite set S of regular elements of R. (4) R is weak strong GCD if gcd(S) exists for every set S of regular elements of R.

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Example 2.8.10. (1) Any UFD is a strong GCD domain. (2) A strong GCD domain need not be a UFD. By [8, p. 327], a valuation domain is a strong GCD domain if and only if it its value group is trivial or isomorphic to Z or R. (Valuation domains are discussed in Section 3.1.) However, a valuation domain is a UFD if and only if it is a DVR. Therefore any valuation domain whose value group is isomorphic to R is a strong GCD domain that is not a UFD. (3) A Marot ring is GCD if and only if it is weak GCD. Likewise, a Marot ring is strong GCD if and only if it is weak strong GCD. Remark 2.8.11. In Bourbaki, GCD domains are called pseudo-Bézoutian domains and strong GCD domains are called pseudo-principal domains [23, Exercise VII.1.21–23]. From Propositions 2.8.7, 2.7.19, and 2.8.5 and Corollary 2.8.8, we deduce the following. Proposition 2.8.12. Let R be a ring. The following conditions are equivalent. (1) (2) (3) (4)

R is a v-PIR. R is a t-Marot strong GCD ring. R is a t-Marot weak strong GCD ring. R is a v-Marot weak strong GCD ring.

Consequently, a strong GCD domain is equivalently a v-PID. Proof. Suppose that R is a v-PIR. Then R is v-Bézout, hence t-Marot, by Proposition 2.8.5. Let S be a subset of R containing a regular element of R. Since R is a v-PIR, the ideal I v is principal, where I = RS, and therefore gcd(S) = gcd(I ) exists by Proposition 2.8.7. Therefore R is also strong GCD. It follows that (1) ⇒ (2) ⇒ (3) ⇒ (4). Suppose that (4) holds. Let I be a regular ideal of R. Since R is v-Marot, there exists a regularly generated ideal J of R such that I v = J v . Since R is weak strong GCD, gcd(c J ) exists for all regular c ∈ R, and therefore J v is principal by Corollary 2.8.8 since R is v-Marot. Therefore I v is principal as well. Thus R is a v-PIR. Therefore (4) ⇒ (1).  Similarly, one has the following. Proposition 2.8.13. Let R be a ring. The following conditions are equivalent. (1) (2) (3) (4) (5) (6)

R is r-GCD. R is t-Bézout. R is v-Bézout. R is a t-Marot GCD ring. R is a t-Marot weak GCD ring. R is a v-Marot GCD ring.

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Consequently, a GCD domain is equivalently a t-Bézout (or v-Bézout) domain. Open Problem 2.8.14. Must a v-Marot weak GCD ring be t-Marot (or, equivalently, t-Bézout)? One has the following implication diagram. v-PIR

strong GCD

weak strong GCD

r-GCD

GCD

weak GCD

Moreover, for the t-Marot rings, all of the horizontal arrows in the diagram are reversible. Propositions 2.8.12 and 2.8.13 generalize to arbitrary semistar operations as follows. Proposition 2.8.15. Let  be a semistar operation on a ring R. The following conditions are equivalent. (1) (2) (3) (4) (5)

R is a -PIR. R is a -Dedekind ring with Cl (R) = 0. R is a v-PIR and   v. R is completely integrally closed with trivial v-class group and   v. R is a v-Marot (weak) strong GCD ring and   v.

Proposition 2.8.16. Let  be a semistar operation on a ring R. The following conditions are equivalent. (1) (2) (3) (4) (5) (6) (7)

R is a t -PIR. R is a t -Dedekind ring with Clt (R) = 0. R is a -PIR and   t . R is a t-PIR and t  t. R is an r-UFR and t  t. R is a v-PIR and t  v. R is a v-Marot (weak) strong GCD ring and t  v.

Proposition 2.8.17. Let  be a semistar operation on a ring R. The following conditions are equivalent. (1) (2) (3) (4) (5) (6)

R is -Bézout. R is t -Bézout. R is a t -Prüfer ring with Clt (R) = 0. R is t-Bézout and t  t. R is r-GCD and t  t. R is a t-Marot (weak) GCD ring and t  t.

Next we characterize the weak GCD rings. First, we need the following lemma, which follows readily from Proposition 2.8.7.

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135

Lemma 2.8.18. Let R be a ring, and let a and b be regular elements of R. Then ab(a, b)−1 = (a) ∩ (b), and the following conditions are equivalent. (1) (a, b)v is principal (in which case gcd(a, b) exists and (a, b)v = (gcd(a, b))). (2) (a, b)v = (a, b) p and gcd(ca, cb) exists for all regular c ∈ R (in which case gcd(ca, cb) = c gcd(a, b)). (3) (a, b)−1 is principal. (4) (a) ∩ (b) is principal. (5) lcm(a, b) exists. Moreover, if the conditions above hold, then (ab) = (gcd(a, b) lcm(a, b)). From the lemma above, we obtain the following. Proposition 2.8.19. Let R be a ring. The following conditions are equivalent. (1) R is a weak GCD ring and I p = I v for every ideal I of R generated by finitely many regular elements of R. (2) I p is principal and I p = I v for every ideal I of R generated by finitely many regular elements of R. (3) I v is principal for every ideal I of R generated by finitely many regular elements of R. (4) gcd(a, b) exists and (a, b)v = (a, b) p for all regular a, b ∈ R (in which case gcd(ca, cb) = c gcd(a, b) for all regular c ∈ R). (5) (a, b)v is principal for all regular a, b ∈ R. (6) The intersection of any two regular principal ideals of R is principal. (7) The lcm of any two regular elements of R exists. Corollary 2.8.20. Let R be a t-Marot ring. The following conditions are equivalent. (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

R is r-GCD. R is GCD. R is weak GCD. R is v-Bézout. R is t-Bézout. The gcd of any two regular elements of R exists. The lcm of any two regular elements of R exists. The intersection of any two regular principal ideals of R is principal. (a, b)v is principal for all regular a, b ∈ R. gcd(ca, cb) = c gcd(a, b) exists for all regular a, b, c ∈ R.

Corollary 2.8.21. Let D be an integral domain. The following conditions are equivalent. (1) (2) (3) (4)

D is a GCD domain. D is v-Bézout. D is t-Bézout. D is a PVMD with trivial t-class group.

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(5) (6) (7) (8) (9)

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The gcd of any two elements of D exists. The lcm of any two elements of D exists. The intersection of any two principal ideals of D is principal. (a, b)v is principal for all a, b ∈ D. gcd(ca, cb) = c gcd(a, b) exists for all a, b, c ∈ D.

If I v is principal for every regularly generated ideal I of a ring R, then clearly R is weak strong GCD. However, the converse appears to be false. Open Problem 2.8.22. Let R be a weak strong GCD ring. Must I v be principal for every regularly generated ideal I of R? Remark 2.8.23. Let  be a semistar operation on a ring R. In this remark, we discuss -Prüfer rings with trivial -class group in the case where  need not be of r-finite type. Let us say that R is -GCD if R is a -Prüfer ring and Cl (R) = 0. By Proposition 2.8.17, a -Bézout ring is equivalently a t -GCD ring. The v-GCD domains are discussed in Sections 1.2 and 1.3. A completely integrally closed v-GCD domain is equivalently a strong GCD domain. More generally, a completely integrally closed v-GCD ring is equivalently a v-Marot (weak) strong GCD ring. Even though every t -Prüfer ring is -Prüfer, somewhat counterintuitively, every -GCD ring is t -GCD, and thus the -GCD rings are less general than the t -GCD rings. Indeed, it is easy to see that, if R is -Prüfer with trivial -class group, then R is -Bézout, or equivalently, t -Prüfer with trivial t -class group. However, since the inclusion Clt (R) ⊆ Cl (R) may be proper, even if R is t -Bézout, the converse does not hold in general, and in fact there exist valuation (hence t-Bézout) domains that do not have trivial v-class group. Indeed, by [8, p. 327], a one dimensional valuation domain has trivial v-class group if and only if its value group is isomorphic to Z or R. A ring R is an H() ring if every regular -invertible (fractional) ideal of R is t -invertible. (Such rings are defined for domains in [47].) For example, an H ring is equivalently an H(v) ring. Also, if   t is an r-finite type semistar operation on a ring R, then, trivially, R is an H() ring. In particular, every ring is an H(d) ring and an H(t) ring. To describe the -GCD rings, it is convenient to say that R is a quasi-H() ring if Cl (R) = Clt (R). We also say that R is a quasi-H ring if R is a quasi-H(v) ring, that is, if Clv (R) = Clt (R). Clearly, every H() ring is a quasi-H() ring. However, the converse does not hold, even for  = v. (This fact corrects an error I made in [47, Proposition 3.11].) For example, by [187, Example 2.1], the ring E of all entire functions is a completely integrally closed Bézout domain with all v-closed ideals principal, and therefore Clv (E) = Clt (E) = 0 and E is a strong GCD domain, but E is not an H domain since it is not a Krull domain. Note also that, if Cl (R) = 0, then R is quasi-H(). It is easy to see that R is a -GCD ring if and only if R is a quasi-H() -Bézout ring. This fact generalizes Proposition 2.8.17 since trivially R is a quasi-H(t ) ring.

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In particular, for example, a domain is a v-GCD domain if and only if it is a quasi-H GCD domain. However, the ring E of all entire functions is a quasi-H GCD domain that is not an H GCD domain.

2.9 r-UFRs, Factorial Rings, and UFDs In this section, we study the UFDs and natural generalizations to rings with zerodivisors. One of our main goals in this section is to show that a UFD is equivalently a t-PID. We also obtain several equivalent characterizations of the r-UFRs. Definition 2.9.1. Let R be a ring and r and element of R. (1) r is prime if (r ) is a prime ideal of R. (2) r is (strongly) irreducible if r is a non-unit of R such that r = ab implies that either a is a unit or b is a unit. Lemma 2.9.2. Let R be a ring and r an element of R. (1) r is prime if and only if r is not a unit and, for all a, b ∈ R, if r divides ab, then either r divides a or r divides b. (2) If r is regular, then the following conditions are equivalent. (a) r is irreducible. (b) (r ) is maximal among the proper principal ideals of R. (c) r is a non-unit, and for all a, b ∈ R, if r = ab, then r divides a or r divides b (or equivalently (r ) = (a) or (r ) = (b)). (3) If r is regular and prime, then r is irreducible. The three conditions on r appearing in statement (2) of the lemma above, for possibly non-regular elements r , are alternative and inequivalent notions of irreducibility. They are worthy rivals for the notion of irreducibility we are using in this section, which is the strongest of all of the notions of irreducibility appearing in the literature. Fortunately, for regular elements r all three definitions agree, and since here we are considering only factorizations of regular elements, we do not need any other notions of irreducibility. Definition 2.9.3. Elements a and b of a ring R are said to be associate if (a) = (b), and strongly associate if a = ub for some unit u of R. Lemma 2.9.4. Let R be a ring, and let a, b be elements of R. (1) a and b are associate if and only if a divides b and b divides a. (2) If a and b are strongly associate, then a and b are associate. (3) If either a or b is regular, then a and b are associate if and only if a and b are strongly associate. (4) A gcd or lcm of a subset of R, if it exists, is uniquely determined up to associates.

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Proof. Statements (1), (2), and (4) are clear. Suppose that a and b are associate and a is regular. Then a = cb = cda for some c, d ∈ R, so that 1 = cd since a is regular. Therefore a and b are strongly associate. This proves (3).  Recall that an r-UFR is a Krull ring with trivial t-class group, or equivalently a t-PIR. We show in Theorem 2.9.25 that an r-UFR is equivalently a ring in which every regular non-unit is a product of prime elements. For this reason, the notion of an r-UFR is in our view the most natural notion of a “unique factorization ring with zerodivisors.” However, the following definition, first introduced in [10], provides another generalization of the notion of a UFD to rings with zerodivisors. Definition 2.9.5. A ring R is factorial if every regular non-unit of R factors as a product of (necessarily regular) irreducible elements of R and any such factorization is unique in the sense that if p1 p2 · · · pm = q1 q2 · · · qn , where the pi and q j are (necessarily regular) irreducible elements of R, then m = n and, after some reordering of the q j , the elements pi and qi of R are (strongly) associate for all i. Example 2.9.6. (1) A UFD is equivalently a factorial integral domain. (2) Any total quotient ring is factorial. (3) A direct product R1 × R2 × · · · × Rn of rings R1 , R2 , . . . , Rn is factorial if and only if R1 , R2 , . . . , Rn are factorial. (4) Any factorial ring is a weak strong GCD ring. However, a factorial ring need not be a GCD ring. (5) The ring R of Example 2.7.13 is a factorial Dedekind ring that is not an integral domain. It is factorial because the regular elements of R factor uniquely in the form uπ n for units u ∈ R and nonnegative integers n. Moreover, the ring R does not have trivial (t-)class group and therefore is not an r-UFR. Although the notion of a prime element is relevant for studying the r-UFRs, it is not general enough for studying the factorial rings. To this end we make the following definition. Definition 2.9.7. An element r of a ring R is r-prime, or regularly prime, if r is a regular non-unit of R and, if r divides ab, where a, b ∈ R are regular, then either r divides a or r divides b. Remark 2.9.8. An element r of a ring R is r-prime if and only if the principal ideal (r ) is a proper regular ideal of R that is prime for its regular elements in the sense of [107, p. 36]. Clearly, in any ring R, we have the irreversible implications regular and prime ⇒ r-prime ⇒ regular and irreducible. Moreover, if R is an integral domain, then the implications above reduce to

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nonzero prime ⇔ r-prime ⇒ irreducible. It is well-known that regular prime factorizations in rings, when they exist, are unique. More generally, we have the following. Proposition 2.9.9. Let R be a ring. Regularly prime factorizations in R, when they exist, are unique. More precisely, if p1 p2 · · · pm = q1 q2 · · · qn , where the pi ∈ R are r-prime and the q j ∈ R are irreducible, then m = n and, after some reordering of the q j , the elements pi and qi of R are (strongly) associate for all i. Proof. The proof is by induction on min{m, n}. The statement is vacuously true for min{m, n} = 0. Suppose that a = p1 p2 · · · pm = q1 q2 · · · qn , where the pi are r-prime and the q j are irreducible, and where, say, m, n > 0. Since pm is r-prime and divides q1 q2 . . . qn and the q j are regular, it follows that pm divides q j for some j. By rearranging the q j if necessary, we may suppose without loss of generality that pm divides qn . Since pm and qn are both irreducible and regular, it follows that pm and qn are strongly associate. Thus, without loss of generality we may assume that pm = qn . Then, since pm = qn is regular, we may cancel it from both sides of the factorization of a to get p1 p2 · · · pm−1 = q1 q2 · · · qn−1 . By the inductive hypothesis, then, we have m − 1 = n − 1 and we may rearrange the qi so that pi is strongly associate to qi for all i  m − 1. Since also m = n and pm = qm , the proposition follows.  Definition 2.9.10. A ring is r-atomic, or regularly atomic, if all of its regular non-units are (irreducible or) products of irreducible elements. A domain that is r-atomic is said to be atomic. It is well-known that a UFD is equivalently an atomic domain in which every nonzero irreducible element is prime. Indeed, this follows readily from the fact that regular prime factorizations, when they exist, are unique. More generally, we have the following. Proposition 2.9.11. Let R be a ring. The following conditions are equivalent. (1) R is factorial. (2) R is an r-atomic ring in which every regular irreducible element is r-prime. (3) Every regular non-unit of R is a product of r-prime elements. Proof. Suppose that (1) holds. Clearly, R is r-atomic. Let p ∈ R be regular and irreducible. Suppose that p divides ab, where a, b ∈ R are regular, say, pc = ab, where c ∈ R. It follows that c is also regular. Since R is r-atomic, we may factor a, b, and c as products of irreducible elements of R. Moreover, since R is factorial, p must appear up to associates either in the factorization of a or in the factorization of b, so p divides a or p divides b. Therefore p is r-prime. Thus (1) ⇒ (2). Clearly, (2) ⇒ (3), and (3) ⇒ (1) by Proposition 2.9.9.  Definition 2.9.12. (1) A ring is said to satisfy ACCRP if it satisfies the ascending chain condition on regular principal ideals.

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(2) A ring is said to satisfy ACCP if it satisfies the ascending chain condition on principal ideals. Example 2.9.13. (1) (2) (3) (4)

A domain satisfies ACCP if and only if it satisfies ACCRP. A ring that satisfies ACCP also satisfies ACCRP. Any Noetherian ring satisfies ACCP. Any r-Noetherian or Mori ring satisfies ACCRP.

Example 2.9.14. It is well-known that any domain satisfying ACCP is atomic. Famously, in 1968 P. M. Cohn asserted the converse to be true [35, Proposition n 1.1], but a counterexample, namely, k[X 1/3 , X 1/(2·5) , X 1/(4·7) , . . . , X 1/(2 · pn+1 ) , . . .], where k is any field and pn denotes the nth prime, was found six years later [89]. Proposition 2.9.15. Any ring that satisfies ACCRP is r-atomic. Proof. Suppose that a ring R is not r-atomic. Let a be any regular non-unit of R that cannot be written as a product of irreducible elements of R. Since a is not irreducible, we can write a = b1 a1 with a1 and b1 regular non-units of R, and clearly either a1 or b1 , say, a1 , cannot be written as a product of irreducible elements of R. Since a1 is not irreducible, we can write a1 = b2 a2 with a2 and b2 regular non-units of R, and either a2 or b2 , say, a2 , cannot be written as a product of irreducible elements of R. Continuing in this manner, for all integers n > 1 we can write an−1 = bn an with an and bn regular non-units of R. Then (a)  (a1 )  (a2 )  · · · is a strictly increasing ascending chain of regular principal ideals of R, so R does not satisfy ACCRP.  Lemma 2.9.16. Let R be a ring, and let a, p ∈ R with p irreducible. Then gcd(a, p) exists and is given by  gcd(a, p) =

p

if p divides a

1

otherwise.

Proof. Since p is irreducible, the only divisors of p are associates of p and units. Therefore gcd(a, p), if it exists, is either 1 or p. If p divides a, then p is a common divisor of p and a, and any common divisor of p and a divides p, whence gcd( p, a) = p. On the other hand, if p does not divide a, then any common divisor of p and a is a unit, whence gcd( p, a) = 1.  Lemma 2.9.17. Every regular irreducible element of a weak GCD ring is r-prime. Proof. Let R be a weak GCD ring, let p ∈ R be regular and irreducible, and suppose that p divides ab, where a, b ∈ R are regular. Since p is regular and irreducible, by the previous lemma one has either gcd(a, p) = p or gcd(a, p) = 1. In the former case, p divides a, and in the latter case, p divides b since b = gcd(ab, pb) by Proposition 2.8.7(5). Therefore p is r-prime. 

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Corollary 2.9.18. Every irreducible element of a GCD domain is prime. Next, at long last, we prove that a UFD is equivalently a t-PID. Theorem 2.9.19. Let D be an integral domain. The following conditions are equivalent. (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12)

D is a UFD. D is a Mori strong GCD domain. D is a Mori GCD domain. D satisfies ACCP and is a GCD domain. D satisfies ACCP and every irreducible element of D is prime. D is an atomic GCD domain. D is atomic and every irreducible element of D is prime. D is a Mori PVMD with trivial t-class group. D is a Krull domain with trivial t-class group. Every regular t-closed (fractional) ideal of D is principal. D is a t-PID. D is an r-UFR.

Proof. Let D be a UFD. Then D is a strong GCD domain, and therefore every v-closed (fractional) ideal of D is principal. Therefore, since D also satisfies ACCP, the domain D satisfies the ascending chain condition on v-closed ideals and is therefore Mori. Therefore (1) ⇒ (2). The implications (2) ⇒ (3) ⇒ (4) are obvious. Also, one has (4) ⇒ (5) and (6) ⇒ (7) by Corollary 2.9.18, and one has (4) ⇒ (6) and (5) ⇒ (7) since ACCP implies atomic. By Proposition 2.9.11 (or by the fact that prime factorizations in any domain are unique when they exist), one has (7) ⇒ (1). Finally, it follows from Corollaries 2.6.20 and 2.8.21 that (3) ⇔ (8) ⇔ (9) ⇔ (10) ⇔ (11) ⇔ (12).  Our goal for the remainder of this section is to generalize the characterizations of the UFDs in Theorem 2.9.19 to factorial rings and r-UFRs. Lemma 2.9.20. If R is a factorial ring, then R is a weak strong GCD ring that satisfies ACCRP. Moreover, if R is factorial and v-Marot, then R is an r-UFR. Proof. Choose a set P of representatives, up to associates, of the regular irreducible

elements of R. Any regular a ∈ R can be written uniquely in the form a = u a p∈P p v p (a) , where each v p (a) is a nonnegative integer, almost all zero, and u a ∈ R is a unit. Then it is clear that, for any set S of regular elements of R, one has gcd(S) = p∈P p v p (S) , where v p (S) = min{v p (s) : s ∈ S}. By a similar argument, R also satisfies ACCRP. Suppose now that R is factorial and v-Marot, and let I be a regular ideal of R. Then there exists a regularly generated ideal J of R such that J v = I v . Moreover, J v is principal by Corollary 2.8.8 since gcd(c J ) exists for all regular c. Therefore I v is principal, for every regular ideal I of R. Moreover, since R satisfies ACCRP and every regular v-closed ideal of R is principal, it follows that R satisfies the ascending

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chain condition on the regular v-closed ideals of R, so R is Mori and hence also TV. Therefore I t = I v is principal for every regular ideal I of R, whence R is an r-UFR.  Combining Proposition 2.9.11 and Lemmas 2.9.17 and 2.9.20, we obtain the following. Proposition 2.9.21. A factorial ring is equivalently an r-atomic weak GCD ring. Proof. If R is a factorial ring, then by definition R is r-atomic, and R is a weak strong GCD ring, hence a weak GCD ring, by Lemma 2.9.20. Conversely, if R is an r-atomic weak GCD ring, then by Lemma 2.9.17 every regular irreducible element of R is r-prime, and therefore R is factorial by Proposition 2.9.11.  Proposition 2.9.22. An r-UFR is equivalently a v-Marot (or t-Marot) factorial ring. Proof. Any r-UFR is an r-GCD ring and is therefore a t-Marot weak GCD ring by Proposition 2.8.13. Moreover, any r-UFR is Mori, hence r-atomic. Therefore any r-UFR is a t-Marot factorial ring by Proposition 2.9.21. Conversely, any v-Marot factorial ring is an r-UFR by Lemma 2.9.20.  Combining several of our previous results, we obtain the following. Theorem 2.9.23. Let R be ring. The following conditions are equivalent. (1) (2) (3) (4) (5) (6)

R is factorial. R satisfies ACCRP and is a weak GCD ring. R is an r-atomic weak GCD ring. R satisfies ACCRP and every regular irreducible element of R is r-prime. R is r-atomic and every regular irreducible element of R is r-prime. Every regular non-unit of R factors as a product of r-primes.

Moreover, each of the equivalent conditions above in conjunction with the condition that R is t-Marot is equivalent to each of the following conditions, all of which are equivalent. (7) (8) (9) (10) (11) (12) (13) (14) (15) (16)

R is an r-UFR, that is, R is a Krull ring with trivial t-class group. R is a t-PIR. Every regular t-closed (fractional) ideal of R is principal. R is a Mori PVMR with trivial t-class group. R is a TV (or Mori) v-PIR. R is a v-Marot (or t-Marot) and Mori strong GCD ring. R is a v-Marot (or t-Marot) and Mori GCD ring. R is a Mori t-Bézout ring. R satisfies ACCRP, and R is a t-Marot GCD ring (or a t-Bézout ring). R is a t-Marot r-atomic GCD ring.

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Example 2.9.24. (1) There exist factorial Krull rings that do not have trivial t-class group. Of course, such rings are neither v-Marot nor an r-UFR. In fact, a factorial Dedekind ring satisfies v  t  d but need not have trivial (t-)class group. An example is provided by the ring R of Example 2.7.13. Indeed, as we have seen, R is Dedekind and factorial, but the class group is finite cyclic of order m > 1. Also note that the unique regular maximal ideal P of R is d-closed, hence t-closed, but it is not principal, so indeed R is not an r-UFR. (2) We show in Example 2.10.24 of the next section that there exist factorial rings that are not integrally closed. Exercise 2.2.11 outlines a proof that, just as a PID is a UFD, in an r-PIR every regular non-unit factors as a product of prime elements. By the following theorem, the latter condition uniquely characterizes the r-UFRs. Theorem 2.9.25 (cf. [11, Theorem]). Let R be a ring. The following conditions are equivalent. (1) (2) (3) (4)

R is an r-UFR. R is r-atomic and every regular irreducible element of R is prime. Every regular non-unit of R is a product of (regular) prime elements. R is a factorial ring in which every r-prime element is prime. To prove the theorem, we need the following definition and lemma.

Definition 2.9.26. An ideal of a ring R is t-maximal if it maximal among the proper t-closed ideals of R. Lemma 2.9.27. For any ring R, one has the following. (1) Every t-maximal ideal of R is prime. (2) Every proper t-closed ideal of R is contained in some t-maximal ideal of R. The proof of the lemma is left as an exercise; alternatively, see Lemmas 3.2.7 and 3.2.8 in Section 3.2. Proof of Theorem 2.9.25. Suppose that (1) holds. Since R is Mori, R is r-atomic. Let r be a regular irreducible element of R. Since (r ) is a proper regular t-closed ideal of R, by the lemma one has (r ) ⊆ p for some regular t-maximal ideal p of R. Then, since p is principal, one has p = ( p) for some p ∈ R, and (r ) ⊆ ( p). Therefore, r = ap for some a ∈ R, whence a is a unit since p is not. Therefore (r ) = ( p) and, since p is prime, p and r are prime. Therefore (1) ⇒ (2). Moreover, that (2) ⇔ (3) is clear, and one has (3) ⇔ (4) by Proposition 2.9.11. Finally, we prove (3) ⇒ (1). Suppose that (3) holds. Let p be a regular prime of R. For any a ∈ R, let v p (a) = sup{e ∈ Z0 : p e divides a} ∈ Z0 ∪ {∞}.

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One can verify that v p (ab) = v p (a) + v p (b) for all a, b ∈ R. Moreover, if a ∈ R is regular, then a = p n b for a unique nonnegative integer n and a unique regular b ∈ R − ( p), and then a ∈ ( p n ) − ( p n+1 ), whence v p (a) = n < ∞. Therefore for all x = a/b ∈ T (R) with a, b ∈ R and b regular, one can unambiguously let v p (x) = v p (a) − v p (b), so that v p defines a function T (R) −→ Z ∪ {∞} such that v p (x y) = v p (x) + v p (y) for all x, y ∈ T (R). Moreover, for all x ∈ T (R), one has x ∈ R if and only if v p (x)  0 for all regular primes p of R. Now, let I be any regular ideal of R. We claim that I −1 is principal. Let c be any regular element of I . We may factor c as a product up1e1 p2e2 · · · pkek , where u is a unit, the pi are non-associate primes, and the ei are positive integers. For any regular prime p, let f p = inf{v p (a) : a ∈ I } ∈ Z0 . If p is not associate to pi for any i = 1, 2, 3, . . . , k, then clearly f p = v p (c) = 0; otherwise, if p is associate to pi , then 0  f p  ei . We claim that I −1 = − fp − fp − fp ( p1 1 p2 2 · · · pk k ) is principal. Let x ∈ T (R). Then x ∈ I −1 if and only if x I ⊆ R, if and only if v p (x) + v p (y)  0 for all y ∈ I and all regular primes p, if and only if v p (x) + f p  0 for all regular primes p, if and only if v pi (x)  − f pi for all i and v p (x)  0 for all p not associate to pi for any i, if and only if − fp − fp − fp x ∈ ( p1 1 p2 2 · · · pk k ). This proves the claim. It follows that I v is principal for every regular ideal I of R. Moreover, R is factorial, so R satisfies ACCRP, and therefore R satisfies the ascending chain condition on the regular v-closed ideals of R, that is, R is Mori. Therefore R is TV, whence I t = I v is principal for every regular ideal I of R. Therefore (3) ⇒ (1).  Theorems 2.9.23 and 2.9.25 show that, although the factorial rings are more general than the r-UFRs, the latter class of rings is more natural. Another fact that suggests this is that factorial rings need not be integrally closed (while all r-UFRs are completely integrally closed). On the other hand, the factorial rings are at least completely r-integrally closed, in the following sense. Definition 2.9.28. (1) A ring R is r-integrally closed if every regular element of T (R) that is integral over R lies in R. (2) A ring R is completely r-integrally closed if every regular element of T (R) that is almost integral over R lies in R. Proposition 2.9.29. Any factorial ring is completely r-integrally closed. Proof. Let R be a factorial ring suppose that x = a/b ∈ T (R) is regular and almost integral over R, where a, b ∈ R are both regular. Then there exists a regular element c of R such that cx n ∈ R for each positive integer n. Let cn = cx n for all n, so that ca n = cn bn for all n. For any regular irreducible (or r-prime) element p of R and any regular d ∈ R, let v p (d) denote the power of p appearing in the irreducible factorization of d in R. Then one has v p (c) + nv p (a) = v p (ca n ) = v p (cn ) + nv p (b)  nv p (b)

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and therefore v p (a) − v p (b)  −v p (c)/n for all n. Taking a limit as n approaches  infinity we see that v p (a) − v p (b)  0 for all p, and therefore a/b lies in R. Note that, by Example 2.9.6(5), a Dedekind factorial ring, and therefore a completely integrally closed factorial ring, need not be an r-UFR. In the next section we show that a ring R is factorial if and only if I pt is principal for every regularly generated ideal I of R. We also provide an example of a factorial ring that is not integrally closed. Finally, we mention two other well-known generalizations of the UFDs to rings with zerodivisors. (1) A BG-UFR, or unique factorization ring in the sense of Bouvier and Galovich [24, 77], is a ring in which every nonzero non-unit factors as a product of irreducible elements, uniquely up to strong associates and reordering of the factors. A ring satisfies this condition if and only if it is a UFD, a SPIR, or a local ring whose maximal ideal m satisfies m2 = 0 [24, 77], where a SPIR, or special principal ideal ring, is a local principal ideal ring whose maximal ideal is nilpotent, or equivalently a local Artinian principal ideal ring, or equivalently still, a proper quotient of a DVR. Note that any local ring whose maximal ideal is nilpotent is a total quotient ring. Therefore, any BG-UFR is an r-UFR. (2) A UFR, or unique factorization ring, in the sense of Fletcher [59, 60], is equivalently a ring in which every non-unit (even 0) is a product of prime elements [10, Theorem 2.4]. Proposition 2.9.11 implies that any UFR is factorial, while Theorem 2.9.25 implies that any UFR is in fact an r-UFR. A UFR is equivalently a ring that is isomorphic to a finite direct product of UFDs and principal ideal rings (or PIRs), which may be further decomposed as a finite direct product of UFDs and SPIRs. Thus, a UFR is equivalently a ring that is isomorphic to a finite direct product of UFDs and quotients of DVRs (or quotients of PIDs). From Theorem 2.9.25 we can see that, analogically speaking, UFR is to PIR as r-UFR is to r-PIR. One has the following implications. UFD or SPIR ⇒ BG-UFR ⇒ r-UFR ⇒ factorial and UFD or PIR ⇒ UFR ⇒ r-UFR ⇒ factorial. However, a UFR need not be a BG-UFR, and a BG-UFR need not be a UFR. While the r-UFR property repairs some deficiencies in the notion of a factorial ring, it also is not nearly as restrictive as the UFR or BG-UFR properties. Remark 2.9.30. Let us say that a ring R is a t-UFR if R is a Krull ring such that for every I ∈ Freg (R) there exists an x ∈ T (R) such that I t = (x)t . Every r-UFR is a t-UFR but not conversely. Moreover, a domain is a t-UFR if and only if it is a UFD. Thus the t-UFRs are yet another generalization of the UFDs to rings with zerodivisors. See Exercise 3.7.13.

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2.10 Weak Counterparts of -Dedekind Rings and -Prüfer Rings Although Theorem 2.9.23 is a very general result, conditions (7)–(16) may not hold for factorial rings that are not t-Marot. In this section, we attempt to repair this deficiency in the interest of better understanding the factorial rings. Recall that a Kaplansky fractional ideal of R is regularly generated if it is generated by regular elements of its total quotient ring K . We say that a Kaplansky fractional ideal of R is finitely regularly generated if it is generated by finitely many regular elements of K . Note that a Kaplansky fractional ideal is finitely regularly generated if and only if it is finitely generated and regularly generated. Definition 2.10.1. Let  and  be semistar operations on a ring R. We write  rg   if I  = I  for all regularly generated (fractional) ideals I of R. The relation rg is an equivalence relation on the poset of all semistar operations on R that is refined by the equivalence relation . Proposition 2.10.2. Let  and  be semistar operations on a ring R. One has  t rg t if and only if I  = I  for all finitely regularly generated (fractional) ideals I of R. Let  be a semistar operation on a ring R. It is natural to loosen the definitions of -Noetherian, -Prüfer, -Dedekind, -Bézout, and -PIR rings by replacing all regular ideals in their definitions with regularly generated ideals. We call these their “weak” counterparts. Here are the definitions. Definition 2.10.3. Let  be a semistar operation on a ring R. (1) R is weak -Dedekind if every regularly generated ideal of R is -invertible. (2) R is weak -Prüfer if every finitely regularly generated ideal of R is -invertible. (3) R is a weak -PIR if every the -closure of any regularly generated ideal of R is principal. (4) R is weak -Bézout if the -closure of every finitely regularly generated ideal of R is principal. (5) R is weak -Noetherian if R satisfies the ascending chain condition on the ideals of R that are the -closure of some regularly generated ideal of R. (6) R is weak Krull if R is weak t-Dedekind. (7) R is a weak PVMR if R is weak t-Prüfer. (8) R is weak Dedekind if R is weak d-Dedekind. (9) R is weak Prüfer if R is weak d-Prüfer. (10) R is weak Bézout if R is weak d-Bézout. (11) R is a weak PIR if R is a weak d-PIR. (12) R is weak Mori if R is weak t-Noetherian. (13) R is weak TV if t rg v, that is, if I t = I v for all regularly generated ideals I of R.

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The strongest of the conditions above is the weak PIR condition. (It implies them all.) We show in Theorem 2.10.16 that any weak t-PIR is factorial, and therefore any weak PIR is factorial, but by Example 2.10.24 a weak PIR need not be integrally closed, and therefore a factorial ring need not be integrally closed. Nevertheless, we have the following. Proposition 2.10.4. Any weak v-Prüfer ring is r-integrally closed, and any weak v-Dedekind ring is completely r-integrally closed. Proof. Let R be a ring with total quotient ring K and x ∈ K be regular. Suppose that R is weak v-Prüfer. If x is integral over R, then R[x] = (1, x, x 2 , x 3 , . . .) is a finitely regularly generated fractional ideal of R and is therefore v-invertible. One has x ∈ (R[x] : K R[x]) ⊆ (R[x]v : K R[x]v ) = R by Lemma 2.3.27, and therefore R is r-integrally closed. Suppose, on the other hand, that R is weak v-Dedekind. If x is almost integral over R, then R[x] = (1, x, x 2 , x 3 , . . .) is a regularly generated fractional ideal of R and is therefore v-invertible, so, again, one has x ∈ R. Thus R is completely r-integrally closed.  Proposition 2.10.5. Let R be a ring with total quotient ring K . The following conditions are equivalent. (1) (2) (3) (4) (5)

R is completely r-integrally closed. (I : K I )rg = R for all I ∈ Freg (R). (I v : K I v )rg = R for all I ∈ Freg (R). ((I I −1 )−1 )rg = R for all I ∈ Freg (R).  (I I −1 ) p = R for all I ∈ Freg (R), where p : I −→ I p = {x R : x ∈ K reg , x R ⊇ I } = ((I −1 )rg )−1 is the closure operation defined in Lemma 2.7.16.

Proof. Suppose that R is completely r-integrally closed. Let I ∈ Freg (R), and let x ∈ (I : K I ) be regular. Then R[x] ⊆ (I : K I ) ∈ Freg (R). Choosing a regular c ∈ R such that c(I : K I ) ⊆ R, we have c R[x] ⊆ R and therefore x is almost integral over R, whence x ∈ R. Therefore R ⊆ (I : K I )rg ⊆ R, so (I : K I )rg = R. Thus (1) ⇒ (2), and from Lemma 2.3.27 it follows that (2) ⇒ (3) ⇔ (4) ⇒ (5). Suppose that (5) holds, and let x ∈ K be regular and almost integral over R, so that I = R[x] lies in Freg (R). Then x ∈ (I v : K I v )rg = ((I I −1 )−1 )rg ⊆ (((I I −1 )−1 )rg )v = R, whence x ∈ R. Therefore R is completely r-integrally closed. Thus (5) ⇒ (1).  Corollary 2.10.6. A v-Marot ring is completely integrally closed if and only if it is completely r-integrally closed. Open Problem 2.10.7. Must a completely r-integrally closed ring be weak v-Dedekind? Let us say that  is a generalized semistar operation on a ring R if  is a closure operation on K(R) such that a I  = (a I ) for all I ∈ K(R) and all a ∈ K reg . The operation p of Lemma 2.7.16 is the largest unital generalized semistar operation on R. We extend Definitions 2.8.1, 2.10.3, 2.4.35, and 2.10.1 to all unital generalized

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semistar operations  and  on R in the obvious way. Note that t  t if and only if  I  = I  for all finitely generated regular I ∈ K(R), and, likewise, t rg t if and  only if I  = I  for all finitely generated regularly generated I ∈ K(R). Just as any Mori ring is TV, any weak Mori ring is weak TV. Moreover, by Proposition 2.8.19, one has the following. Proposition 2.10.8. Let R be a ring, and let  be a unital generalized semistar operation on R. 1. 2. 3. 4. 5. 6. 7. 8.

R is GCD if and only if R is p-Bézout. R is strong GCD if and only if R is a p-PIR. R is weak GCD if and only if R is weak p-Bézout. R is weak strong GCD if and only if R is a weak p-PIR. R is -Bézout if and only if R is a GCD ring such that t  pt . R is a -PIR if and only if R is a strong GCD ring such that   p. R is weak -Bézout if and only if R is a weak GCD ring such that t rg pt . R is a weak -PIR if and only if R is a weak strong GCD ring such that  rg p.

Although by Proposition 2.8.7(6) a weak v-PIR is necessarily weak strong GCD, the converse appears likely to be false. Open Problem 2.10.9. Must a weak strong GCD ring be a weak v-PIR (or equivalently, v-Marot)? The following results are the “weak” counterparts of results proved earlier in this chapter. The proofs are left as an exercise. Proposition 2.10.10. Let  be a semistar operation on a ring R. The following conditions are equivalent. (1) (2) (3) (4)

R is weak -Noetherian. Every regularly generated I ∈ K(R) is strictly -finite. R is weak t -Noetherian. Every regularly generated I ∈ K(R) is strictly t -finite.

Moreover, if any of the conditions above holds, then  rg t . Proposition 2.10.11. Let  be an r-finite type semistar operation on a ring R, or more generally a semistar operation on R such that  rg t . Then R is weak -Dedekind if and only if R is weak -Prüfer and weak -Noetherian. Proposition 2.10.12. Let R be a ring and  a semistar operation on R. (1) (2) (3) (4)

R is weak -Dedekind if and only if R is weak v-Dedekind and  rg v. R is weak t -Prüfer if and only if R is a weak PVMR and t rg t. R is weak t -Dedekind if and only if R is weak Krull and t rg t. R is weak -Prüfer if R is weak v-Prüfer and  rg v.

Corollary 2.10.13. Let R be a ring and    semistar operations on R.

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(1) R is weak  -Dedekind if and only if R is weak -Dedekind and  rg  . (2) R is weak t -Prüfer if and only if R is weak t -Prüfer and t rg t . We also define a natural “weak” counterpart to semistar class groups. Definition 2.10.14. Let  be a semistar operation on a ring R. The weak -class group Clwk (R) of R is the subgroup of Cl (R) generated by the image in Cl (R) of the -invertible -closed I ∈ K(R) such that I = J  for some regularly generated J ∈ K(R). Proposition 2.10.15. Let  be a semistar operation on a ring R. (1) R is a -PIR if and only if R is a weak -PIR and -Marot. (2) R is -Bézout if and only if R is weak -Bézout and t -Marot. (3) If R is -Dedekind, then R is weak -Dedekind, and the converse holds if R is -Marot. (4) If R is -Prüfer, then R is weak -Prüfer, and the converse holds if R is t -Marot. (5) If R is -Noetherian, then R is weak -Noetherian, and the converse holds if R is t -Marot. (6) If R is -Marot, then Clwk (R) = Cl (R). The following theorem repairs the aforementioned deficiencies of Theorem 2.9.23. Theorem 2.10.16. Let R be a ring. The following conditions are equivalent. (1) R is factorial. (2) R is a weak pt -PIR, that is, I pt is principal for all regularly generated ideals I of R. (3) R is a weak p-PIR and p rg pt . (4) R is a weak strong GCD ring and p rg pt . (5) R is a weak GCD ring and satisfies ACCRP. (6) R is an r-atomic weak GCD ring. Moreover, each of the equivalent conditions above in conjunction with the condition that p rg v (or pt rg t) is equivalent to each of the following conditions, all of which are equivalent. (7) R is factorial and p rg v. (8) R is a weak t-PIR, that is, I t is principal for all regularly generated ideals I of R. (9) R is a weak Mori weak v-PIR. (10) R is a weak TV weak v-PIR. (11) R is a weak Krull ring R with Cltwk (R) = 0. Proof. We first show that (1) ⇒ (2). Suppose that R is factorial. Let I be any regularly generated ideal of R, generated by S = I rg . Then, since R is a weak strong GCD ring, one has I ⊆ (gcd(S)), whence I pt ⊆ (gcd(S)) pt = (gcd(S)) since gcd(S) is regular. Now, gcd(S) factors as a product p1e1 p2e2 · · · pkek , where the pi are nonassociate regular irreducibles and the ei are positive integers. For each i, there exists

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an si ∈ S such that the power of pi in the irreducible factorization of si is ei , for otherwise p1e1 p2e2 · · · pkek pi is a common divisor of S. It follows that q1 q2 · · · ql gcd(S) = gcd(s1 , s2 , . . . , sk ) for some regular irreducibles q j , none of which are associate to any pi . For each j we may choose some t j ∈ S that is not divisible by q j . Then, since R is a weak GCD ring and therefore weak p-Bézout, it follows that (gcd(S)) = (gcd(s1 , s2 , . . . , sk , t1 , t2 , . . . , tl )) = (s1 , s2 , . . . , sk , t1 , t2 , . . . , tl ) p ⊆ I pt and therefore I pt = (gcd(S)) is principal. Therefore (1) ⇒ (2). Suppose that (2) holds. Let I be a regularly generated ideal of R. Then I pt = (a) is principal, so that I p = (I pt ) p = (a) p = (a) = I pt . Therefore (3) holds. It follows that (2) ⇔ (3). Moreover, one has (3) ⇔ (4) by Proposition 2.10.8(4). Now, suppose that (2)–(4) hold. We show that R satisfies ACCRP. Let (a1 ) ⊆ ⊆ · · · be an ascending chain of regular principal ideals of R. Then the union (a2 )  ∞ (ai ) is a regularly generated ideal of R, so that I pt = (a) is principal. I = i=1 However, since (ai ) p = (ai ) for all i, one has I pt = I , so that I = (a) is principal. A standard argument, then, shows that the chain (a1 ) ⊆ (a2 ) ⊆ · · · stabilizes. Therefore R satisfies ACCRP. Thus, one has (4) ⇒ (5). Moreover, one has (5) ⇒ (6) by Proposition 2.9.15, and (6) ⇔ (1) by Proposition 2.9.21. It follows that conditions (1)–(6) are equivalent. Finally, by the relevant definitions, conditions (8)–(11) are easily seen to be equivalent, and conditions (2) and (8) are equivalent if pt rg t (hence if p rg v), and therefore conditions (7) and (8) are also equivalent.  Open Problem 2.10.17. Is a completely r-integrally closed weak Mori ring necessarily (or equivalently) a factorial ring? Proposition 2.10.18. Let  be a semistar operation on a ring R. The following conditions are equivalent. (1) R is a weak -PIR. (2) R is a weak -Dedekind ring with Clwk (R) = 0. (3) R is a weak v-PIR and  wk v. Proposition 2.10.19. Let  be a semistar operation on a ring R. The following conditions are equivalent. (1) (2) (3) (4)

R is a weak t -PIR. t (R) = 0. R is a weak t -Dedekind ring with Clwk R is a factorial ring and t wk t. R is a weak Krull ring with trivial weak t-class group and t wk t.

Proposition 2.10.20. Let  be a semistar operation on a ring R. The following conditions are equivalent. (1) (2) (3) (4)

R is weak -Bézout. R is weak t -Bézout. t (R) = 0. R is a weak t -Prüfer ring with Clwk R is weak t-Bézout and t wk t.

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(5) R is a weak PVMR with trivial weak t-class group and t wk t. (6) R is weak GCD ring and t wk pt (or t wk t wk pt ). Corollary 2.10.21. Let R be a ring. The following conditions are equivalent. (1) (2) (3) (4)

R is weak v-Bézout. R is weak t-Bézout. R is a weak PVMR with trivial weak t-class group. R is a weak GCD ring and pt wk t.

Proposition 2.10.22. Let  be a semistar operation on a ring R. Then R is a weak -PIR if and only if R is a weak -Dedekind ring with Clwk (R) = 0. Moreover, the following conditions are equivalent. (1) R is weak -Bézout. (2) R is weak t -Bézout. t (R) = 0. (3) R is a weak t -Prüfer ring with Clwk Corollary 2.10.23. Let R be a ring. (1) R is weak Krull if and only if R is a weak Mori weak PVMR. (2) R is weak Dedekind if and only if R is weak Mori and weak Prüfer. (3) A ring R is weak Bézout if and only if R is a weak Prüfer ring with Cldwk (R) = 0. As a consequence of the results above, we have for any r-finite type semistar operations    on a ring R the following full lattice of implications. The upward direction in the lattice is the “impose Clwk (R) = 0” axis, the upward-right direction is the “impose  rg  ” axis, and the upward-left direction is the “impose weak -Noetherian” axis. weak  -PIR weak  -PIR

weak  -Dedekind

weak  -Bézout

weak  -Dedekind

weak  -Bézout

weak  -Prüfer

weak  -Prüfer Consequently, the following full implication lattice generalizes to commutative rings the corresponding full implication lattice for integral domains. The upward direction in the lattice is the “impose Cltwk (R) = 0” axis, the upward-right direction is the “impose t rg d” axis, and the upward-left direction is the “impose weak Mori” axis.

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weak PIR

factorial, pt rg t

weak Dedekind

weak Bézout

weak Krull

weak GCD, pt rg t

weak Prüfer

weak PVMR The table below contrasts some of the characterizations we have noted in this section with analogous results proved in earlier sections. Ring

Characterization 1

t-Prüfer weak t-Prüfer t-Bézout weak t-Bézout v-Bézout weak v-Bézout t-Dedekind weak t-Dedekind v-Dedekind weak v-Dedekind t-PIR weak t-PIR v-PIR weak v-PIR

PVMR weak PVMR t-Marot GCD weak GCD, pt rg t t-Bézout weak t-Bézout Krull weak Krull completely integrally closed r-UFR (or t-Marot factorial) factorial, pt rg v v-Marot (weak) strong GCD weak strong GCD, pt rg v

Characterization 2

PVMR with Clt = 0 weak PVMR with Cltwk = 0 Mori PVMR weak Mori weak PVMR Krull with Clt = 0 weak Krull with Cltwk = 0 CIC with Clv = 0 weak v-Dedekind with Clvwk = 0

Finally, the following example, due to T. G. Lucas (private communication), shows that a weak PIR (hence a factorial ring) need not be integrally closed. Example 2.10.24. Let k be a field, and let D = k[X, X Y, Y 2 , Y 3 ], so that D = k[X, Y ]. Let D  = D[1/ X ] = k[X, 1/ X, Y ] and P = X 1 (D  ) be the set of all height one primes of D  . Since D  is a UFD, every height one prime of D  is principal, and the group (D  )× of units of D  consists of the elements of D  that do not lie in any height one prime. Moreover, the units of D  = k[X, 1/ X ][Y ] are the units of k[X, 1/ X ], which are the elements of the form cX n for some nonzero c ∈ k and  ×  integer n. Therefore (D ) = D − p∈P p = {cX n : c ∈ k − {0}, n ∈ Z}. as the quoFor each p ∈ P, let κ(p) be the quotient field of D  /p. This is the same tient field of D/(p ∩ D) and the quotient field of D/(p ∩ D). Let M = p∈P κ(p), which is a D  -module, hence also a D-module. Let R = D(+)M be the idealization of M over D, as defined in Definition 2.7.11. The regular elements of R all have the  form (r, m) where r ∈ D − p∈P p and m ∈ M is arbitrary. Therefore, a necessary

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and sufficient condition for (r, m) ∈ R to be regular is that r = cX n for some nonzero c ∈ k and nonnegative integer n. It follows that T (R) = D  (+)M is the total quotient ring of R. Moreover, the ring R = D(+)M is the integral closure of R, so the ring R is not integrally closed. (For example, the zerodivisor (Y, 0) of T (R) is integral over R but not in R.) Now, for all r ∈ (D  )× and all m ∈ M one has (r, m) = (1, (1/r )m)(r, 0) and (1, (1/r )m) is a unit of R and therefore (r, m)R = (r, 0)R. Thus if (r, m) is a regular element of R, then (r, m)R = (r, 0)R = (X n , 0)R for some nonnegative integer n. It follows that any ideal of R that can be generated by regular elements is generated by the smallest power of (X, 0) in the ideal. We conclude that the set of all regularly generated ideals of R is equal to {((X, 0)R)n : n = 0, 1, 2, . . .} and is totally ordered, and in particular every regularly generated ideal of R is principal. Therefore R is a weak PIR that is not integrally closed.

2.11 Further Characterizations of -Dedekind and -Prüfer Rings In this final section of Chapter 2, we prove several alternative ideal-theoretic characterizations of the -Dedekind and -Prüfer rings. In general, for two ideals I, J of a ring R, the elements I + J and I ∩ J are lattice-theoretically defined, respectively, as sup{I, J } and inf{I, J }, representing a lattice-theoretic “gcd” and “lcm” of I and J of sorts, where “to divide” is reinterpreted lattice-theoretically as “to contain.” However, since the ideal lattice of a ring is an ordered monoid under multiplication, there is also a literal sense of divisibility of ideals: an ideal I divides an ideal J if there exists an ideal H such that J = H I . This condition obviously implies I ⊇ J , but the converse need not hold. We have the following. Proposition 2.11.1. A ring R is Dedekind if and only if I ⊇ J is equivalent to I dividing J for all regular ideals I and J of R. Thus, we say that a Dedekind ring is a ring in which, for regular ideals, “to contain” is “to divide.” In particular, a Dedekind domain is equivalently a domain where “to contain is to divide.” More generally, we have the following. Theorem 2.11.2. Let R be a ring and  a semistar operation on R. Then R is -Dedekind if and only if for all I, J ∈ Freg (R) one has I  ⊇ J  if and only if there exists a regular ideal H of R such that J  = (H I ) . Proof. Suppose that R is -Dedekind. Let I, J ∈ Freg (R) with I  ⊇ J  . Let H = I −1 J . Then H ⊆ H  = (I −1 J  ) ⊆ (I −1 I  ) = R and (H I ) = (I −1 J I ) = ((I I −1 ) J ) = (R J ) = J  . Thus, the required condition holds. Conversely, suppose that the given condition holds, and let I ∈ Freg (R). We must show that I is -invertible. Since I contains a regular element a of R, by replacing I

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with a −1 I we may assume without loss of generality that I ⊇ R. Since I  ⊇ R  , by hypothesis there is an H ∈ Freg (R) with H ⊆ R and R = R  = (H I ) . Therefore I is -invertible.  Note that for R to be -Dedekind it is enough that the condition in Theorem 2.11.2 hold for all regular ideals I and J of R. Let M be an ordered monoid. It is natural to say that a divides b in M, where a, b ∈ M, if b = cad for some c, d ∈ M with c  1 and d  1. We apply this notion to the ordered monoid Freg (R) . Let I, J ∈ Freg (R) . Then there exists H ∈ Freg (R) with H ⊆ R and J = (H I ) if and only if I divides J in the ordered monoid Freg (R) . Therefore, we have the following. Corollary 2.11.3. Let R be a ring and  and semistar operation on R. Then R is -Dedekind if and only if for all I, J ∈ Freg (R) one has I ⊇ J if and only if I divides J in the ordered monoid Freg (R) , if and only if the ordered monoid Freg (R) is a group. Thus, we say that a -Dedekind ring is equivalently a ring where, for regular ideals,“to -contain” is “to -divide.” The following result provides an analogous characterization of the -Prüfer rings. Theorem 2.11.4. Let R be a ring and  a semistar operation on R. Then R is -Prüfer if and only if, for all I, J ∈ Freg (R) with I finitely generated, one has I  ⊇ J  if and only if there exists a regular ideal H of R such that J  = (H I ) . Proof. Suppose that R is -Prüfer. Let I, J ∈ Freg (R) with I  ⊇ J  and I finitely generated. Let H = I −1 J . Then H ⊆ H  = (I −1 J  ) ⊆ (I −1 I  ) = R and (H I ) = (I −1 J I ) = ((I I −1 ) J ) = (R J ) = J  . Thus, the required condition holds. Conversely, suppose that the given condition holds, and let I ∈ Freg (R) be finitely generated. We must show that I is -invertible. Since I contains a regular element a of R, by replacing I with a −1 I we may assume without loss of generality that I ⊇ R. Since I  ⊇ R  , by hypothesis there is an H ∈ Freg (R) with H ⊆ R and R = R  = (H I ) . Therefore I is -invertible. Note that Freg (R) in Theorems 2.11.2 and 2.11.4 can be replaced with Ireg (R). Lemma 2.11.5. For any ring R, one has (I + J )(I + J )−1 I J ⊆ (I + J )(I ∩ J ) ⊆ I J for all I, J ∈ K(R). Proof. One has (I + J )(I + J )−1 I J = (I + J )(I −1 ∩ J −1 )I J ⊆ (I + J )((I −1 I J ) ∩ (J −1 J I )) ⊆ (I + J )(J ∩ I ) ⊆ I J + J I = I J . Theorem 2.11.6. Let R be a ring and  and semistar operation on R. The following conditions are equivalent.

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R is weak -Prüfer. ((I + J )(I ∩ J )) = (I J ) for all finitely regularly generated ideals I, J of R. ((I + J )(I ∩ J )) = (I J ) for all regularly generated I, J ∈ Freg (R). If I, J ∈ Freg (R) are regularly generated and -invertible, then I + J is -invertible. (5) If I, J ∈ Freg (R) are finitely regularly generated and -invertible, then I + J is -invertible.

(1) (2) (3) (4)

Proof. (1) ⇒ (2). Let I, J be finitely regularly generated ideals of R. Then I + J is finitely regularly generated and therefore -invertible, whence (I J ) = ((I + J )(I + J )−1 I J ) ⊆ ((I + J )(I ∩ J )) ⊆ (I J ) , so that (I J ) = ((I + J )(I ∩ J )) . (2) ⇒ (3). Let I, J ∈ Freg (R) be regularly generated. We may suppose that I, J ⊆ R. Since (I + J )(I ∩ J ) ⊆ I J , it suffices to show that I J ⊆ ((I + J )(I ∩ J )) . Let x ∈ I J . Then x ∈ G H , where G and H are finitely generated subideals of I and J , respectively. We may suppose without loss of generality that G and H are finitely regularly generated by adding in finitely many regular elements of I and J , respectively. By (4), then, it follows that x ∈ (G H ) = ((G + H )(G ∩ H )) ⊆ ((I + J )(I ∩ J )) . (3) ⇒ (4). If I, J ∈ Freg (R) are regularly generated and -invertible, then (I J ) = ((I + J )(I ∩ J )) is -invertible, whence I + J is also -invertible. (4) ⇒ (5). Clear. (5) ⇒ (1). The implication holds by induction, since every principal regular fractional ideal is -invertible. Definition 2.11.7. Let R be a ring and  and semistar operation on R. The ring R is generalized -Prüfer if for every finitely generated I ∈ Freg (R) there exist finitely generated and -invertible I1 , I2 , . . . , In ∈ Freg (R) such that I  = (I1 + I2 + · · · + I n ) . Example 2.11.8. Let    be semistar operations on a ring R. (1) If R is -Prüfer, then R is generalized -Prüfer. (2) If R is generalized  -Prüfer, then R is generalized -Prüfer. (3) By Proposition 2.7.9, a ring is generalized d-Prüfer if and only if it is quasiMarot. (4) By Proposition 2.7.23, if R is t -Marot, then R is generalized t -Prüfer. (5) Any PVMR, quasi-Marot ring, or t-Marot ring is generalized t-Prüfer. Theorem 2.11.9 (cf. [3, Theorem 2.2]). Let R be a ring and  and semistar operation on R. Each of the following conditions implies the next. (1) R is -Prüfer. (2) ((I + J )(I ∩ J )) = (I J ) for all finitely generated regular ideals I, J of R. (3) ((I + J )(I ∩ J )) = (I J ) for all I, J ∈ Freg (R).

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(4) ((I + J )(I ∩ J )) = (I J ) for all I, J ∈ Freg (R) . (5) If I, J ∈ Freg (R) are -invertible, then I + J is -invertible. (6) If I, J ∈ Freg (R) are finitely generated and -invertible, then I + J is -invertible. (7) R is weak -Prüfer. Moreover, if R is generalized -Prüfer, then conditions (1)–(6) are equivalent, and if R is t -Marot, then conditions (1)–(7) are equivalent. Proof. As in the proof of Theorem 2.11.6, it is easy to verify that each of the given conditions implies the next. Moreover, if R is generalized -Prüfer, then clearly (6) ⇒ (1), and if R is t -Marot, then (7) ⇒ (1) by Proposition 2.10.15(4). Corollary 2.11.10. Let R be a ring and  and semistar operation on R. Then R is -Prüfer if and only if R is generalized -Prüfer and ((I + J )(I ∩ J )) = (I J ) for all I, J ∈ Freg (R) (or, equivalently, for all I, J ∈ Freg (R) ). Since I  J = (I J ) (which is the operation of -multiplication) and sup{I, J } = (I + J ) and inf{I, J } = I ∩ J for all I, J in the ordered monoid Freg (R) , we have the following corollary. Corollary 2.11.11. Let R be a ring and  and semistar operation on R. Then R is -Prüfer if and only if R is generalized -Prüfer and for all I, J ∈ Freg (R) one has sup{I, J }  inf{I, J } = I  J in the ordered monoid Freg (R) (or, equivalently, in the ordered monoid Ireg (R) ).

Exercises Section 2.1 1. a) Let R be a ring. Prove that the following conditions are equivalent. 1) R is a field. 2) R is the only nontrivial ideal of R. 3) (0) is the only prime ideal of R. 4) Every R-module is free. 5) Every nontrivial R-module is faithful. b) Prove Proposition 2.1.3. 2. Verify Example 2.1.2. 3. Prove Proposition 2.1.5. 4. Verify Example 2.1.7 5. Prove Proposition 2.1.11. 6. Show that an overring of a ring R is equivalently a Kaplansky fractional ideal I of R containing R such that I 2 = I . 7. Let R be a subring of a ring T , and let H, I, J ∈ Mod R (T ). Prove the following.

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157

a) I + J = {x + y : x ∈ I, y ∈ J } is equal to the R-submodule R(I ∪ J ) of T generated  n by I ∪ J . b) I J = k=0 ak bk : n ∈ Z>0 and ak ∈ I and bk ∈ J for all k . c) H ⊆ J implies H + (I ∩ J ) = (H + I ) ∩ J . This expresses the fact that the poset Mod R (T ) is a modular lattice. d) H (I + J ) = H I + H J . e) (H + I )(H + J )(I + J ) = (H + I + J )(H I + H J + I J ). 8. Let R be a subring of a ring T . One says that I ∈ Mod R (T ) is T -invertible if I is invertible in the monoid Mod R (T ), that is, if I J = R for some J ∈ Mod R (T ). Let I, J ∈ Mod R (T ). Prove the following. a) (I + J )m+n = (I + J )m (I n + J n ) for all nonnegative integers m, n such that n  m + 1. b) If I + J is T -invertible, then (I + J )n = I n + J n for all nonnegative integers n. 9. Prove Proposition 2.1.20. 10. Let R be a subring of a ring T . Let {Iλ : λ ∈ } be a subset of Mod R (T ). Prove the following.  a) The intersection λ∈ Iλ is the infimum  in the poset Mod R (T ) of the collection {Iλ : λ ∈ }; in other words, λ∈ Iλ is the largest R-submodule of T contained in Iλ forall λ ∈ .   b) The R-module sum λ∈ Iλ = R λ∈ Iλ is the supremum  in the poset Mod R (T ) of the collection {Iλ : λ ∈ }; in other words, λ∈ Iλ is the smallest R-submodule of T containing Iλ for all λ ∈ . c) The R (T ) is a complete lattice. poset Mod d)  J λ∈ Iλ =  λ∈ J Iλ for all J ∈ Mod R (T ).  e) λ∈ Iλ =  λ∈ aλ : a λ ∈ Iλ for all λ ∈  if  is finite. f) λ∈ Iλ = M⊆ is finite λ∈M Iλ . g) One has  λ∈

Iλ =

⎧ ⎨ ⎩

 λ∈: aλ  =0

⎫ ⎬

aλ : aλ ∈ Iλ for all λ ∈  and aλ = 0 for all but finitely many λ ∈  . ⎭

11. Let R be a ring. Show that the poset F(R) of all fractional ideals of R is complete if and only if R is a total quotient ring, if and only if K(R) = F(R), if and only if K(R) = I(R). 12. Let R be a ring. Show that the posets F(R) and Freg (R) are lattice-ordered, that is, they possess suprema and infima of all nonempty finite subsets, and they are also bounded complete, that is, they possess suprema of all nonempty subsets that are bounded above, and (equivalently) they possess infima of all nonempty subsets that are bounded below. Verify that suprema and infima are computed in F(R) and in Freg (R) exactly as they are in K(R), namely, as module sums and intersections, respectively. 13. Prove Proposition 2.1.17. 14. Let R be a ring and I, J ∈ K(R). Prove the following. a) I J ⊆ R if and only if J ⊆ I −1 . b) If I ⊆ J , then I −1 ⊇ J −1 .

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I ⊆ (I −1 )−1 . ((I −1 )−1 )−1 = I −1 . I −1 J −1 ⊆ (I J )−1 . ((I J )−1 )−1 ⊆ (I −1 J −1 )−1 . (H. Klawa) The containments in parts (e) and (f) may be proper. (Hint: Consider the ring R = Q[X 2 , X 3 ].) Let R be a ring, and let I v = (I −1 )−1 for all I ∈ K(R). Let I, J ∈ K(R). Using the previous exercise, prove the following. a) If I ⊆ J , then I v ⊆ J v . b) I ⊆ I v . c) (I v )v = I v . d) (I v )−1 = I −1 = (I −1 )v . e) R v = R. f) I J ⊆ R if and only if I J v ⊆ R. g) I v J v ⊆ (I J )v . h) (I J )v = (I J v )v = (I v J )v = (I v J v )v . Let R be a subring of a ring T . Generalize the previous two exercises to I, J ∈ Mod R (T ). Show that there exists a one dimensional local Noetherian total quotient ring R having a zerodivisor that is not contained in any minimal prime ideal of R. (Hint: Consider quotients of the ring (Z/4Z)[Y ] or of the ring (Q[X ]/(X 2 ))[Y ].) Prove the Chinese remainder theorem for commutative rings, which states that, if I1 , I2 , . . . , In are ideals of R such that

n Ii + I j = R for all i = j, then R/Ii is surjective with kernel the canonical ring homomorphism R −→ i=1 I

1 I2 · · · In = I1 ∩ I2 ∩ · · · ∩ In and induces an isomorphism R/I1 I2 · · · In −→ n i=1 R/Ii of rings. Verify Examples 2.1.28 and 2.1.33 without using Theorems 2.1.34 or 2.1.35. Prove Corollary 2.1.24 directly, without appealing to Proposition 2.1.23. Prove statements (1)–(7) of Proposition 2.1.23, and then use the proposition to prove Corollary 2.1.24. a) Verify Example 2.1.10. b) Describe all of the Kaplansky fractional ideals of Z. Let p be a prime ideal of a ring R. Show that the total quotient ring T (Rp ) of the localization Rp of R at p is the localization of T (R) at a multiplicative subset of R. (Hint: Consider the set U = {a ∈ R : a/1 is a regular element of Rp }.) Let R be a ring. A ring S is a ring of fractions of R if S is a ring containing R and for every a ∈ S there exists a regular element b of R such that ab ∈ R. Let S be a ring containing R. Prove the following. a) S is a ring of fractions of R if and only if (R : R x) is a regular ideal of R for all x ∈ S, if and only if (R : R J ) is a regular ideal of R for all finitely generated ideals J of S. b) If S is a ring of fractions of R, then there exists a unique ring homomorphism S −→ T (R) that is the identity on R. Moreover, this property of T (R) characterizes T (R) uniquely up to a unique isomorphism that is the identity on R. c) d) e) f) g)

15.

16. 17.

18.

19. 20. 21. 22. 23.

24.

Exercises

159

c) S is a ring of fractions of R if and only if there exists a (necessarily unique) ring homomorphism ϕ : S −→ T (R) such that ϕ is the identity on R and ϕ−1 (R) = R. d) Let S be a ring containing R. The following conditions are equivalent. 1) S is an R-torsion-free ring of fractions of R. 2) For every a ∈ S there exists a regular element b of S lying in R such that ab ∈ R. 3) There exists a (necessarily unique) injective ring homomorphism S −→ T (R) that is the identity on R. 4) S is a ring of fractions of R and the unique ring homomorphism S −→ T (R) that is the identity on R is injective. 25. Prove the following, without using Theorems 2.1.34 or 2.1.35. a) Any von Neumann regular ring is reduced. b) Any quotient of a von Neumann regular ring by an ideal is von Neumann regular. c) An arbitrary (possibly infinite) direct product of von Neumann regular rings is von Neumann regular. d) An arbitrary direct product of fields is von Neumann regular. e) Any localization of a von Neumann regular ring at a multiplicative subset is von Neumann regular. f) A local von Neumann regular ring is equivalently a field. g) Let R be a ring. The following conditions are equivalent. 1) R is von Neumann regular. 2) The localization Rp of R at p is a field for every prime ideal p of R. 3) The localization Rm of R at m is a field for every maximal ideal m of R. 4) R is reduced and zero dimensional. 5) I 2 = I for every ideal I of R. 6) Every ideal of R is a radical ideal. 26. Let R be a ring. Prove the following. a) A prime ideal p of R is minimal if and only if for every x ∈ p there exists a y ∈ R − p such that yx n = 0 for some positive integer n, if and only if for every x ∈ p there exists a y ∈ R − p such that x y is nilpotent. b) Suppose that R is reduced. i) A prime ideal p of R is minimal if and only if for every x ∈ p there exists a y ∈ R − p such that x y = 0. ii) The set Z (R) of all zerodivisors of R is equal to the union of the minimal prime ideals of R. iii) If I is a finitely generated ideal of R, then ann(I ) = (0) if and only if I is not contained in any minimal prime of R. c) A prime ideal of a ring R that is minimal over a zerodivisor of R need not be non-regular. (Hint: Consider the ring R = Z[X ]/(X 3 − 1) and p = (x + 1), where x denotes the image of X in R.) 27. Let R be a ring. An ideal I of R is said to be pure if R/I is flat as an R-module. Prove the following.

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a) For any ideal I of R, the following conditions are equivalent. 1) I is pure. 2) I ∩ J = I J for every ideal J of R. 3) I ∩ J = I J for every finitely generated ideal J of R. (Hint: Use the fact that an R-module M is flat if and only if the tensor product of the exact sequence 0 −→ J −→ R −→ R/J −→ 0 with M is exact for every (finitely generated) ideal J .) b) The following conditions are equivalent. 1) R is von Neumann regular. 2) Every finitely generated ideal of R is principal and generated by an idempotent. 3) Every R-module is flat. 4) Every ideal of R is pure. 5) I ∩ J = I J for all ideals I and J of R. (Hint: To prove (1) ⇒ (2), let I = (a, b) with a, b ∈ R, and let c = ax and d = by, where a 2 x = a and b2 y = b, and show that e = c + d − cd is an idempotent with I = (e). To prove (2) ⇒ (3), use the fact that an R-module M is flat if and only if the canonical map J ⊗ R M −→ J M is injective for any finitely generated ideal J .) 28. Let R be a ring. Using Exercises 25 and 27, prove that the following conditions are equivalent. 1) R is√zero dimensional. 2) R/ 0 is von √ Neumann regular. is flat. 3) Every R/ 0-module √ √ 4) I ∩ J = I J + 0 for all ideals I and J of R containing 0. 29. A ring R is said to have few zerodivisors if the set of all zerodivisors of R is a finite union of prime ideals. Let R be a ring with total quotient ring K . Prove the following. a) The set Z (R) of zerodivisors of a ring R is a union of prime ideals and contains all minimal primes of R. b) The prime ideals of R contained in Z (R) are in one-to-one correspondence with the prime ideals of K . c) An ideal of R is a maximal ideal of zero if it is maximal with respect to containing only zerodivisors. Every maximal ideal of zero is prime, and Z (R) is the union of the maximal ideals of zero. d) The maximal ideals of zero of R are in one-to-one correspondence with the maximal ideals of K . e) The minimal primes of R are in one-to-one correspondence with the minimal primes of K . f) The following conditions are equivalent. 1) R has few zerodivisors. 2) R has only finitely many maximal ideals of zero. 3) The total quotient ring of R is semilocal. g) If R has few zerodivisors, then every overring of R has few zerodivisors. h) If R is Noetherian, then R has few zerodivisors.

2.11 Further Characterizations of -Dedekind and -Prüfer Rings

161

i) Every overring of a Noetherian ring has few zerodivisors. 30. Let R be a reduced ring that has only finitely many minimal prime ideals (e.g., any reduced Noetherian ring). Let p1 , p2 , . . . , pn be the distinct minimal primes of R. Show that the total quotient ring T (R) of R is isomorphic to T (R/p1 ) × T (R/p2 ) × · · · × T (R/pn ) and is therefore isomorphic to a finite direct product of fields. 31. A ring R is said to have few zerodivisors if the set of all zerodivisors of R is a finite union of prime ideals. Let R be a ring with total quotient ring K . Prove the following. a) The total quotient ring of a ring with few zerodivisors is zero dimensional. b) The following conditions are equivalent for any ring R. 1) R is a reduced total quotient ring with few zerodivisors. 2) R is a reduced zero dimensional ring with finitely many minimal prime ideals. 3) R is isomorphic to a finite direct product of fields. Section 2.2 1. Let R be ring K , and let I ∈ K(R). Prove the following.  quotient  a ring with total a) I {J : J ∈ J } ⊆ {I J : J ∈ J } for any subset J of K(R), with equality holding if I is invertible. b) (I J : K H ) ⊇ I (J : K H ) and (J : K I H ) ⊇ (I −1 J : K H ) ⊇ I −1 (J : K H ) for all H, J ∈ K(R), with equalities holding if I is invertible. 2. Prove Proposition 2.2.26. 3. Prove that, if R is a semilocal ring, then a Kaplansky fractional ideal of R is invertible if and only if it is free of rank one. 4. Verify all of the claims made in Example 2.2.27. 5. a) Show that the following conditions are equivalent for any integral domain D. 1) D is a DVR. 2) D is a UFD with at most one irreducible element up to associates. 3) D is a local Dedekind domain. b) Let D be a DVR with quotient field K and maximal ideal (π). Show that K(D) = {(0), K } ∪ {(π n ) : n ∈ Z} and that the poset K(D) is isomorphic to the poset Z ∪ {±∞}. 6. Let k be a field, let R = k[X 2 , X 3 ], and let S = k[X ]. Prove the following. a) R is equal to the set of all polynomials f ∈ k[X ] with coefficient of X equal to 0. b) m = (X 2 , X 3 ) is a maximal ideal of R that is not invertible. c) (m−1 )−1 = m. d) (X 2 R ∩ X 3 R)S = X 5 S and X 2 S ∩ X 3 S = X 3 S. e) S is not flat over R. 7. Show that a ring R is r-Bézout if and only if (a, b) is principal for all a, b ∈ R with a regular. 8. Show that any overring of a Prüfer ring is Prüfer.

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9. A ring R is regular locally Prüfer if Rp is Prüfer for every regular maximal ideal p of R. Show that if R is regular locally Prüfer, then R is Prüfer. 10. Let R and S be rings. Prove the following. a) For any ring R, let R × denote the group of units of R. Then (R × S)× = R× × S×. b) For any ring R, let R reg denote the set of all regular elements of R. Then (R × S)reg = R reg × S reg . c) T (R × S) = T (R) × T (S). d) The ideals of R × S are precisely the subsets of R × S of the form I × J , where I is an ideal of R and J is an ideal of S. e) Let I be an ideal of R and J be an ideal of S. Then I × J is regular (resp., finitely generated, principal, invertible) if and only if the ideals I and J are regular (resp., finitely generated, principal, invertible). f) R × S is an r-PIR (resp., Dedekind, r-Bézout, Prüfer) if and only if the rings R and S are r-PIRs (resp., Dedekind, r-Bézout, Prüfer). 11. In this exercise we generalize the fact that a PID is a UFD to rings with zerodivisors. Let R be a ring. An element p of R is said to be prime if ( p) is a prime ideal of R. A non-unit p of R is said to be (strongly) irreducible if p = ab implies that a is a unit or b is a unit, for all a, b ∈ R. Two elements a and b of R are said to be (strongly) associate if a = ub for some unit u of R. Prove the following. a) A regular element p of a ring R is irreducible if and only if ( p) is maximal among the proper principal ideals of R. b) A regular element p of an r-PIR is irreducible if and only if ( p) is maximal. c) In an r-PIR, every regular non-unit factors as a product of irreducible elements. d) In an r-PIR, every regular irreducible element is prime. e) Prime factorizations of regular elements in any ring, when they exist, are unique up to associates and reordering the factors. f) In an r-PIR, every regular non-unit factors, uniquely up to associates and reordering the factors, as a product of primes. 12. Show that, for any field k, every fractional ideal of the domain k[[X 2 , X 3 ]] is divisorial. √ 13. Show that every fractional ideal of the domain Z[ 5] is divisorial. 14. Let D be a Bézout domain. Show that a D-module is flat if and only if it is Dtorsion-free. (Hint: For a module M over a ring R to be flat, it is necessary and sufficient that the R-module homomorphism I ⊗ R M −→ R ⊗ R M induced by the inclusion I −→ R is injective for every finitely generated ideal I of R.) 15. Let R be a ring. Prove the following. a) Let I be a regular fractional ideal of R. A Kaplansky fractional ideal J of R is isomorphic as an R-module to I if and only if there exist regular elements a and b of R such that a I = b J , if and only if there exists a principal regular fractional ideal H of R such J = H I (in which case J is also regular and fractional).

2.11 Further Characterizations of -Dedekind and -Prüfer Rings

163

b) A Kaplansky fractional ideal of R is free of rank one if and only if it is regular and principal. 16. Let R be a ring, and let I and J be invertible Kaplansky fractional ideals of R. Prove the following. a) I Rp is invertible, hence regular and principal, in Rp for every prime ideal p of R, and therefore I is locally free of rank one. b) I ⊗ R J ∼ = I J. c) I is finitely generated and projective and therefore also finitely presented. d) I −1 ∼ = Hom R (I, R) and the canonical map I ⊗ R Hom R (I, R) −→ R is an isomorphism. (Hint: Localization commutes with Hom R (M, −) if M is finitely presented.) Section 2.3 1. Let R be a ring. Show that if R[X ] is integrally closed, then R is integrally closed and reduced. 2. Show that any UFD is completely integrally closed. 3. Show that any Prüfer domain is integrally closed. 4. Verify Examples 2.3.7 and 2.3.14. 5. Verify the unproved assertions in Example 2.3.22. 6. Let D be an integral domain with  quotient field K and I ∈ K(R). Show that I v is equal to the intersection {a D : a ∈ K , a D ⊇ I } of all of the principal fractional ideals of D containing I . 7. Showthat, if a is a regular non-unit of a completely integrally closed ring R, n then ∞ n=1 (a) consists only of zerodivisors of R. Deduce that  if a isn a nonzero non-unit of a completely integrally closed domain D, then ∞ n=1 (a) = (0). 8. Let R = Z(2) + X Q[X ]. Prove that the set of all principal ideals of R is linearly ordered by inclusion, and therefore R is a Bézout domain. 9. Let Z[ j] = Z[X ]/(X 2 − 1), where j is the image of X in Z[X ]/(X 2 − 1). Prove the following. a) Z[w] ∼ = Z[Y ]/(Y 2 − Y ) is the (complete) integral closure of Z[ j], where 1+ j w= 2 . b) Z[w] is isomorphic to Z × Z and is an r-PIR. 10. Show that, if p is any prime ideal of a ring R, then either p = pv (that is, p is divisoral) or else p = pp−1 . Conclude that, if p is a v-invertible prime ideal of R, then either p is divisorial or else pv = p−1 = R. 11. Let T be an extension of a ring R. We say that T is integral over R (resp., almost integral over R) if every element of T is integral over R (resp., almost integral over R). Let R ⊆ S ⊆ T be rings. Prove the following. a) The integral closure of R in T is the smallest subring of T containing R that is integrally closed in T . b) The integral closure of R in T is the largest subring of T containing R that is integral over R. c) The complete integral closure of R in T is the largest subring of T containing R that is almost integral over R.

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d) T is integral over R if and only if T is integral over S and S is integral over R. e) If T is almost integral over R, then T is almost integral over S and S is almost integral over R. f) If T is integral over S and S is almost integral over R, then T is almost integral over R. 12. Show that the complete integral closure of a ring R in an extension T of R is integrally closed in T . In particular, the complete integral closure of any ring is integrally closed. 13. Prove Proposition 2.3.19. 14. Let T be an extension of a ring R. Let us say that a ∈ T is T -fractionally integral over R if R[a] is T -fractional. Let us also that the T -fractional integral closure of R in T is the set of all elements of T that are T -fractionally integral over R. Prove the following. a) a ∈ T is T -fractionally integral over R if and only if there exists an I ∈ Mod R (T ) such that I T = T and I a n ⊆ R for all positive integers n. b) The T -fractional integral closure of R in T is a subring of T containing R. c) The T -fractional integral closure of R in T is equal to  {(I :T I ) : I ∈ Mod T (R) is T -fractional and T -regular}. d) Suppose that I is T -fractional for every finitely generated T -regular I ∈ Mod R (T ). If a ∈ T is almost integral over R, then a is T -fractionally integral over R. e) a ∈ T (R) is T (R)-fractionally integral over R if and only if a is almost integral over R. 15. Let R be a ring with total quotient ring K . a) Show that K [X ] is the complete integral closure of the ring R + X K [X ]. In particular, R + X K [X ] is completely integrally closed if and only if R = K . b) Show that R + X K [X ] is the integral closure of the ring R + X K [X ]. In particular, R + X K [X ] is integrally closed if and only if R is integrally closed. c) Conclude that if R is integrally closed but not a total quotient ring, then R + X K [X ] is integrally closed but not completely integrally closed. 16. Let R be a ring with total quotient ring K . Let S be a multiplicatively closed set of regular ideals of R.  The generalized quotient ring of R with respect to S is defined as RS = {(R : K J ) : J ∈ S}. The saturation of S is the set S = {I ⊆ R : I ⊇ J for some J ∈ S}. Prove the following. a) RS is an overring of R. b) U −1 R is a generalized quotient ring of R for any multiplicative subset U of R containing only non-zerodivisors. c) The intersection of any collection of generalized quotient rings of R is a generalized quotient ring of R.

2.11 Further Characterizations of -Dedekind and -Prüfer Rings

17. 18.

19.

20. 21.

165

 d) For any ideal I of R, the set IS = {(I : K J ) : J ∈ S} is an ideal of RS such that I RS ⊆ IS . e) The set S is multiplicatively closed and IS = IS for every ideal I of R. In particular, RS = RS . f) IS = RS if and only if I ∈ S, for every ideal I of R. g) Let I and J be ideals of R. Then (I : K J )S ⊆ (IS : K JS ), and if J is finitely generated, then (I : K J )S = (IS : K JS ) = (IS : K J RS ). h) Let vS and tS denote the v and t operations on RS . If J is a finitely generated ideal of R, then (J v )S ⊆ (JS )vS = (J RS )vS . i) If J is an ideal of R, then J t ⊆ (J RS )tS , whence the extension RS of R is semistar t-compatible. The ring of algebraic integers is the integral closure of Z in C (or in any algebraic closure of Q). Is the ring of all algebraic integers completely integrally closed? Let T be an extension of a ring R. Define vT : Mod R (T ) −→ Mod R (T ) by I vT = (I −1T )−1T for all I ∈ Mod R (T ). Generalize Lemma 2.3.23, Proposition 2.3.24, Definition 2.3.26, Lemma 2.3.27, and Corollary 2.3.28 to this more general setting. Let R be a ring whose integral closure has only finitely many minimal primes (which holds, for example, if R is Noetherian, by [106, Theorem 2.4]). Show that R is reduced if and only if the integral closure of R is isomorphic to a finite direct product of integrally closed integral domains. Conclude that the integral closure of a reduced Noetherian ring is isomorphic to a finite direct product of integrally closed integral domains. (Hint: Supposing that R is reduced, first show that T (R) is isomorphic to a finite direct product of fields. Also note that every idempotent in T (R) is integral over R.) Let R be a ring. Show that R[X ] is an r-PIR if and only if R is isomorphic to a finite direct product of PIDs. Prove that an integral domain D is integrally closed if and only if the polynomial ring D[X ] is integrally closed.

Section 2.4 1. a) Let  and  be semistar operations on a ring R. Show that    if and only if every -closed fractional ideal of R is  -closed. b) Show that I ∈ K(R) is v-closed if and only if I is -closed for every semistar operation  on R. c) Show that I ∈ K(R) is t-closed if and only if I is -closed for every finite type semistar operation  on R. 2. Prove Lemma 2.4.9. 3. Prove Proposition 2.4.7 and Corollary 2.4.8. 4. How many semistar operations are there on the ring Q[X ]/(X 2 )? 5. Let R be a ring with total quotient ring K . Prove the following. a) A semistar operation on R is equivalently a closure operation  on K(R) such that a I  ⊆ (a I ) for all a ∈ K . b) The self-map

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  : I −→

6.

7.

8.

9.

10.

11.

R if I ⊆ R I otherwise

of K(R) is a closure operation on K(R) such that a I  ⊆ (a I ) for all a ∈ R, but  is a semistar operation on R if and only if R = K . Let R be a ring. Prove that a self-map  of K(R) is a semistar operation on R if and only if R  = R, the map  is order-preserving and expansive, and the operation (I, J ) −→ (I J ) of -multiplication on K(R) is associative. Let R be a ring and  a self-map of K(R). Prove that the following conditions are equivalent. 1)  is a semistar operation on R. 2) R  = R, and H J ⊆ I  if and only H J  ⊆ I  for all H, I, J ∈ K(R). 3) R  = R and (I  : K J ) = (I  : K J  ) for all I, J ∈ K(R). Let S be a partially ordered set and x −→ x − a self-map of S such that x  (x − )− , and x  y implies y −  x − , for all x, y ∈ S. Prove the following. a) ((x − )− )− = x − for all x ∈ S. b) The self-map x −→ x  = (x − )− of S is a closure operation on S. c) Suppose that S is also an ordered commutative monoid and that x y  1 if and only if x  y − for all x, y ∈ S. Then  is a nucleus on S. d) For any ring R, the operation I −→ I v = (I −1 )−1 on K(R) is a semistar operation on R and in fact is the largest semistar operation on R. Let  be a semistar operation on a ring R. Two ideals I and J of R are said to be -coprime if (I + J ) = R. Prove the following. a) If I and J are -coprime ideals of R, then (I J ) = (I ∩ J ) = I  ∩ J  . b) If I1 , I2 , . . . , In are pairwise -coprime ideals of R, then (I1 I2 · · · In ) = (I1 ∩ I2 ∩ · · · ∩ In ) = I1 ∩ I2 ∩ · · · ∩ In . c) If I and J are -coprime ideals of R, then J  ⊇ (I H ) implies J  ⊇ H  , for any ideal H of R. d) If I and I  are both -coprime to an ideal J of R, then so is I I  . e) If I and J are -coprime, then I m and J n are -coprime for any nonnegative integers m, n. Let  be a finite type semistar operation on a ring R. a) Show that the union of a chain (that is, a set totally ordered by inclusion) of -closed ideals of R is also a -closed ideal of R. b) Using Zorn’s lemma, show that any proper -closed ideal of R is contained in an ideal that is maximal among all proper -closed ideals of R. c) Show that the union of an ascending chain of regular principal ideals of R is -closed. d) A nonempty set I of ideals of R is said to be directed if for all  I, J ∈ I there exists an H ∈ I such that I, J ⊆ H . Show that theunion I of a directed set of ideals of R is an ideal of R, and moreover I is -closed if I is -closed for all I ∈ I. Let  be a semistar operation on a ring R.

Exercises

12. 13. 14. 15.

16. 17.

18.

167

a) Show that the intersection of a chain (that is, a set totally ordered by inclusion) of prime ideals of R is prime. b) Using Zorn’s lemma, show that if I is a regular ideal of R that is contained in some -closed prime ideal of R, then there exists a minimal -closed prime ideal of R containing I . Prove Lemma 2.4.13 and Proposition 2.4.12. Prove Proposition 2.4.18. Let  be a semistar operation on a ring R. Show that t is a finite type semistar operation on R, and then prove Proposition 2.4.24(1). Let R be a ring with total quotient ring K . Let {Rλ : λ ∈ } be an indexed collection of rings lying between R and K whose intersection is R. Prove the following.  a)  : I −→ I  = {I Rλ : λ ∈ } is a semistar operation on R. b) Suppose that the collection {Rλ : λ ∈ } is locally finite, that is, that every regular element of R is a non-unit in only finitely many of the Rλ . Then  is of r-finite type. Prove Proposition 2.4.19. A ring R is said to be divisorial if v  d on R. Let R be a ring. Prove the following. a) R is divisorial if and only if d is the only star operation on R. b) If R is Dedekind, then R is divisorial. c) If R is divisorial or r-Noetherian, then every semistar operation on R is of r-finite type. d) Not every semistar operation on Z is of finite type. Let R be a ring with total quotient ring K , and let ∗ be a star operation on R. a) Show that, if R is an integral domain, then the largest semistar operation e∗ on R that restricts to ∗ is acts by  e∗ : I −→

I ∗ if I is fractional K otherwise.

b) Find an example of star operation ∗ on a ring R such that e∗ is not given as in part (a). 19. Let  be a semistar operation on a ring R. a) Show that  and w are stable semistar operations on R. b) Show that w is of finite type. c) Prove Proposition 2.4.24(2). d) Prove Proposition 2.4.24(3). 20. Prove Proposition 2.4.27. 21. Let R be a ring with total quotient ring K , and let I, J, J  ∈ K(R). Prove the following. a) I is v(J )-closed if and only if I = I v ∩ (J : K H ) for some H ∈ K(R). b) v(J J  ) = v(J ) if J  is invertible.

168

22.

23. 24.

25.

26. 27. 28. 29.

30.

2 Semistar Operations on Commutative Rings

c) v(J ) = v if and only if J is divisorial and R is the largest subring of K containing J as an ideal. Let R be a ring with total quotient ring K , and let  be a semistar operation on R. a) Show that r = v((0)) : I −→ I v ∩ ann K (ann K (I )) is the largest reduced semistar operation on R. b) Show that red = inf{, r } is the largest reduced semistar operation on R less than or equal to . Let D be a DVR. Show that |Semistar(D)| = 2 if D is a field and |Semistar (D)| = 1 otherwise. A special principal ideal ring, or SPIR, is a local principal ideal ring whose maximal ideal is nilpotent. It is known that a SPIR is equivalently a ring that is a proper quotient of some DVR. Let D be a DVR with maximal ideal m, and let R denote the SPIR D/mn , where n is some positive integer. Prove the following. a) The ideal lattice I(R) of R is isomorphic to the poset {1, 2, . . . , n + 1}. b) I v(J ) = I + J for all I, J ∈ K(R) = I(R). c) The map I(R) −→ Semistar(R) acting by J −→ v(J ) is a poset isomorphism. d) The poset Semistar(R) is isomorphic to the poset {1, 2, . . . , n + 1} and thus |Semistar(R)| = n + 1. e) d = v((0)) and e = v((1)) are the only stable semistar operations on R. f) If S is a SPIR with maximal ideal n and n is the least positive integer such that nn = (0), then Semistar(S) is isomorphic to the poset {1, 2, . . . , n + 1}. Let K 1 , K 2 , . . . , K n be fields, and let R = K 1 × K 2 × · · · × K n . Prove the following. a) I v(J ) = I + J for all I, J ∈ K(R) = I(R). b) The map I(R) −→ Semistar(R) acting by J −→ v(J ) is a poset isomorphism. c) The poset Semistar(R) is isomorphic to the power set 2{1,2,...,n} of {1, 2, . . . , n} and thus |Semistar(R)| = 2n . d) Every semistar operation on R is stable and of finite type. Find two rings R and S such that |Semistar(R)| = |Semistar(S)| < ∞ but Semistar(R) and Semistar(S) are not isomorphic as posets. Prove Proposition 2.4.38. Prove Proposition 2.4.30. a) Prove Proposition 2.4.34. b) Prove Proposition 2.4.39. c) Prove Proposition 2.4.42. (∗∗) Characterize the rings R for which the map K(R) −→ Semistar(R) acting by J −→ v(J ) is surjective.

Section 2.5 1. Prove Propositions 2.5.3, 2.5.4, and 2.5.10. 2. Prove statements (2)–(4) of Proposition 2.5.12, and prove Corollary 2.5.13. 3. Prove Lemma 2.5.15 and Proposition 2.5.16.

Exercises

4. 5. 6. 7.

169

Prove Proposition 2.5.22 and Corollaries 2.5.23–2.5.25. Verify all of the claims made in Remark 2.5.26. Show that Cl(R) = 0 for any semilocal ring R. Let  be a semistar operation on a ring R. a) Let Prin (R) = {(x) : x ∈ T (R) and ∃y ∈ T (R) ((x y) = R)}. Show that Prin (R) is a subgroup of Inv (R). b) Let Cl (R) = Inv (R)/ Prin (R) and

Cl (R) = Inv (R)/ Prin (R).

Show that there is a commutative square Cl (R)

Cl (R)

Cl (R)

Cl (R)

of group homomorphisms such that the horizontal arrows are injective and the vertical arrows are surjective. c) State and prove analogues of Corollaries 2.5.23 and 2.5.24 for Cl (R) and Cl (R). 8. Let R be a ring. Prove the following. a) Let I ∈ K(R). If  is a nonempty set of semistar operations on R such that I is -invertible for all  ∈ , then I is inf -invertible. b) Let I ∈ K(R) be v-invertible. Let  I = inf{ : I is  -invertible}. Then I is -invertible, where  is a semistar operation on R, if and only if    I . c) Suppose that R is v-Prüfer. If  is a nonempty set of semistar operations on R such that R is -Prüfer for all  ∈ , then R is inf -Prüfer. d) Suppose that R is v-Prüfer. Let Prüf(R) = sup{ I : I ∈ Freg (R), I f.g.}. Then R is -Prüfer, where  is a semistar operation on R, if and only if Prüf(R)  . e) R is a PVMR if and only if Prüf(R)  t, if and only if Prüf(R)  t. 9. Let R and S be rings with total quotient rings K and L, respectively. Prove the following. a) The total quotient ring of R × S is K × L. b) Every R × S-submodule of K × L is of the form I × J , where I is an Rsubmodule of K and J is an S-submodule of L. Moreover, I × J is fractional (resp., regular, finitely generated, principal) if and only if I and J are. c) (I × J : K ×L I  × J  ) = (I : K I  ) × (J : L J  ) for all R-submodules I, I  of K and all S-submodules J, J  of L.

170

10.

11.

12.

13.

14.

2 Semistar Operations on Commutative Rings

d) (I × J )v = I v × J v , (I × J )t = I t × J t , and (I × J )w = I w × J w for all R-submodules I of K and all S-submodules J of L. Let R and S be rings with total quotient rings K and L, respectively. Let  be a  semistar operation on R and  a semistar operation on S. Define (I × J )× =   I × J ⊆ K × L for all R-submodules I of K and all S-submodules J of L. Prove the following. a)  ×  is a semistar operation on R × S. b) Every semistar operation on R × S has the form  ×  for a unique semistar operation  on R and a unique semistar operation  on R. c) Semistar(R × S) is isomorphic to the poset Semistar(R) × Semistar(S), where Semistar(R) × Semistar(S) is given the coordinatewise partial ordering (so that (1 , 1 )  (2 , 2 ) if and only if 1  2 and 1  2 ). d) ( ×  )t = t × t ,  ×  =  ×  , and ( ×  )w = w × w . e) R × S is a ( ×  )-PIR (resp., ( ×  )-Bézout, ( ×  )-Dedekind, ( ×  )-Prüfer) if and only if R is a -PIR (resp.,-Bézout, -Dedekind, -Prüfer) and S is a  -PIR (resp.,  -Bézout,  -Dedekind,  -Prüfer).    f) Inv× (R × S) = Inv (R) × Inv (S) and Cl× (R × S) = Cl (R) ×  Cl (S).    g) Inv× (R × S) = Inv (R) × Inv (S) and Cl× (R × S) = Cl (R) ×  Cl (S). Let R and S be rings. Using the previous two exercises, show that R × S is an r-PIR (resp., r-Bézout, Dedekind, Prüfer, r-UFR, r-GCD, Krull, PVMR) if and only if both R and S are. Let D be a Dedekind domain. a) Prove that every nonzero ideal of D can be written as a product of prime ideals of D and that such a representation is unique up to reordering the factors. Conclude that Freg (D) is the free abelian group generated by the nonzero prime ideals of D. b) Prove that, for every nonzero element a of a nonzero ideal I of D, there exists an element b of I such that I = (a, b). (Hint: Show that the ring D/a D is semilocal with all ideals invertible.) a) Let  be a semistar operation on a ring R. Show that, if I ∈ K(R) is -invertible, then I v and I t are -invertible. Show by example that I v and I t may be -invertible while I is not. b) Provide a specific example of a v-invertible ideal of some integral domain that is not t-invertible. Let R be a ring with total quotient ring K . An R-module M is said to be invertible if there exists an R-module N such that M ⊗ R N is isomorphic to R. Prove the following. a) The tensor product (over R) of two invertible R-modules is invertible. b) Any invertible fractional ideal of R is an invertible R-module. (Hint: Use Exercise 2.2.16(b).) c) Any invertible R-module M is finitely generated. (Hint: Suppose that f : n xi ⊗ yi with M ⊗ R N −→ R is an isomorphism, and write f −1 (1) = i=1  xi ∈ M and yi ∈ N for all i. Let M = (x1 , . . . , xn ). Show that M  = M by

Exercises

171

showing the map M  ⊗ R N −→ M ⊗ R N is surjective and then tensoring with M to show that the map M  −→ M is surjective.) d) Any invertible R-module M is finitely generated and projective and therefore finitely presented. e) An R-module M is invertible if and only if it is finitely presented and locally free of rank one, if and only if the canonical R-module homomorphism M ⊗ R Hom R (M, R) −→ R is an isomorphism. (Hint: If M is finitely presented, then the R-module homomorphism Hom R (M, N )p −→ Hom Rp (Mp , Np ) is an isomorphism for every R-module N .) f) The set of all isomorphism classes [M] of invertible R-modules M is a group under the well-defined operation ([M], [N ]) −→ [M ⊗ R N ]. This group, denoted Pic(R), is called the Picard group of R. g) A regular fractional ideal of a ring R is invertible if and only if it is projective, if and only if it is invertible as an R-module. h) There is an injective group homomorphism Cl(R) −→ Pic(R). (Hint: Use Exercise 2.2.15.) i) The Z-submodule of Q consisting of all rational numbers whose denominator is squarefree is locally free of rank one but is not invertible. 15. This exercise uses the previous exercise and Exercise 2.1.29. Let R be a ring with few zerodivisors, and let K = T (R). Prove the following. a) [43, Exercise 4.13]. Suppose that R is semilocal. If M and N are finitely presented R-modules such that Mp ∼ = Np for every prime ideal of R, then M∼ = N . Thus, an R-module M is invertible if and only if and only if it is isomorphic to an invertible ideal of R. b) M is flat and M ⊗ R K ∼ = K for any invertible R-module M. c) Any invertible R-module is isomorphic to an invertible fractional ideal of R. d) An R-submodule of K is invertible if and only if it is invertible as an R-module. e) The inclusion Cl(R) −→ Pic(R) of the previous exercise is an isomorphism. Section 2.6 1. Verify Proposition 2.6.17. 2. Verify the implications (4) ⇒ (5) ⇒ (1) in Proposition 2.6.19. 3. Let  be an r-finite type semistar operation on a ring R. Show that R is -Dedekind if and only if R is -Prüfer and Mori. 4. Show that an H ring is v-Prüfer if and only if it is a PVMR. 5. Prove that the following conditions on a ring R are equivalent. 1) R is r-Noetherian. 2) Every regular ideal of R is finitely generated. 3) Every regular prime ideal of R is finitely generated. 4) Every nonempty set of regular ideals of R has a maximal element. 6. Show that a domain D is d-coherent if and only if every finite intersection of principal ideals of D is finitely generated. 7. Find an example of an r-Noetherian ring that is not Noetherian.

172

2 Semistar Operations on Commutative Rings

8. The Hilbert basis theorem states that R[X ] is Noetherian if and only if R is Noetherian, for any ring R. a) Find three different proofs of the Hilbert basis theorem in the literature and write up in your own words your favorite of the three. b) Using the Hilbert basis theorem, show that R[X ] is r-Noetherian if and only if R is Noetherian, for any ring R. 9. Let  be a semistar operation on a ring R. Prove the following. a) R is -Noetherian if and only if R is quasi--Noetherian and  is of r-finite type. b) R is -Noetherian if and only if R is quasi-t -Noetherian. 10. Show by example that a regular v-finite ideal of a ring R need not be strictly v-finite. 11. Let D be a DVR with quotient field K = D. Show that the domain D + X K [X ] is an integrally closed divisorial domain that is not Dedekind. 12. Show by example that an integrally closed Mori domain need not be Krull. 13. Show by example that the implications Noetherian ⇒ Mori ⇒ TV ⇒ H for integrally closed integral domains are irreversible. 14. Show that a direct product R1 × R2 × · · · × Rn of rings R1 , R2 , . . . , Rn is divisorial (resp., w-divisorial, DW, TD, TV ,TW, H, r-Noetherian, Mori) if and only if the rings R1 , R2 , . . . , Rn are divisorial (resp., w-divisorial, DW, TD, TV ,TW, H, r-Noetherian, Mori). √ √ 15. Show that the domain Z[2i, 2 2, 2i 2] is not divisorial. 16. (∗∗) Let k be a field and let a and b be relatively prime positive integers. Without using results in [113], show that the domain k[[X a , X b ]] is divisorial. 17. (∗∗) Let ξ be any algebraic integer. Without using results in [113], show that the domain Z[ξ] is divisorial. 18. (∗∗) Must a quasi-v-Noetherian domain be v-Noetherian? Must a quasi-vNoetherian v-Dedekind (completely integrally closed) domain be a Krull domain? 19. (∗∗) In [172, Theorem 1.14] it is shown that if R is a Mori domain, then (R  ) is completely integrally closed. Is it true that if R is a Mori ring, then (R  ) is completely integrally closed? Section 2.7 1. Show that a Kaplansky fractional ideal of a ring is generated by finitely many regular elements if and only if it is finitely generated and regularly generated. 2. Let    be semistar operations on a ring R. Show that every  -Marot ring is -Marot. 3. Prove statements (1)–(5) of Lemma 2.7.16, and prove Proposition 2.7.19. 4. Prove Proposition 2.7.4 and Corollary 2.7.5. 5. Prove Proposition 2.7.9 and Corollary 2.7.10.

Exercises

173

6. Show that a direct product R1 × R2 × · · · × Rn of rings R1 , R2 , . . . , Rn is Marot (resp., quasi-Marot, v-Marot, t-Marot) if and only if the rings R1 , R2 , . . . , Rn are Marot (resp., quasi-Marot, v-Marot, t-Marot). 7. Prove Propositions 2.7.22 and 2.7.23. 8. Let  be an r-finite type semistar operation on a ring R. Show that R is -Noetherian and -Marot if and only if for every regular ideal I of R there exist regular elements a1 , a2 , . . . , an of R such that I  = (a1 , a2 , . . . , an ) . 9. Let R be a ring with total quotient ring K . Let us say that a closure operation  on K(R) is regular if cI  = (cI ) for all regular c ∈ K . Prove the following. a) The poset of all regular closure operations on K(R) is complete. b) For any J ∈ K(R), the operation u(J ) : I −→ (J : K (J : K I )rg ) is the largest regular closure operation  on K(R) such that J  = J . c) For any regular closure operation  on K(R), one has  = inf{u(J ) : J ∈ K(R) }. d) The operation p of Lemma 2.7.16 is equal to u(R). 10. Let R be a Marot ring. Prove that a proper regular ideal p of R is prime if and only if ab ∈ p implies a ∈ p or b ∈ p for all regular elements a and b of R. 11. Let p be any prime ideal of a ring R such that pp−1 = p. Prove the following. a) There exist elements a of p and b of R − p such that p = (a R : R b R). (Hint: / p.) Choose a ∈ p and u ∈ p−1 such that au ∈ b) p is minimal over a R. c) If p is regular and R is Marot, or more generally if prg p−1  p, then a may be chosen to be regular. 12. A conucleus on an ordered monoid M is an interior operation ◦ on M such that x ◦ y ◦  (x y)◦ for all x, y ∈ M. Let R be a ring with total quotient ring K . Let us say that a cosemistar operation on R is a unital conucleus on the ordered monoidK(R). Let S be a submonoid of the monoid K • . For all I ∈ K(R), let I ◦S = a∈I ∩S a R = (I ∩ S)R. Prove the following. a) The self-map ◦ S of K(R) is a cosemistar operation on R. b) For all I ∈ K(R), one has I ◦S = I if and only if I is generated by elements of S, and one has I ◦S = (0) if and only if I ∩ S = ∅. c) (a R)◦S = a R for all a ∈ S. d) S is saturated if ab ∈ S implies a, b ∈ S, for all a, b ∈ K . If S is saturated, then (a R)◦S = a R if and only if a ∈ S, for all a ∈ K . e) For all I ∈ K(R) one has (I −1 )◦S ⊆ (I ◦S )−1 . f) If S = K reg , then ◦ S : I −→ I rg for all I ∈ K(R). Use parts (a)–(f) to write down several properties of the map rg : I −→ I rg . 13. Let R be a ring. Prove the following. a) The idealization R(+)R of the free R-module R over R is isomorphic to R[X ]/(X 2 ). b) For any positive integer n, the idealization R(+)R n of the free R-module R n over R is isomorphic to R[X 1 , X 2 , . . . , X n ]/(X 1 , X 2 , . . . , X n )2 . 14. [13]. Let R be a ring and M an R-module. An element of r of R is said to be regular on M if r m = 0 implies m = 0, for all m ∈ M. Let M reg denote the set of all elements of R that are regular on M. Prove the following.

174

15. 16. 17.

18. 19.

2 Semistar Operations on Commutative Rings

a) The prime ideals of R(+)M are precisely the ideals p(+)M, where p is a prime ideal ideal of R. b) The units of R(+)M are the elements (r, m) such that r is a unit of R and m ∈ M is arbitrary. c) M reg is a multiplicatively closed subset of R. d) The regular elements R(+)M are the elements (r, m) such that r ∈ R reg ∩ M reg and m ∈ M. e) Let U = R reg ∩ M reg . The total quotient ring T (R(+)M) of R(+)M is isomorphic to U −1 (R(+)M) ∼ = U −1 R(+)U −1 M. f) The integral closure R(+)M of R(+)M is equal to (R ∩ U −1 R)(+)U −1 M. (Hint: (0, m)2 = (0, 0), so (0, m) is integral over R(+)M, for all m ∈ U −1 M.) In particular, R(+)M is integrally closed if and only if R is integrally closed in U −1 R and U −1 M = M. g) Let  be a set of prime ideals of R and K (p) an overring of R/p for every  reg K (p). Then M = R − p and U = R reg − p ∈ . Let M = p∈ p∈  p∈ p. Moreover, if every element of U is a unit in K (p) for all p ∈  (which holds, for example, if K (p) is the quotient field of R/p for all p ∈ ), then U −1 M = M. h) U −1 M = M if and only if every regular ideal of R(+)M has the form J (+)M, where J is an ideal of R containing an element of U . i) Suppose that U −1 M = M, and let K = U −1 R. i) T (R(+)M) = K (+)M. ii) R(+)M = (R ∩ K )(+)M. iii) For any regular (r, m) ∈ K (+)M, one has (r, m)(R(+)M) = (r, 0) (R(+)M) and r = u/v for some u, v ∈ U . iv) The regular ideals of R(+)M are precisely the ideals a(+)M, where a is an ideal of R with a ∩ U = ∅. Moreover, a(+)M is finitely generated if and only if a is finitely generated. v) For any regular ideal I of R(+)M, one has I rg = a(+)M, where a is the (regularly generated) ideal a of R generated by {u ∈ U : (u, 0) ∈ I }. vi) For any regular Kaplansky fractional ideal I of R(+)M, one has I rg = a(+)M, where a is the (regularly generated) Kaplansky fractional ideal a of R generated by {u/v : u, v ∈ U, (u/v, 0) ∈ I }. Use the previous exercise to verify all of the unsubstantiated claims in Examples 2.7.13 and 2.7.14. Adapt the proof of Proposition 6.6.2 of Chapter 6 to provide a direct proof of Proposition 2.7.15. a) Show that an arbitrary direct product of additively regular rings is additively regular. b) Using condition (5) of Theorem 2.1.34, show that, if the total quotient ring T (R) of a ring R is zero dimensional, then R is additively regular. [166]. Let R be an additively regular ring. Show that, if I and  J1 , J2 , . . . , Jn are  regular ideals of R, then I ⊆ nk=1 Jk if and only if I rg ⊆ nk=1 Jk . Without using Theorem 2.7.7, prove that each of the following conditions implies the next.

2.11 Further Characterizations of -Dedekind and -Prüfer Rings

175

1) R is Noetherian. 2) R has few zerodivisors. 3) R is additively regular. 4) R is Marot. n pi , where Z (R) (Hint: For the implication (2) ⇒ (3), we may let Z (R) = i=1 is the set of all zerodivisors of R, and where the pi are distinct prime ideals, none of which is contained in the  union of the Let x = a/b with a, b ∈ R and others. k n pi − i=k+1 pi , and show that one can choose b regular. Suppose that a ∈ i=1 n k y ∈ i=k+1 pi − i=1 pi .) 20. a) Let us say that a ring R is strictly t-Marot if for every regular ideal I of R there exists a regularly generated ideal J ⊆ I of R such that I t = J t . Let I be a regular fractional ideal of a strictly t-Marot ring R. Prove the following conditions equivalent. 1) I is invertible. 2) I is finitely generated and locally free as an R-module. 3) I is finitely generated and flat as an R-module. 4) I is strictly v-finite and flat as an R-module. b) Show that a regular fractional ideal of a Mori strictly t-Marot ring is invertible if and only if it is flat. c) Conclude that a nonzero fractional ideal of a Mori domain is invertible if and only if it is flat. 21. (∗∗) Let  be a semistar operation on a ring R. Let us say that R is strictly -Marot if for every regular ideal I of R there exists a regularly generated ideal J ⊆ I of R such that I  = J  . Does there exist -Marot ring that is not strictly -Marot, for some semistar operation ? Section 2.8 1. Prove Proposition 2.8.3. 2. Let S and T be subsets of a ring R. Prove the following. a) gcd(∅) = 0 and lcm(∅) = 1 exist. b) gcd(S ∪ {1}) = 1 and lcm(S ∪ {0}) = 0 exist. c) gcd(S ∪ {0}) = gcd(S) if and only if gcd(S) exists. d) lcm(S ∪ {1}) = lcm(S) if and only if lcm(S) exists. e) gcd(S ∪ T ) = gcd(gcd(S), gcd(T )) if gcd(S), gcd(T ), and gcd(gcd(S), gcd(T )) exist. f) lcm(S ∪ T ) = lcm(lcm(S), lcm(T )) if lcm(S), lcm(T ), and lcm(lcm(S), lcm(T )) exist. 3. Let S be a subset of a ring R. Prove that if the ideal S R = (d) is principal, then d is a gcd of S. Show by example that a gcd of S might exist even though the ideal S R is not principal. 4. Give direct proofs, from first principles, of the following. a) Any UFD is a strong GCD domain. b) Any strong GCD domain is completely integrally closed. c) Any Bézout domain is a GCD domain. d) Any GCD domain is integrally closed.

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5. Verify the examples in Example 2.8.2. 6. Let  be a semistar operation on a ring R. Show that R is -Bézout if and only if (a, b) is principal for all a, b ∈ R with a regular. 7. Prove Corollary 2.8.8. 8. Prove Proposition 2.8.13. 9. Prove Propositions 2.8.15–2.8.17. 10. Prove Lemma 2.8.18 and Proposition 2.8.19. 11. Let D be a UFD. Show that I v and I t are principal for all fractional ideals I of D. 12. Let R be a ring in which every regular non-unit factors as a product of prime elements. Show that every prime ideal of R that is minimal among the regular prime ideals of R is principal. Conclude that every height one prime of a UFD is principal. 13. Let a and b be regular elements of a ring R. a) Suppose that m = lcm(a, b) exists. Prove that m is regular and ab/m is a gcd of a and b. b) Let R = k[X 2 , X 3 ], where k is a field. Show that gcd(X 2 , X 3 ) = 1 exists and yet lcm(X 2 , X 3 ) does not exist. c) Show that in the domain Z[2i] one has gcd(2, 2i) = 1 and yet lcm(2, 2i) does not exist. √ d) Find two elements in the Dedekind domain Z[ −5] whose gcd exists even though their lcm does not. e)  Let R = Z[2X, X 2 , X 3 ] be the subring of Z[X ] consisting of all polynomials ∞ k k=0 ck X in Z[X ] such that c1 is even. Show that gcd(2, 2X ) = 1 exists and yet lcm(2, 2X ) does not exist. 14. a) Let p be an invertible prime ideal of a ring R. Prove that gcd(p) = p if p = ( p) is principal and gcd(p) = 1 otherwise. Prove in the latter case that, if R is v-Marot, then there exists a regular c ∈ R such that gcd(cp) does not exist. b) In a Dedekind ring, every regular ideal is a product of prime ideals. Use this fact and part (a) to prove that a v-Marot ring R is an r-PIR if and only if R is a Dedekind ring such that gcd(cS) exists for all regular c ∈ R whenever gcd(S) exists, for any subset S of R. 15. Let a and b be elements of a ring R, and let S be a subset of R. Prove the following. a) gcd(a, b) exists if and only if (a) and (b) have a least upper bound in the poset Principal(R) of all principal ideals of R. b) lcm(a, b) exists if and only if (a) and (b) have a greatest lower bound in the poset Principal(R). c) gcd(S) exists if and only if the set {(a) : a ∈ S} has a least upper bound in the poset Principal(R). d) lcm(S) exists if and only if the set {(a) : a ∈ S} has a greatest lower bound in the poset Principal(R).

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16. A poset is said to be lattice-ordered if every pair of elements has a least upper bound and a greatest lower bound. A poset is said to be complete if every subset has a least upper bound and a greatest lower bound. Prove the following. a) The following conditions are equivalent. 1) D is a GCD domain. 2) The poset Principal(D) of all principal ideals of D is lattice-ordered. 3) Every pair of elements of Principal(D) has a least upper bound. 4) Every pair of elements of Principal(D) has a greatest lower bound. b) The following conditions are equivalent. 1) R is a strong GCD domain. 2) gcd(S) exists for every subset S of R. 3) lcm(S) exists for every subset S of R. 4) The poset Principal(D) is complete. 5) Every subset of Principal(D) has a least upper bound. 6) Every subset of Principal(D) has a greatest lower bound. 17. A generalized GCD domain is an integral domain D in which the intersection of any two invertible ideals is invertible. Let D be an integral domain. Prove that the following conditions are equivalent. 1) D is a generalized GCD domain. 2) Any finite intersection of invertible ideals of D is invertible. 3) Any finite intersection of nonzero principal ideals of D is invertible. 4) (a) ∩ (b) is invertible for all nonzero a, b ∈ D. 5) I v is invertible for every nonzero finitely generated fractional ideal I of D. Section 2.9 1. 2. 3. 4. 5. 6. 7.

Prove Lemma 2.9.2. Show that a v-PID need not be a t-PID. Show that any domain satisfying ACCP is atomic. Verify the unproved assertions in the proof of Lemma 2.9.20. Prove Theorem 2.9.23. Prove Lemma 2.9.27 Verify the implications UFD or SPIR ⇒ BG-UFR ⇒ r-UFR and UFD or PIR ⇒ UFR ⇒ r-UFR .

8. Let Z[ε] = Z[X ]/(X 2 ), where ε is the image of X in Z[X ]/(X 2 ). Prove the following. a) 2 and 2 + ε are regular and irreducible, but not r-prime, in Z[ε], and therefore Z[ε] is not factorial. b) Z[ε] is not r-integrally closed, and in fact 21 (2 + ε) ∈ Q[ε] is regular and integral over Z[ε]. c) Z + εQ[ε] is the integral closure of Z[ε] and is an r-PIR.

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d) Z + εQ[ε] is the smallest r-integrally closed ring containing Z[ε]. 9. Show that the direct product R1 × R2 × · · · × Rn of rings R1 , R2 , . . . , Rn is factorial (resp., v-Marot, an r-UFR) if and only if the rings R1 , R2 , . . . , Rn are factorial (resp., t-Marot, r-UFRs). 10. It is known that, if D is an integrally closed domain, then the divisorial fractional ideals of D[X ] are the fractional ideals f I D[X ], where I is a divisorial ideal of D and f ∈ K (X ). Distil a proof of this result from [9] and present it as simply as possible. 11. Let D be an integrally closed domain with quotient field K . Use the previous exercise to prove the following. a) (I D[X ] : K (X ) J D[X ]) = (I : K J )D[X ] for all I, J ∈ Freg (D). b) (I D[X ])v = I v D[X ] and (I D[X ])t = I t D[X ] for all I ∈ Freg (D). c) I ∈ Freg (D) is t-invertible (resp., v-invertible, invertible, finitely generated) if and only if I D[X ] is t-invertible (resp., v-invertible, invertible, finitely generated) in D[X ]. d) D[X ] is Noetherian (resp., Mori, TV, H) if and only if D is. e) The association I −→ I D[X ] induces a group isomorphism Clt (D) −→ Clt (D[X ]). f) D[X ] is Krull (resp., UFD, GCD, PVMD, completely integrally closed, v-domain) if and only if D is. g) Clv (D[X ]) is trivial if and only if Clv (D) is trivial. 12. Let D be an integral domain with quotient field K . Prove the following. a) The t-maximal ideals of D[X ] all have the form pD[X ] for some t-maximal ideal p of D or f K [X ] ∩ D[X ] for some irreducible polynomial f ∈ K [X ]. (Note that the latter ideals need not all be t-maximal. A domain D for which all ideals of the form f K [X ] ∩ D[X ] for some irreducible polynomial f ∈ K [X ] are t-maximal is said to be UMT domain. It is known that a PVMD is equivalently an integrally closed UMT domain.) b) D[X ] is of finite t-character if and only if D is. c) D[X ] is of finite character if and only if D is a field. d) D[X ] is DW if and only if D is a field. (Hint: Suppose that D[X ] is DW. Let a ∈ D be nonzero. Show that (a, X )−1 = D[X ], so that (a, X ) = (a, X )w = D[X ].) 13. [37]. Let D be an integral domain with quotient field K = D, and let T = D + X K [X ]. Prove the following. a) The fractional ideals of T are the fractional ideals f I T = f (I + X K [X ]), where f ∈ K (X ) and I is a D-submodule of K . b) The ideals of T are the ideals f I T = f (I + X K [X ]), where I is a Dsubmodule of K and f ∈ K [X ] satisfies f (0) ∈ I −1 . c) The finitely generated fractional ideals of T are the ideals f I T , where I is a finitely generated ideal of D and f ∈ K (X ). d) The finitely generated ideals of T are the ideals f I T , where I is a finitely generated ideal of D and f ∈ T . e) (I T : K (X ) J T ) = (I : K J )T for all I, J ∈ Freg (D). f) (I T )v = I v T and (I T )t = I t T for all I ∈ Freg (D).

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g) I ∈ Freg (D) is t-invertible (resp., v-invertible, invertible, finitely generated) if and only if I T is t-invertible (resp., v-invertible, invertible, finitely generated) in T . h) The association I −→ I D[X ] induces a group isomorphism Clt (D) −→ Clt (T ). i) T is integrally closed (resp., a PVMD, Prüfer, GCD, Bézout, a v-domain) if and only if D is. j) T is not completely integrally closed; its complete integral closure is K [X ]. k) T is not atomic since X ∈ T cannot be written as a product of irreducible elements of T . l) Clv (T ) is trivial if and only if Clv (D) is trivial. 14. (∗∗) Generalize any parts of the previous four exercises to rings with zerodivisors. Section 2.10 1. 2. 3. 4. 5.

6. 7.

8.

9. 10. 11.

Prove Proposition 2.10.2. Show that any weak PIR satisfies all 13 conditions in Definition 2.10.3. Prove Proposition 2.10.12 and Corollary 2.10.13. Prove Proposition 2.10.15. Let R be a quasi-Marot ring. a) Show that rg is an equivalence relation on the poset of all semistar operations on R. b) Show that, if R is Marot, then    if and only if  rg  for any semistar operations  and  on R. Prove Proposition 2.10.22 and Corollary 2.10.23. Let  be a semistar operation on a ring R. a) Let I ∈ K(R) be regularly generated. Show that I is strictly -finite if and only if I  = H  for some H ∈ K(R) generated by finitely many regular elements of R in I . b) Prove Proposition 2.10.10. c) Prove Proposition 2.10.11. A poset is said to be lattice-ordered if every pair of elements has a least upper bound and a greatest lower bound. A poset is said to be complete if every subset has a least upper bound and a greatest lower bound. Show that the following conditions are equivalent. 1) R is a weak GCD (i.e., a weak t-Bézout) ring. 2) The poset Prin(R) of all principal regular fractional ideals of R is latticeordered. 3) Every pair of elements of Prin(R) has a least upper bound. 4) Every pair of elements of Prin(R) has a greatest lower bound. Let R be a ring with total quotient ring K . Show that R is r-integrally closed if and only if (I : K I )rg = R for all finitely generated I ∈ Freg (R). Use Exercise 2.7.14 to prove the unsubstantiated claims in Example 2.10.24. Find an example of two semistar operations  and  on a ring R such that  rg  even though    .

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Section 2.11 1. Let  be a semistar operation on a ring R. a) Verify that sup{I, J } = (I + J ) and inf{I, J } = I ∩ J for all I, J in the poset K(R) . b) Show that every subset (not necessarily finite) of K(R) has a least upper bound in K(R) and every subset of K(R) has a greatest lower bound in K(R) . 2. Show that lcm(a, b) gcd(a, b) = ab for all positive integers a, b, and generalize the result to any Bézout domain. 3. a) Verify directly that any r-PIR satisfies the equivalent condition of Proposition 2.11.1. b) Show that the ring Z[X ] is not Dedekind by verifying directly that it does not satisfy the equivalent condition of Proposition 2.11.1. c) Modify the proof of Theorem 2.11.2 to give a direct proof of Proposition 2.11.1. 4. Modify the proof of Theorem 2.11.2 to give a direct proof the following fact: A ring R is Prüfer if and only if, for all I, J ∈ Freg (R) with I finitely generated, one has I ⊇ J if and only if there exists H ∈ I(R) with J = H I . 5. Verify Corollaries 2.11.3 and 2.11.11. 6. Let R be a ring in which the set of all regular ideals of R is totally ordered. Show that R is a Prüfer ring with at most one regular maximal ideal. 7. Show that, in any Prüfer ring, the intersection of two finitely generated regular ideals is finitely generated. 8. A generalized GCD domain is an integral domain D in which the intersection of any two invertible ideals is invertible. Prove the following. a) A generalized GCD domain is equivalently a PVMD D such that Invt (D) = Inv(D), or equivalently such that Clt (D) = Cl(D). b) All Prüfer domains and all GCD domains are generalized GCD domains. 9. Generalize Theorem 2.11.2 to weak -Dedekind rings. 10. Show that, if R is Prüfer ring, then R/I is a weak Prüfer ring for any regular ideal I of R. Conclude that, if R is Prüfer ring, then R/p is a Prüfer domain for any regular prime ideal p of R.

Chapter 3

Semistar Operations on Commutative Rings: Local Methods

This chapter studies commutative rings by combining the theory of semistar operations developed in the previous chapter with more standard techniques of commutative algebra, specifically the tools of prime ideals, localization, large localization, valuation domains, Manis valuation rings, essential primes, weak Bourbaki associated primes, and strong Krull primes. Most of the results herein are known for integral domains, and some of them are also known for general commutative rings [10, 11, 14, 59, 60, 77, 93, 106–108, 118, 135, 141, 183].

3.1 Valuation Domains Besides localization and a variant known as large localization, introduced in Section 3.5, the main tools used in this chapter are that of valuation domains and generalizations to commutative rings known as (Manis) valuation rings, introduced in Section 3.6. In this section, we introduce valuation domains. Definition 3.1.1. Let D be an integral domain with quotient field K . (1) D is a valuation domain if for all nonzero x ∈ K either x or x −1 lies in D. (2) A prime ideal p of D is essential if Dp is a valuation domain. Example 3.1.2. (1) Trivially, any field is a valuation domain. (2) Recall that a DVR is a local PID. Any DVR D is a valuation domain. Indeed, any nonzero element x of the quotient field of D has the form uπ k for some unit u of D and some integer k, where (π) is the unique maximal ideal of D, and then x ∈ D if and only if k  0 while x −1 ∈ D if and only if k  0. (3) By example (2) and Proposition 2.2.16, every maximal ideal of a Dedekind domain is essential. © Springer Nature Switzerland AG 2019, corrected publication 2020 J. Elliott, Rings, Modules, and Closure Operations, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-24401-9_3

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Valuation domains are an extremely useful tool for studying integral domains. It is well-known, for example, that all valuation domains are integrally closed, and the integral closure of any integral domain is equal to the intersection of all of its valuation overrings. A proof of that fact, along with a generalization to integral closure of ideals, is provided in Section 4.4. The theory of valuation domains is quite rich. Both [70, Chapter 2] and [178, Chapter 6] are excellent introductions, while [192, Chapter VI and Appendices 2–5] provides a more extensive treatment of the theory. In this chapter, we need only a few very basic equivalent characterizations of the valuation domains and DVRs. The following result, especially the equivalence of (1) and (5), which is due to Krull, is fundamental. Theorem 3.1.3. Let D be an integral domain. The following conditions are equivalent. (1) (2) (3) (4) (5) (6)

D is a valuation domain. The set of all principal ideals of D is totally ordered. The set of all ideals of D is totally ordered. The set of all Kaplansky fractional ideals of D is totally ordered. D is a local Bézout domain. D is a local Prüfer domain.

Proof. The equivalence of (1)–(4) is straightforward, and clearly (2) and (3) together imply (5). To prove (5) ⇒ (2), suppose that D is a local Bézout domain, and let a and b be elements of D, not both zero. Then (a, b) = (c) for some nonzero c ∈ D, and writing a = ca  and b = cb we see that (a  , b ) = (1). Since D is local, it follows that either a  or b is a unit (for otherwise (a  , b ) is contained in the maximal ideal of D). Therefore a divides b or b divides a, so (a) ⊇ (b) or (b) ⊇ (a). Thus (5) ⇒ (2). Finally, (5) and (6) are equivalent, since by Proposition 2.2.7 any invertible ideal of a local ring is principal.  Corollary 3.1.4. Let p be a prime ideal of an integral domain D. The following conditions are equivalent. (1) p is essential, that is, Dp is a valuation domain. (2) Dp is a Bézout domain. (3) Dp is a Prüfer domain. Proposition 3.1.5. Let D be a domain. The following conditions are equivalent. (1) (2) (3) (4)

D is a DVR. D is a UFD with at most one irreducible element up to associates. D is a Noetherian valuation domain. The group Prin(D) of all nonzero principal fractional ideals of D is either trivial or isomorphic to Z.

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Proof. Statements (1) and (2) are easily seen to be equivalent, and clearly (1) implies (3). Moreover, (3) implies (1) since if D is a Noetherian valuation domain then D is a Noetherian Bézout domain, hence a PID, hence a DVR since D is local. Finally, we show that (2) and (4) are equivalent. We may suppose without loss of generality that D is not a field. Suppose that (2) holds and that π ∈ D is irreducible. It is easy to see, then, that Prin(D) is the infinite cyclic group {(π n ) : n ∈ Z}. Conversely, suppose that Prin(D) is infinite cyclic. Then the cylic group Prin(D) has precisely two possible generators, say, (π) and (π)−1 = (π −1 ). We may suppose without loss of generality that π ∈ D. Then every principal ideal of D is a nonnegative power of (π), so that (π) is the unique maximal proper principal ideal of D. In particular, the nonzero principal ideals of D are precisely the (distinct) ideals (1), (π), (π 2 ), . . .. Moreover, (a) = (π n ) if and only if a and π n are associated, for all a ∈ D and all nonnegative integers n. Therefore, every nonzero element of D can be written uniquely in the form uπ n for some unit u of D and some nonnegative integer n. Moreover, if π = ab, where, say, a = uπ n and b = vπ m , then 1 = uvπ n+m−1 , so n + m = 1, so either n = 0 or m = 0, so either a is a unit or b is a unit. Therefore π is irreducible. It follows that D is a UFD with only one irreducible element up to associates.  Definition 3.1.6. Let K be a field. A valuation on K is a function v : K −→  ∪ {∞} on K , where  is a totally ordered abelian group (where  is written additively, and where one sets γ + ∞ = ∞ + γ = ∞ = ∞ + ∞ and γ < ∞ for all γ ∈ ), satisfying the following three conditions for all x, y ∈ K : (1) v(0) = ∞ and v(1) = 0. (2) v(x y) = v(x) + v(y). (3) v(x + y)  min{v(x), v(y)}. For any valuation v on K , the set K v = {x ∈ K : v(x)  0} is a subring of K , and the set pv = {x ∈ K : v(x) > 0} is the unique maximal ideal of the domain K v . The value group of the valuation v is the totally ordered abelian group v(K × ) ⊆ . Note that v(x) = ∞ if and only if x = 0, since x = 0 implies that v(x) + v(1/x) = 0, whence v(x) = −v(1/x) = ∞. Since the value group v(K × ) of a valuation v : K −→  ∪ {∞} is a subgroup of , one can assume without loss of generality that a given valuation v on a field K is surjective, or equivalently that  is the value group of v. The following is another well-known result due to Krull.

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Proposition 3.1.7. A valuation domain is equivalently a domain D with quotient field K such that there exists a valuation v : K −→  ∪ {∞} on K such that D = K v . Moreover, when the latter condition holds, pv is the unique maximal ideal of D, and the map Prin(D) −→ v(K × ) acting by (x) −→ v(x) is a (well-defined) isomorphism of ordered abelian groups. Proof. Suppose that D is a valuation domain with maximal ideal m and quotient field K , and let  be the totally ordered group Prin(D) of all nonzero principal fractional ideals of D, ordered by ⊇. Then it is easy to check that they map v : K −→  ∪ {∞} acting by x −→ (x), where ∞ = (0) is a valuation on K with D = K v and m = pv . Conversely, suppose that v : K −→  ∪ {∞} is a valuation on K such that D = K v . Then D is a valuation domain, since if x = a/b ∈ K = T (D) is nonzero, then v(a)  v(b) or v(a)  v(b), so v(a/b)  0 or v(b/a)  0, so x ∈ D or x −1 ∈ D. Moreover, it is easy to check that the given map Prin(D) −→ v(K × ) is a well-defined isomorphism of ordered abelian groups.  Corollary 3.1.8. If v is a valuation on a field K , then K v is a valuation domain, and the value group v(K × ) of v is isomorphic as an ordered abelian group to Prin(K v ). Because of the proposition and corollary above, if V is a valuation domain, then the ordered group Prin(V ) is called the value group of V . By Propositions 3.1.5 and 3.1.7, we have the following. Corollary 3.1.9. A DVR is equivalently a valuation domain whose value group is order isomorphic to {0} or Z. Definition 3.1.10. Let K be a field. (1) A valuation v on K is discrete rank one if its value group is isomorphic to Z. (2) A discrete rank one valuation v : K −→  ∪ {∞} is normalized if  = Z and v is surjective. Corollary 3.1.11. Let D be an integral domain with quotient field K = D. Then D is a DVR if and only if there exists a discrete rank one valuation v on K such that D = K v . Moreover, in either case, there exists a unique such v on K that is normalized. The corollary above is well-known and fundamental to the theory of DVRs, Dedekind domains, and Krull domains. Indeed, the condition is often taken to be the definition of a DVR, and it explains where the name came from. Example 3.1.12. Let D be a UFD, and let  denote a complete set of non-associate primes of D. (For example, one may let D = Z and  the set of all prime numbers, or D = k[X ], where k is a field, and  the set of all monic  irreducible polynomials in k[X ].) For all nonzero x ∈ K , we may write x = u x π∈ π vπ (x) for a unique unit u x of D and unique integers vπ (x) for π ∈ , and we define vπ (0) = ∞. One can show that the functions vπ : K −→ Z ∪ {∞} are normalized discrete rank one valuations on K . Moreover, one has K vπ = D(π) , and therefore D(π) is a DVR, for every π ∈ .

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The proposition below provides further well-known equivalent characterizations of the DVRs. We do not need the result in our exposition; the interested reader may consult [178, Proposition 6.3.4] for a proof. Proposition 3.1.13. Let D be a local domain. The following conditions are equivalent. (1) (2) (3) (4) (5)

D is a DVR. D is Noetherian with principal maximal ideal. D is Noetherian and the only overrings of D are D and its quotient field. D is Noetherian, at most one dimensional, and integrally  closed. n The maximal ideal m of D is principal and satisfies ∞ n=0 m = (0).

The following result characterizes the valuation domains that are divisorial. The proof is left as an exercise. Proposition 3.1.14. Let V be a valuation domain with maximal ideal m, quotient field K = V , and value group  = Prin(V ). The following conditions are equivalent. (1) (2) (3) (4)

V is divisorial. m is divisorial. m is principal. m2 = m.

Moreover, if V is not divisorial, then the fractional ideals of V that are not divisorial are precisely the fractional ideals cm for c ∈ K × , and therefore they are in natural one-to-one correspondence with . Finally, the following well-known result characterizes the valuation domains that are one dimensional. See Exercise 3.1.19 for an outline of the proof. Proposition 3.1.15. Let V be a valuation domain. The following conditions are equivalent. (1) V is at most one dimensional. (2) The value group of V is order isomorphic to a subgroup of R. (3) V is completely integrally closed. In 1957, Samuel extended the technique of valuation domains to rings with zerodivisors in several ways [174], and in 1967 Manis singled out one of the most useful such notions, now known as (Manis) valuation pairs [140, 141], that can be used to extend the results of this section to rings with zerodivisors. This is carried out in Sections 3.6 and 3.10.

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3.2 -Prime Ideals and Applications to -Prüfer Domains In this section, we study the t -Prüfer domains (known in the literature as PMDs), where  is any semistar operation on an integral domain. Our main tools are localization and Theorem 3.2.3 below. Definition 3.2.1. Let R be a ring and O a collection of overrings of R whose intersection is R.  (1) O denotes the semistar operation on R given by I O = S∈O I S for all I ∈ K(R). (2) If  R is a domain and  ⊆ Spec R is a set of prime ideals of R such that R = {Rp : p ∈ }, then we set  = O , where O = {Rp : p ∈ }.  Definition 3.2.2. A domain D is essential if D = {Dp : p ∈ Spec D is essential}. Theorem 3.2.3. Let R be a ring and O a collection of overrings of R whose intersection is R. (1) If every S ∈ O is flat over R, then the semistar operation O is stable. (2) If I ∈ K(R) is O -invertible, then I S is invertible in S for all S ∈ O; moreover, the converse holds if I is finitely generated and every S ∈ O is flat over R. (3) If R is O -Prüfer, then S is Prüfer for all S ∈ O; moreover, the converse holds if every S ∈ O is flat over R. Proof. Statement (1) is Proposition 2.4.20. Then (I I −1 )O = R, so that, for Let I ∈ K(R). Suppose O -invertible.  that I is −1 −1 T S ⊆ (I I S)S ⊆ S, so that I S I −1 S = S all S ∈ O, one has S = T ∈O I I and thus I S is invertible in the ring S. Conversely, suppose that I is finitely generated, I S is invertible for all S ∈ O, and every S ∈ O is flatover R. Then, since I −1 S = (I S)−1by Corollary 2.2.12, we have (I I −1 )O = S∈O I I −1 S =  −1 = S∈O S = R, so I is O -invertible. This proves (2). S∈O (I S)(I S) Next, we prove (3). Suppose that R is O -Prüfer, let S ∈ O, and let I = (x1 , . . . , xn )S be a finitely generated regular fractional ideal of S. Let J = (x1 , . . . , xn )R, which is a finitely generated regular fractional ideal of R with J S = I . Since R is O -Prüfer, J is O -invertible, and therefore I = J S is invertible in S by (2). Therefore every finitely generated regular fractional ideal of S is invertible, so S is Prüfer. Conversely, suppose that all overrings in O are Prüfer and flat over R, and let I be a finitely generated regular fractional ideal of R. For all S ∈ O, since S is a Prüfer ring, the finitely generated regular fractional ideal I S of S is invertible. Therefore,  by (2), I is O -invertible. Thus, R is O -Prüfer. This proves (3). Since a local Prüfer domain is equivalently a valuation domain and every invertible ideal of a local ring is principal, we obtain the following corollary. Corollary  3.2.4. Let D be an integral domain and  a subset of Spec D such that D = {Dp : p ∈ }.

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(1)  is a stable semistar operation on D. (2) If I ∈ Freg (D) is  -invertible, then I Dp is principal (or equivalently, invertible) in Dp for all p ∈ , and the converse holds if I is finitely generated. (3) D is a  -Prüfer domain if and only if Dp is a valuation domain for all p ∈ , if and only if all of the primes in  are essential. In particular, any such domain D is an essential domain. (4) Therefore, a domain D is essential if and only if it is  -Prüfer for some set   ⊆ Spec D such that D = {Dp : p ∈ }, in which case all of the primes in  are essential. (5) In particular, every essential domain is a v-domain. Remark 3.2.5. Nagata in 1952–1955 famously provided an example of a one dimensional local completely integrally closed domain, hence a v-domain, that is not a valuation domain and is therefore not essential. Thus a v-domain need not be essential. We would like to apply Corollary 3.2.4 to various distinguished subsets of Spec(D). For this purpose, the following definition is crucial. Definition 3.2.6. Let R be ring, let  be a semistar operation on R, and let p be an ideal of R. (1) (2) (3) (4)

p is -prime if p is a -closed prime ideal of R. p is -maximal if p is maximal among the proper -closed ideals of R. The set of all -prime ideals of R is denoted -Spec(R). The set of all -maximal ideals of R is denoted -Max(R).

Note that an ideal I of a ring R is d-prime if and only if I is a prime ideal of R, and I is d-maximal if and only if I is a maximal ideal of R. It is well-known that every maximal ideal of a ring is prime, and every proper ideal of a ring is contained in at least one maximal ideal. The two lemmas below generalize these two facts to arbitrary semistar operations. Lemma 3.2.7. Let  be a semistar operation on a ring R. Every -maximal ideal of R is prime, and therefore -Max(R) ⊆ -Spec(R). Proof. Let p be a -maximal ideal of R. Suppose that ab ∈ p but a ∈ / p. Let I = p + a R, so that bI ⊆ p, and also I  = R since I  is a -closed ideal of R properly containing p. Therefore b R = bI  ⊆ p = p, whence b ∈ p. Therefore p is prime.  Lemma 3.2.8. Let  be a semistar operation on a ring R. (1) Suppose that  is of finite type. Then every proper -closed ideal of R is contained in a -maximal ideal of R. In particular, -Spec(R) is empty if and only if -Max(R) is empty, if and only if R has no proper -closed ideals, if and only if (0) = R.

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(2) Suppose that  is of r-finite type. Then every proper regular -closed ideal of R is contained in a regular -maximal ideal of R. In particular, -Spec(R)reg is empty if and only if -Max(R)reg is empty, if and only if R has no proper regular -closed ideals, and if and only if R is a total quotient ring. Proof. We prove the first statement using Zorn’s lemma. Let H be a proper -closed ideal of R. Let I be the union of a chain {Iλ }λ∈ of proper -closed ideals of R containing H . If J ⊆ I is finitely generated, then J is contained in some Iλ and so J  ⊆ Iλ ⊆ I . Taking the union over all such finitely generated J ⊆ I , we see that I  ⊆ I , and therefore I is a proper -closed ideal of R containing H . The first statement, then, follows from Zorn’s lemma. Statement (2) is proved similarly.  One reason that localization is particularly suited to the study of integral domains is that, for any integral domain D, the localization map D −→ Dp is injective, so, in particular,  Dp is an overring of D, for  every prime ideal p of D, and moreover one has D = {Dp : p ∈ Max(D)} = {Dp : p ∈ Spec(D)}. This well-known and very useful fact can be translated into the language of semistar operations as a fact about the trivial semistar operation d, namely, that d = Max(D) = Spec(D) , where here we use the notation of Definition 3.2.1. This generalizes as follows. Theorem 3.2.9.  Let  be a semistar operation on an integral domain D. Then, one has D = {Dp : p ∈ t -Max(D)} and w= t -Max(D) , and also w -Max(D) = t -Max(D). In other words, one has I w = {I Dp : p ∈ t -Max(D)} for all I ∈ K(D), and I ∈ I(D) is w -maximal if and only if I is t -maximal. Proof. We may suppose  without loss of generality that  = t is of finite type. First,  = {I  Dp : p ∈ -Max(D)} for all I ∈ F(D), from which it folwe claim that I  lows that D = {Dp : p ∈ -Max(D)}. To prove  the claim, it suffices to assume that I is a -closed ideal of D and show that {I Dp : p ∈ -Max(R)} ⊆ I . Suppose x ∈ K lies in the given intersection, where K is the quotient field of D. For each p ∈ -Max(D), we can write bp x = ap with ap ∈ I and bp ∈ D − p. Then, since bp ∈ (I : D x) − p, we have (I : D x) Ę p, for all p ∈ -Max(R). It follows from Lemma 3.2.8 that (I : D x) = D. But since (I : K x) is -closed, we have D = (I : D x) = ((I : K x) ∩ D) ⊆ (I : K x) ∩ D = (I : K x) ∩ D = (I : D x), and so x ∈ I. D = (I : D x) and therefore  : p ∈ -Max(D)}, by Proposition 2.4.20 it follows that Now, since D = {D p  I −→ I c = {I Dp : p ∈ -Max(D)} defines a stable semistar operation c on D. By our claim proved above, we have (I  )c = I  for all I ∈ F(D) and therefore c  . But then since c is stable we have also c   = w . Now, if J  = D, or equivalently if J w = D, where J ∈ F(D), then we claim that J c = D. For suppose that J c = D. Then there exists x ∈ D − J Dp for some -maximal ideal p of D. Then (J : D x) ⊆ p, so D = (D : D x) = (J w : D x) = (J : D x)w ⊆ p, which is a contradiction. Now, let I, J ∈ F(D) with J finitely generated and J  = D. Then since J c = D, we have (I : K J ) ⊆ (I : K J )c ⊆ (I c : K J ) = (I c : K J c ) = (I c : K D) = I c . Taking the union over all such J , we see that I w ⊆ I c . Therefore w  c, so equality holds.

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Finally, since w = c and  are both of finite type and J  = D if and only if J = D, we see from Lemma 3.2.8 that p ∈ I(D) is -maximal if and only if p is w -maximal. Indeed, if p is w -maximal, then pw = D, so p = D, so p is contained in some -maximal ideal of D. Likewise, if p is -maximal, then p = D, so pw = D, so p is contained in a w -maximal ideal of D. It follows, then, that the -maximal  ideals and w -maximal ideals of D coincide with one another. w

By the theorem above and Proposition 2.5.16, one has the following. Corollary 3.2.10. Let  be a semistar operation on an integral domain D. (1) (2) (3) (4)

d = Max(D) = Spec(D) . w = t -Max(D) . D is t -Prüfer if and only if D is w -Prüfer, if and only if D is t -Max(D) -Prüfer. In particular, one has w = t-Max(D) , and D is a PVMD if and only if D is w-Prüfer, if and only if D is t-Max(D) -Prüfer. Combining Corollaries 3.2.4 and 3.2.10, we obtain the following.

Theorem 3.2.11. Let  be a semistar operation on an integral domain D. The following conditions are equivalent. (1) D is a t -Prüfer domain. (2) Dp is a valuation domain for every p ∈ t -Spec(D), that is, every t -prime ideal of D is essential. (3) Dp is a valuation domain for every p ∈ t -Max(D), that is, every t -maximal ideal of D is essential.  Moreover, since D = {Dp : p ∈ t -Max(D)}, every t -Prüfer domain is essential. Corollary 3.2.12. Let D be an integral domain. The following conditions are equivalent. (1) D is a PVMD. (2) Dp is a valuation domain for every p ∈ t-Spec(D). (3) Dp is a valuation domain for every p ∈ t-Max(D). Corollary 3.2.13. Let D be an integral domain. The following conditions are equivalent. (1) D is a Prüfer domain. (2) Dp is a valuation domain for every prime ideal p of D. (3) Dp is a valuation domain for every maximal ideal p of D. Thus, a Prüfer domain is equivalently a domain that is locally a valuation domain, and a PVMD is equivalently a domain that is “t-locally” a valuation domain. By the results in this section, the following implications hold: PVMD ⇒ essential ⇒ v-domain ⇒ integrally closed. In fact, none of the implications above is reversible [68].

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3.3 Associated Primes, Strong Krull Primes, and t-Primes In this section, we consider again rings with zerodivisors, studying, in particular, the t-primes and various notions of associated primes of a ring. This proves useful for the study of the PVMDs, Krull domains, Dedekind domains, and their generalizations to rings with zerodivisors, which we undertake in Sections 3.4–3.11. Let R be a ring with total quotient ring K . Recall that an R-module M is R-torsionfree if and only if the map M −→ K ⊗ R M is injective. Moreover, an R-algebra S is R-torsion-free as an R-module if and only if every regular element of R is a regular element of S. (For example, any flat R-algebra is R-torsion-free.) In this situation, the total quotient ring L of S is a K -algebra, and therefore it makes sense for any I ∈ K(R) to compare I −1 , I t , I v , etc., as computed in K , with (I S)−1 , (I S)v , (I S)t , etc., respectively, as computed in L. Generally, whether we intend these to be computed in K or in L is clear from the context. Definition 3.3.1. Let R be a ring, and let S be an R-torsion-free R-algebra. (1) S is a t-compatible extension of R if I t S ⊆ (I S)t , or equivalently if (I S)t = (I t S)t , for all I ∈ Freg (R). (2) S is a semistar t-compatible extension of R if I t S ⊆ (I S)t , or equivalently if (I S)t = (I t S)t , for all I ∈ K(R). (3) S is a t-linked extension of R if I t = R implies (I S)t = S, for all I ∈ Freg (R). (4) S is a semistar t-linked extension of R if I t = R implies (I S)t = S, for all I ∈ K(R). Proposition 3.3.2. Let R be ring, and let S be an R-torsion-free R-algebra. Each of the following conditions implies the next. (1) (2) (3) (4) (5) (6) (7) (8) (9)

S is flat as an R-module. I −1 S = (I S)−1 for all finitely generated I ∈ Freg (R). (I −1 S)v = (I S)−1 for all finitely generated I ∈ Freg (R). S is a t-compatible extension of R. If J is a regular t-closed ideal of S and J ∩ R is regular, then J ∩ R is a t-closed ideal of R. S is a t-linked extension of R. I t S ⊆ (I S)t , or equivalently (I S)t = (I t S)t , for all t-invertible I ∈ Freg (R). There is a group homomorphism Invt (R) −→ Invt (S) induced by the map I −→ (I S)t from Freg (R)t to Freg (S)t . There is a group homomorphism Clt (R) −→ Clt (S) induced by the map I −→ (I S)t from Freg (R)t to Freg (S)t .

In fact, (4) and (5) are equivalent and (6) and (7) are equivalent. Proof. The implication (1) ⇒ (2) follows immediately from Proposition 2.2.11, and (2) ⇒ (3) is obvious. Suppose that (3) holds. Let J be a finitely generated regular subideal of a regular ideal I of R. Then (J −1 S)v = (J S)−1 , so J v = (J −1 )−1 ⊆

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(J −1 S)−1 = ((J −1 S)v )−1 = ((J S)−1 )−1 = (J S)v = (J S)t ⊆ (I S)t . Taking the union over all such J , we see that I t S ⊆ (I S)t . Therefore (3) ⇒ (4). Next, suppose that (4) holds and J is a regular t-closed ideal of S such that J ∩ R is regular. Then (J ∩ R)t ⊆ ((J ∩ R)S)t ∩ R ⊆ J t ∩ R = J ∩ R, and therefore J ∩ R is a t-closed ideal of R. Therefore (4) implies (5). Conversely, suppose that (5) holds and I is a regular ideal of R. Then (I S)t is t-closed ideal of S such that (I S) ∩ R ⊇ I is regular, so (I S)t ∩ R is a regular t-closed ideal of R and therefore I t ⊆ (I S)t ∩ R. Thus (5) implies (4), so (4) and (5) are equivalent. Finally, the implications (4) ⇒ (6) ⇔ (7) ⇒ (8) ⇒ (9) are easily verified.  Example 3.3.3. (1) By Exercise 1.1.10, the overring D  = Z[X/2] of the domain D = Z[X ] is not a t-linked extension of Z[X ]. However, trivially there exists a group homomorphism Prin(D) = Invt (D) −→ Prin(D  ) = Invt (D  ) induced by the map I −→ (I D  )t = I D  from Freg (D)t to Freg (D  )t . Thus, condition (8) of the proposition does not imply conditions (6) or (7). (2) [51, Example 4.12] provides an example of a t-linked overring of an integral domain that is not t-compatible. Thus, conditions (6)–(8) of the proposition do not imply conditions (4) or (5). (3) The papers [7, 47, 52] study various conditions, some stronger, some weaker, than t-compatibility and t-linkedness, that imply condition (9) of the proposition. (4) It is straightforward to show that any t-linked overring of a PVMR is a PVMR. (See Exercise 3.3.14.) In general, t-closure does not commute with localization. In fact, if p is a regular t-prime of a domain D, then pDp need not be a regular t-prime of Dp : see Example 3.3.5 below. However, by the following corollary, the converse does hold. Corollary 3.3.4. Let R be a ring, and let p be a regular prime ideal of R. Then pRp is a regular prime of Rp . Moreover, if pRp is t-closed in Rp , then p is t-closed in R. In particular, if pRp ∈ t-Max(Rp ), then p ∈ t-Spec(R). Example 3.3.5. If D is an integral domain, then a prime ideal p of D such that pDp is t-closed (hence t-maximal) in Dp is said to be well behaved, or t-localizing. There are domains with t-primes that are not well behaved. For example, if V is a valuation domain of dimension greater than one and q is any nonzero non-maximal prime ideal of V , then by [189, Section 2] the prime ideal P = q + X Vq [X ] of the domain D = V + X Vq [X ] is a nonzero t-prime of D even though PDP is not t-closed in DP . We note the following analogue of Proposition 3.3.2. Proposition 3.3.6. Let R be ring, and let S be an R-torsion-free R-algebra. Each of the following conditions implies the next.

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(1) (2) (3) (4) (5) (6) (7) (8)

S is flat overring of R. I −1 S = (I S)−1 for all finitely generated I ∈ K(R). (I −1 S)v = (I S)−1 for all finitely generated I ∈ K(R). S is a semistar t-compatible extension of R. If J is a t-closed ideal of S, then J ∩ R is a t-closed ideal of R. S is a semistar t-linked extension of R. I t S ⊆ (I S)t , or equivalently (I S)t = (I t S)t , for all t-invertible I ∈ K(R). There is a group homomorphism Invt (R) −→ Invt (S) induced by the map I −→ (I S)t from K(R)t to K(S)t . (9) There is a group homomorphism Clt (R) −→ Clt (S) induced by the map I −→ (I S)t from K(R)t to K(S)t . In fact, (4) and (5) are equivalent and (6) and (7) are equivalent. The t-prime ideals of a ring R with total quotient ring K generalize the strong Krull primes of the R-module K /R, which in turn generalize the weak Bourbaki primes of K /R, both defined below. Definition 3.3.7. Let R be a ring, M an R-module, and p a prime ideal of R.

(1) p is a Bourbaki associated prime of M if p equals the annihilator ann R (m) = (0 : R m) of some element m of M. (2) p is a weak Bourbaki associated prime of M if p is minimal over the annihilator of some element of M. (3) p is a strong Krull prime, or Northcott attached prime, of M if for every finitely generated ideal I ⊆ p there exists an m ∈ M such that I ⊆ ann R (m) ⊆ p [42]. The sets of all such primes p of R are denoted Ass R (M), wAss R (M), and sK R (M), respectively, or Ass(M), wAss(M), and sK(M), respectively, if the ring R is understood. Proposition 3.3.8. Let R be a ring and M an R-module. One has Ass(M) ⊆ wAss(M) ⊆ sK(M), and the finitely generated ideals of Ass(M), wAss(M), and sK(M) coincide. Proof. We show that wAss(M) ⊆ sK(M); the rest of the proposition then follows. Let p ∈ wAss(M), so there exists an m ∈ M such that p is minimal over ann R (m). Then pRp is the only prime of Rp containing ann R (m)Rp = ann Rp (m) and thus  pRp = ann Rp (m). Let I be any finitely generated ideal contained in p. Then I Rp  is a finitely generated ideal contained in ann Rp (m), and therefore some power I k Rp of I Rp is contained in ann Rp (m). Let k be least such that I k Rp m = 0, so that I k−1 Rp m = 0. Then there exists a nonzero element n of I k−1 Rp m ⊆ Mp such that I Rp n = 0, say, n = (a/v)m, where a ∈ I k−1 and v ∈ R − p. Since I Rp am = 0 and I is finitely generated, there exists a u ∈ R − p such that I uam = 0. Then

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I ⊆ ann R (uam). Moreover, if r ∈ ann R (uam), then r n = 0, whence r ∈ p since n = 0 in Mp . Therefore I ⊆ ann R (uam) ⊆ p. It follows, then, that p ∈ sK(M).  Note that, if M = 0, then wAss(M) = ∅, but Ass(M) may be empty. For Noetherian rings, the sets Ass(M), wAss(M), and sK(M) coincide, but for non-Noetherian rings the latter two sets are the more fundamental of the three. Remark 3.3.9. A prime p of R is said to be a Krull prime of M if for every a ∈ p there exists an m ∈ M such that a ∈ ann R (m) ⊆ p. Every strong Krull prime of M is a Krull prime of M. We do not require use of the Krull primes. Definition 3.3.10. Let R be a ring with total quotient ring K . A conductor ideal of R is an ideal of R of the form (R : R a/b) = (b R : R a R) = (1, a/b)−1 for some a, b ∈ R with b regular. Equivalently, a conductor ideal of R is an ideal of the form ann R (x) for some x ∈ K /R. Clearly, a Bourbaki associated prime of K /R is equivalently a prime conductor ideal of R, a weak Bourbaki associated prime of K /R is equivalently a prime ideal that is minimal over some conductor ideal of R, and a strong Krull prime of K /R is equivalently a prime ideal p such that for every finitely generated ideal I ⊆ p there exists a conductor ideal J of R such that I ⊆ J ⊆ p. Note also that any conductor ideal is regular and therefore any strong Krull prime of K /R is regular. Proposition 3.3.11 (cf. [189, Proposition 1.1]). Let R be a ring with total quotient ring K and let p be a prime ideal of R. Then p ∈ sK(K /R) if and only if p is regular and pRp ∈ t-Max(Rp ). Moreover, if p ∈ sK(K /R), then p ∈ t-Spec(R)reg . In other words, one has sK(K /R) = {p ∈ Spec(R)reg : pRp ∈ t-Spec(Rp )} ⊆ t-Spec(R)reg . Proof. Suppose that p ∈ sK(K /R). Then p contains a conductor ideal and is therefore regular. We prove that H v ⊆ pRp for every finitely generated subideal H of pRp , so that pRp is t-closed and therefore t-maximal. Let H be any finitely generated subideal of pRp . We may write H = (x1 /u, . . . , xn /u)Rp , where u ∈ R − p and xi ∈ p for all i. Let I = (x1 , . . . , xn )R, so that I is a finitely generated subideal of p with I Rp = u H regular in Rp . Since p ∈ sK(K /R), there exists a conductor ideal J = (1, a/b)−1 for some a, b ∈ R with b regular such that I ⊆ J ⊆ p. Extending to Rp , we have I Rp ⊆ J Rp = ((1, a/b)Rp )−1 ⊆ pRp , and applying v-closure on Rp we get (I Rp )v ⊆ (((1, a/b)Rp )−1 )v = ((1, a/b)Rp )−1 and therefore (I Rp )v ⊆ pRp . Therefore, since H = u −1 I Rp , we have H v = u −1 (I Rp )v ⊆ pRp as well, since u is a unit in Rp . Conversely, suppose that p is regular and pRp is t-maximal in Rp . Let I be a finitely generated subideal of p. Since p is regular, there is a c ∈ p with c regular, and by replacing I with I + c R we may assume without loss of generality that I is regular too. Now, (I Rp )v ⊆ pRp . Since then (I Rp )v  Rp , one has I −1 Rp = (I Rp )−1  Rp , so there exists x ∈ I −1 such that x Rp Ę Rp . Write x = a/b with a, b ∈ R and b

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regular, so that a I ⊆ b R since x I ⊆ R. Thus I ⊆ (b R : R a R). Moreover, if (b R : R a R) Ę p, then ua R ⊆ b R for some u ∈ R − p, and then x Rp ⊆ Rp , which is a contradiction. Therefore I ⊆ (b R : R a R) ⊆ p. Thus, since I was an arbitrary finitely generated subideal of p, we have p ∈ sK(K /R). Finally, if p ∈ sK(K /R), then p is regular and pRp is t-closed in Rp , so that p is t-closed in R by Corollary 3.3.4.  Corollary 3.3.12. If R is a ring with total quotient ring K , then one has Ass(K /R) ⊆ wAss(K /R) ⊆ sK(K /R) = {p ∈ Spec(R)reg : pRp ∈ t-Max(Rp )} ⊆ t-Spec(R)reg . Definition 3.3.13. For any integral domain D, let Ess(D) = {p ∈ Spec(D) : Dp is a valuation domain} denote the set of all essential primes of D. Corollary 3.3.14. Every nonzero essential prime of an integral domain D with quotient field K is a strong Krull prime of K /D and is therefore t-prime. In other words, one has Ess(D)reg ⊆ sK(K /D) ⊆ t-Spec(D)reg . Proof. If p is a nonzero prime of D such that Dp is a valuation domain, then every ideal of Dp is t-closed, and therefore pDp is a maximal, hence t-maximal, ideal of  Dp . The result therefore follows from the proposition. By Corollary 3.2.12 and the corollary above, we have the following. Corollary 3.3.15. Let D be an integral domain with quotient field K . One has Ess(D)reg ⊆ sK(K /D) ⊆ t-Spec(D)reg , and equalities hold if and only if D is a PVMD. In other words, every nonzero essential prime of D is a t-prime of D (and in fact a strong Krull prime of K /D), and the converse holds if and only if D is a PVMD. Recall that a ring R is said to be an H ring if every regular v-invertible fractional ideal of R is t-invertible. All Noetherian or Krull rings are Mori, all Mori rings are TV, and all TV rings are H rings. Our next proposition reveals how the t-primes can be useful for controlling the various associated primes of an H ring. Lemma 3.3.16. Let R be a ring with total quotient ring K , and let  be a semistar operation on R, and let p ∈ -Max(R). Then the following conditions are equivalent. (1) (2) (3) (4)

p is v-closed. p is v-maximal. pv = R. p ∈ Ass(K /R).

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195

Proof. Clearly (1) ⇒ (2) ⇒ (3), and one has (4) ⇒ (1) since every conductor ideal, and therefore every element of Ass(K /R), is v-closed. Suppose that (3) holds. Then p−1  R, so there exists x ∈ K − R such that xp ⊆ R. Therefore p ⊆ (R : R x)  R. Moreover, (R : R x) is v-closed, hence -closed, and therefore p = (R : R x) since p is -maximal. Therefore (3) ⇒ (4).  Corollary 3.3.17. Let R be a ring with total quotient ring K . Then, one has v-Max(R) ⊆ Ass(K /R) ⊆ v-Spec(R). Note that Ass(T (R)/R) (and v-Max(R)) and may be empty even if R is a one dimensional Bézout domain: see Exercise 3.3.16. Proposition 3.3.18. Let R be a ring with total quotient ring K . The following conditions are equivalent. (1) (2) (3) (4) (5) (6)

R is an H ring. I v = R if and only if I t = R, for all I ∈ Freg (R). If I v = R, then I is strictly v-finite, for all I ∈ Freg (R). Every regular t-maximal ideal of R is v-closed. Every regular t-maximal ideal of R is v-maximal. t-Max(R)reg ⊆ Ass(K /R).

Proof. The equivalence of (1)–(3) is easy to verify, and the equivalence of (4)–(6) follows from Lemma 3.3.16. To prove (2) ⇒ (4), let p be a regular t-maximal ideal of R. Then pt = p = R, so pv = R. It follows from Lemma 3.3.16, then, that p is v-closed. Finally, we prove that (4) ⇒ (2). Suppose that (4) holds and I ∈ Freg (R) satisfies I v = R. Then I Ę p for every (v-closed) t-maximal ideal p of R, whence  I t = R by Lemma 3.2.8. Corollary 3.3.19. Let R be a ring with total quotient ring K . Then R is an H ring if and only if t-Max(R)reg ⊆ Ass(K /R) ⊆ wAss(K /R) ⊆ sK(K /R) ⊆ t-Spec(R)reg . Moreover, equalities hold if and only if R is an H ring and every regular t-prime ideal of R is t-maximal. In Theorem 3.3.23 below we show that any TV ring is not only an H ring, it is also of finite t-character, which is a useful property defined below. This important and nontrivial result was first proved for integral domains in [104, Theorem 1.3]. Here we generalize that proof to rings with zerodivisors. As with the H property, the finite t-character property is useful in our study, in the next section, of the Krull domains and Dedekind domains. Definition 3.3.20. An indexed collection {Rλ : λ ∈ } of algebras over a ring R is locally finite if every regular element of R is a non-unit in Rλ for only finitely many λ ∈ .

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Example 3.3.21. A collection {Rp : p ∈ } of localizations of a ring R, where  ⊆ Spec(R), is locally finite if and only if every regular element of R lies in only finitely many of the primes in . For example, it is well-known that, if D is a Dedekind domain, then {Dp : p ∈ Max(D)} is locally finite. Definition 3.3.22. Let R be a ring. (1) R is of finite character if {Rp : p ∈ Max(R)} is locally finite, that is, if every regular element of R lies in only finitely many maximal ideals of R. (2) R is of finite t-character if {Rp : p ∈ t-Max(R)} is locally finite, that is, if every regular element of R lies in only finitely many t-maximal ideals of R. Note that, if R is a TD ring (for example, if R is Prüfer), then R is of finite character if and only if R is of finite t-character. However, in general, neither condition implies the other. Theorem 3.3.23 (cf. [104, Theorem 1.3]). Any TV ring is of finite t-character. Proof. Let a be a regular element of R, and let denote the set of all t-maximal ideals of R that contain a. For each p ∈ , let Ip = q∈−{p} q. We claim that Ip Ę p for all p ∈ . Assuming, for the moment, that this claim is true, we complete the  proof. Let I = p∈ Ip , so that a ∈ I Ę p for all p ∈ . Since I contains a but is not contained in any t-maximal ideals of R containing a, it follows that I is not in any  t-maximal ideals of R, and therefore I t = R. Therefore 1 ∈  contained k t Ipi )t for some some finite subset {p1 , . . . , pk } of . If ( p∈ Ip ) , so 1 ∈ ( i=1 k k Ipi and therefore p ⊇ ( i=1 Ipi )t = R, which p ∈  − {p1 , . . . , pk }, then p ⊇ i=1 is a contradiction. Therefore  = {p1 , . . . , pk } is finite. Finally, we prove our claim, namely, that Ip Ę p for all p ∈ . Let J be any ideal of R not contained in p. Then we have (p + J )t = R since p is t-maximal. It follows that (p + J )v = R, and therefore p−1 ∩ J −1 = (p + J )−1 = R. Since p is t-closed, hence v-closed, we have pv  R and therefore p−1  R, so we may choose / J −1 for any ideal J not contained in p. Note that x ∈ p−1 − R, so that x ∈ Ip =



⎛



(q−1 )−1 = ⎝

q∈−{p}

⎞−1 q−1 ⎠

q∈−{p}

−1  because each q ∈  ⊆ t-Max(R) is v-closed (since R is TV) and aλ = λ  −1  λ aλ for any set of ideals aλ . Moreover, the Kaplansky fractional ideal q∈−{p} q−1 is regular and fractional and therefore ⎛ Ip−1 = ⎝



q∈−{p}

⎞v

⎛

q−1 ⎠ = ⎝



⎞t q−1 ⎠ .

q∈−{p}

We claim that x ∈ / Ip−1 , and therefore p−1 Ę Ip−1 , whence Ip Ę p, as claimed. Sup pose otherwise, that is, suppose that x ∈ Ip−1 = ( q∈−{p} q−1 )t . Then x ∈ H v

3.3 Associated Primes, Strong Krull Primes, and t-Primes

197

 −1 for some finitely generated regular R-submodule H of q∈−{p} q , whence  n n qi )−1 for some finite subset {q1 , . . . , qn } of  − {p}. Let x ∈ ( i=1 qi−1 )v = ( i=1 n J = i=1 qi , so that x ∈ J −1 . But since qi Ę p for all i, we have J Ę p since p is prime, and therefore, as shown earlier, we must have x ∈ / J −1 . This is our desired contradiction.  Thus, we have proved the implications Mori ⇒ TV ⇒ H and of finite t-character. In fact, these implications are irreversible, even for Bézout domains. For example, as noted in Example 1.2.8, if D is a nonlocal semilocal PID with quotient field K = D, then D + X K [X ] is a Bézout H domain of finite (t-)character that is not a TV domain, and if D is a valuation domain with principal maximal ideal but is not a DVR, then D is a local Bézout domain that is TV but not Mori. Proposition 3.3.24. Let R be a ring with total quotient ring K . The t-finite primes of Ass(K /R), wAss(K /R), and sK(K /R) coincide. In particular, if R is Mori, then one has Ass(K /R) = wAss(K /R) = sK(K /R). Proof. Let p ∈ sK(K /R) be t-finite. Since p is also t-closed, one has p = I t for some finitely generated ideal I of R. Since p ∈ sK(K /R), one has I ⊆ (R : R x) ⊆ p for some x ∈ K and therefore p = I t = (R : R x)t = (R : R x), whence p ∈ Ass(K /R).  We also note the following. Proposition 3.3.25. Any flat overring of a Mori ring is Mori. Proof. Let S be a flat overring of a Mori ring R, and let I1 ⊆ I2 ⊆ · · · be an ascending chain of regular t-closed ideals of S. Let Jn = In ∩ R for all n. Then J1 ⊆ J2 ⊆ · · · is an ascending chain of regular t-closed ideals of R by Proposition 3.3.2, and therefore the chain stabilizes since R is Mori. Moreover, by Proposition 2.2.10, one has Jn S = (In ∩ R)S = In S ∩ RS = In for all n, and therefore the chain I1 ⊆ I2 ⊆ · · · stabilizes as well.  The proposition implies that any localization of a Mori ring R at a multiplicative subset of regular elements of R is Mori. However, a localization of a Mori ring at an arbitrary prime ideal need not be Mori [18, Example 5.3]. Also, a similar argument shows that a flat overring of a Noetherian (resp., r-Noetherian) ring is Noetherian (resp., r-Noetherian).

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3.4 Krull Domains and Dedekind Domains In this section, we apply our previous results to study the Krull domains and Dedekind domains. All of the results in this section are generalized to rings with zerodivisors in Section 3.7. Thus the reader, especially those familiar with Krull domains and Dedekind domains, may skip this section. Definition 3.4.1. If R is a ring, then X 1 (R) denotes the height one primes of R. Lemma 3.4.2. One has X 1 (D) ⊆ wAss(K /D) for any integral domain D with quotient field K . Proof. If p is a height one prime of D, then p is minimal over any nonzero principal ideal contained in it, and every nonzero principal ideal is a conductor ideal.  Definition 3.4.3. Let D be an integral domain. (1) D is almost Dedekind if it is locally a DVR, that is, if Dp is a DVR for every maximal (or prime) ideal p of D. (2) D is almost t-Dedekind if it is t-locally a DVR, that is, if Dp is a DVR for every t-maximal (or t-prime) ideal p of D. Since the intersection of any collection of completely integrally closed overrings of an integral domain is completely integrally closed, almost Dedekind rings are completely integrally closed and of Krull dimension at most one. By Proposition 2.2.16, a Dedekind domain is equivalently a Noetherian almost Dedekind domain. The existence of a non-Noetherian almost Dedekind domain was conjectured by Krull, and an example was first constructed in 1953 by Nakano [152], the example being the integral closure of Z in the smallest field containing Q and all complex pth roots of unity for all primes p. Since a Krull domain is equivalently a Mori PVMD and a DVR is equivalently a Mori valuation domain, we have the following. Proposition 3.4.4. An almost Dedekind domain is equivalently a locally Mori (or locally Noetherian) Prüfer domain, and an almost t-Dedekind domain is equivalently a t-locally Mori (or t-locally Noetherian) PVMD. The height one primes are critical for studying the Krull domains. This is substantiated by the following result. Proposition 3.4.5. If D is a Krull domain with quotient field K , then X 1 (D) = Ess(D)reg = Ass(K /D) = wAss(K /D) = sK(K /D) = t-Max(D)reg = t-Spec(D)reg , and D is almost t-Dedekind. Proof. By Corollary 2.6.20, every Krull domain is a Mori PVMD. Therefore, by Corollary 3.3.15 and Proposition 3.3.25, if p is a t-prime of D, then Dp is a Mori valuation domain, hence a (t-)Noetherian valuation domain, hence a DVR. Therefore every nonzero t-prime of D has height one, and D is almost t-Dedekind. Now,

3.4 Krull Domains and Dedekind Domains

199

conversely, if p has height one, then, by Lemma 3.4.2 and Corollary 3.3.12, the ideal p is t-prime. Therefore X 1 (D) = t-Spec(D). Consequently, if p is a nonzero t-prime, then p cannot be properly contained in any other t-primes and is therefore t-maximal. Moreover, every Krull domain is Mori and is therefore an H domain, so the result now follows from Corollaries 3.3.15 and 3.3.19.  The following theorem provides several equivalent characterizations of the Krull domains. Theorem 3.4.6. Let D be an integral domain with quotient field K . The following conditions are equivalent. (1) (2) (3) (4) (5) (6) (7) (8)

D is a Krull domain. D is an almost t-Dedekind Mori domain. D is an almost t-Dedekind TV domain. D is almost t-Dedekind and of finite t-character. D is a completely integrally closed Mori domain. D is a completely integrally closed TV domain. D is acompletely integrally closed H domain. D = {Dp : p ∈ X 1 (D)}, where {Dp : p ∈ X 1 (D)} is locally finite and each a DVR. Dp is (9) D = {Dp : p ∈ }, where  ⊆ Spec D, the collection {Dp : p ∈ } is locally finite, and each Dp is a DVR. (10) D = {Dλ : λ ∈ }, where {Dλ : λ ∈ } is a locally finite collection of DVR overrings of D. (11) There exists a family  {vλ }λ∈ of discrete rank one valuations vλ : K −→ Z ∪ {∞} such that D = {Dλ : λ ∈ }, where Dλ = {x ∈ K : vλ (x)  0} and where for every nonzero x ∈ D one has vλ (x) = 0 for all but finitely many λ ∈ . Proof. Since a Krull domain is equivalently a Mori PVMD and Krull implies almost t-Dedekind, which in turn implies PMVD, we have (1) ⇔ (2). Also, we have (2) ⇒ (3) ⇒ (4) by Theorem 3.3.23, and we have (1) ⇔ (5) ⇔ (6) ⇔ (7) by Proposition 2.6.25. Suppose that (4) holds. Then every t-prime ideal of D has height one and is therefore t-maximal, so by Lemma 3.4.2 the t-maximal ideals of D coincide with its primes of height one. Therefore (4) ⇒ (8). Clearly one has (8) ⇒ (9) ⇒ (10) ⇒ (11). Finally, we prove that (11) ⇒ (5). Suppose that (11) holds. Any DVR is Noetherian and integrally closed and therefore completely integrally closed. Moreover, since an intersection of completely integrally closed overrings of an integral domain is completely integrally closed, we conclude that D is completely integrally closed. Therefore, it remains only to show that D is Mori. Now, for all a ∈ K one clearly has a D = {x ∈ K : vλ (x)  vλ (a) for all λ ∈ }. A nonzero fractional ideal of D is v-closed, then, if and only if it is the intersection of principal fractional ideals, if and only if it is of the form I (n) = {x ∈ K : vλ (x)  n λ for  all λ ∈ } = λ∈ Z. (We require the direct sum λ∈ Z here, for some n = (n λ )λ∈ ∈ Z()  rather than the direct product λ∈ Z, because of the local finiteness condition on

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3 Semistar Operations on Commutative Rings: Local Methods

the valuations vλ .) Therefore an ideal I of D is v-closed if and only if I is of the form I (n) for some n ∈ N() , where N = Z0 . In particular, there are only finitely many v-closed ideals containing a given nonzero v-closed  ideal I = I (n), namely, at most (and in fact exactly by Theorem 3.4.9 below) λ∈ (1 + n λ ) of them. In particular, D is Mori.  Remark 3.4.7. Statements (8)–(11) of the theorem are more traditional definitions of the Krull domains. Correspondingly, the following is a more traditional proof that every UFD is Krull. Let  denote a complete  set of non-associate primes of D. For all nonzero x ∈ K , we may write x = u x p∈ p v p (x) for a unique unit u x of D and unique integers v p (x) for p ∈ , and we define v p (0) = ∞. It is clear that the functions v p : K −→ Z ∪ {∞} are normalized discrete rank one valuations on K and the family {v p } p∈ satisfies condition (11) of Theorem 3.4.6. Therefore D is a Krull domain. Corollary 3.4.8. Let D be an integral domain. The following conditions are equivalent. (1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

D is a Dedekind domain. D is an almost Dedekind Noetherian domain. D is an almost Dedekind Mori domain. D is an almost Dedekind TV domain. D is almost Dedekind and of finite character. D is completely integrally closed and divisorial. D is a divisorial Krull domain. D is a TD Krull domain. D is a Krull domain of dimension at most one. D is an integrally closed Noetherian domain of dimension at most one.

Condition (10) of the corollary is often taken to be the definition of a Dedekind domain.  If  is a set then the free abelian group Z() = λ∈ Z is a (partially) ordered group under the coordinatewise partial ordering induced from that on Z. The proof of Theorem 3.4.6 leads readily to the following. (An explicit proof, generalized to rings with zerodivisors, is provided in Section 3.7.) Theorem 3.4.9. Let D be a Krull domain with quotient field K , and for each p ∈ X 1 (D) let vp : K −→ Z ∪ {∞} denote the normalized discrete rank one valuation on K with Dp = {x ∈ K : vp (x)  0}. Then we have the following. (1) A nonzero fractional ideal of D is divisorial if and only if it is of the form I (n) = 1 {x ∈ K : vp (x)  n p for all p ∈ X 1 (D)} for some n = (n p )p∈X 1 (D) ∈ Z(X (D)) . (2) I (m) ⊇ I (n) if and only if m  n, and I (m + n) = (I (m)I (n))v , for all m, n ∈ 1 Z(X (D)) . (3) In particular, the map n −→ I (n) defines an order-reversing isomorphism from 1 the ordered (free abelian) group Z(X (D)) onto the group Freg (D)v = Invv (D) of nonzero divisorial (v-invertible) fractional ideals of D under v-multiplication.

3.4 Krull Domains and Dedekind Domains

201

(4) p = I (ep ) for all p ∈ X 1 (D), where ep ∈ Z(X (D)) is the elementary unit vector with nonzero p-th coordinate. In particular, the group Freg (D)v is the free abelian group on the set X 1 (D). (5) A nonzero  divisorial ideal I = I (n) of D for n = (n p )p∈X 1 (D) is contained in exactly p∈X 1 (D) (1 + n p ) < ∞ many divisorial ideals of D. 1

Corollary 3.4.10. Let D be a Dedekind domain with quotient field K , and for each p ∈ Max(D) let vp : K −→ Z ∪ {∞} denote the normalized discrete rank one valuation on K with Dp = {x ∈ K : vp (x)  0}. Then we have the following. (1) Every nonzero fractional ideal of D is of the form I (n) = {x ∈ K : vp (x)  n p for all p ∈ Max(D)} for some n = (n p )p∈Max(D) ∈ Z(Max(D)) . (2) I (m) ⊇ I (n) if and only if m  n, and I (m + n) = I (m)I (n), for all m, n ∈ Z(Max(D)) . (3) In particular, the map n −→ I (n) defines an order-reversing isomorphism from the ordered (free abelian) group Z(Max(D)) onto the group Freg (D) = Inv(D) of nonzero (invertible) fractional ideals of D under multiplication. (4) p = I (ep ) for all p ∈ Max(D), where ep ∈ Z(Max(D)) is the elementary unit vector with nonzero p-th coordinate. In particular, the group Freg (D) is the free abelian group on the set Max(D). (5) A  nonzero ideal I = I (n) of D for n = (n p )p∈Max(D) is contained in exactly p∈Max(D) (1 + n p ) < ∞ many ideals of D.

3.5 Large Localization Like Section 3.3, this section is devoted to developing some tools for generalizing results on integral domains to rings with zerodivisors. In particular, we introduce the tool of large localization. The remainder of this chapter is focused on using these tools to generalize the results of Sections 3.1–3.4 to commutative rings. In those  sections, we made free use of the ability to consider intersections of the form p∈ Dp for any integral domain D and any subset  of Spec D. The most obvious obstruction to generalizing these results is in reinterpreting such intersections, since a ring with zerodivisors need not be contained in its localizations. The following definition serves this purpose. Definition 3.5.1. Let R be a ring with total quotient ring K , and let p be a prime ideal of R. We let R[p] = {x ∈ K : ux ∈ R for some u ∈ R − p}  = {(R : K u) : u ∈ R − p}  = {(R : K J ) : J ∈ I(R), J Ę p}.

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3 Semistar Operations on Commutative Rings: Local Methods

In other words, R[p] is the inverse image of Rp under the map K −→ K p (or under the map K = T (R) −→ T (Rp )). For any R-submodule I of K , we let [I ]R[p] = {x ∈ K : ux ∈ I for some u ∈ R − p}  = {(I : K u) : u ∈ R − p}  = {(I : K J ) : J ∈ I(R), J Ę p}. In other words, [I ]R[p] is the inverse image of I Rp under the map T (R) −→ T (R)p (or under the map T (R) −→ T (Rp )). The ring R[p] is called the large localization of R at p. Remark 3.5.2. Let R be a ring with total quotient ring K , and let p be a prime ideal of R. The ring R[p] is a limit of the diagram Rp −→ K p ←− K in the category of rings, so that R[p]

K

Rp

Kp

is a pullback square. Here, K p may be replaced with T (Rp ) ⊇ T (R)p = K p . The following proposition collects some elementary properties of large localization. The proof is left as an exercise. Proposition 3.5.3. Let R be a ring with total quotient ring K , and let p be a prime ideal of R. (1) R[p] is an overring R. (2) The map R[p] −→ Rp given by x −→ ux/u, where u is any element of R − p such that ux ∈ R, is a well-defined ring homomorphism. (3) The homomorphisms R −→ R[p] and R[p] −→ Rp become isomorphisms when tensored over R with Rp . In particular, one has R[p] ⊗ R Rp = Rp . (4) Rp is flat over R[p] . (5) [p]R[p] is regular if and only if p is regular, if and only if R[p] = K . Moreover, if p is not regular, then R[p] = K and [p]R[p] = pK . (6) Let q be a prime contained in p. Then [q]R[p] is a prime ideal of R[p] lying  under qRp in Rp and lying over q in R. Moreover, one has R[q  ] = R[q] ⊇ R[p] ,   where R = R[p] and q = [q]R[p] . (7) R[p] = Rp if and only if p contains every zerodivisor of R. (8) If R = R[p] , then p = [p]R[p] and p contains every regular non-unit of R. (9) [p]R[p] contains every regular non-unit of R[p] . (10) For any ideal a of R, one has [a]R[p] = R[p] if and only if a Ę p. (11) [I ∩ J ]R[p] = [I ]R[p] ∩ [J ]R[p] for all I, J ∈ K(R).

3.5 Large Localization

203

  (12) [I J ]R[p] ⊇ [I ]R[p] [J ]R[p] ⊇ I [J ]R[p] for all I, J ∈ K(R), with equalities if I is invertible. (13) [(I : K J )]R[p] ⊆ ([I ]R[p] : K J ) = ([I ]R[p] : K [J ]R[p] ) for all I, J ∈ K(R), with equalities if J is finitely generated. (14) [I −1 ]R[p] ⊆ ([I ]R[p] )−1 = (I R[p] )−1 for all I ∈ K(R), with equalities if I is finitely generated. (15) Let a be an ideal of R. One has a ⊆ [a]R[p] ∩ R, and equality holds if and only if a = aRp ∩ R, if and only if x y ∈ a implies x ∈ a or y ∈ p for all x, y ∈ R. Moreover, one has [[a]R[p] ∩ R]R[p] = [a]R[p] . (16) Let I be an ideal of R[p] . One has I ⊆ [I ∩ R]R[p] = [I ]R[p] , and equalities hold if and only if I is of the form [a]R[p] for some ideal a of R, if and only if I = I Rp ∩ R[p] , if and only if x y ∈ I implies x ∈ I or y ∈ p for all x ∈ R[p] and all y ∈ R. (17) The correspondence a −→ [a]R[p] , with inverse I −→ I ∩ R, defines an order-preserving bijection from the set of all ideals a of R satisfying the condition on a in statement (15) to the set of all ideals I of R[p] satisfying the condition on I in statement (16). (18) The correspondence I −→ I Rp , with inverse J −→ J ∩ R[p] , defines an order-preserving bijection from the set of all ideals I of R[p] satisfying the condition on I in statement (16) to the set of all ideals J of Rp . (19) The correspondence q −→ [q]R[p] , with inverse Q −→ Q ∩ R, defines an order-preserving bijection from the set of all prime ideals q of R contained in p to the set of all prime ideals Q of R[p] contained in [p]R[p] (which is equal to the set of all prime ideals Q of R[p] with Q ∩ R ⊆ p).   For any ring R, one has R = p∈Spec(R)reg R[p] = m∈Max(R)reg R[m] , where Spec(R)reg (resp., Max(R)reg ) is the set of all regular prime ideals (resp., regular maximal ideals) of R. Indeed, the first equality follows from the proposition below, while the second equality follows from the first and the fact if p ⊆ q are prime ideals then R[p] ⊇ R[q] . Proposition 3.5.4. One has R=

 p∈wAss(K /R)

R[p] =

 p∈sK(K /R)

R[p] =

 p∈t-Spec(R)reg

R[p] =



R[p]

p∈t-Max(R)reg

for any ring R with total quotient ring K .  Proof.  If we show that R = p∈wAss(K /R) R[p] , then the other equalities follow. Let S = p∈wAss(K /R) R[p] , and suppose to obtain a contradiction that S properly contains / R, the ideal I is R. Let x ∈ S − R, and let I = (R : R x R) = R ∩ (x R)−1 . Since x ∈ a proper ideal of R, and I is regular since R is regular and x R is fractional. Thus there exists a (regular) prime p in R that is minimal over I . Since I is a conductor ideal of R, we have p ∈ wAss(K /R) and therefore x ∈ R[p] . But then [I ]R[p] = R[p] ∩ [(x R)−1 ]R[p] = R[p] ∩ (x R[p] )−1 = (R[p] : R[p] x R[p] ) = R[p] , contradicting the fact that [I ]R[p] ⊆ [p]R[p] . 

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One of the morals of this chapter is that, for some purposes, large localization is more suitable for the study of commutative rings than ordinary localization. Both have advantages over the other. The main disadvantages of localization over large localization are that Rp need not contain R as a subring, and many important ringtheoretic properties, such as Dedekind, Prüfer, Krull, and PVMD, do not pass from R to Rp , even if p is regular and t-maximal. The main disadvantages of large localization over localization are that R[p] need not be local, [p]R[p] need not be maximal, and R[p] need not be flat over R. Nevertheless, one has the following. Lemma 3.5.5. Let R be a ring and p a prime ideal of R. The overring R[p] of R is a semistar t-compatible extension of R. Proof. Let I be an ideal of R, and let J = I −1 . If I is finitely generated, then by Proposition 3.5.3(14) one has I v = J −1 ⊆ (J R[p] )−1 = (I −1 R[p] )−1 = ([I −1 ]R[p] )−1 = ((I R[p] )−1 )−1 = (I R[p] )v .

In the general case, if H is any finitely generated subideal of I , then H v ⊆  (H R[p] )v ⊆ (I R[p] )t , and therefore I t ⊆ (I R[p] )t . Lemma 3.5.6. Let R be a ring, and let O be a collection of semistar t-compatible overrings of R. Then {S : S ∈ O} is a semistar t-compatible overring of R.  Proof. Let T = {S : S ∈ O}, let I be a finitely generated ideal of R, and let x ∈ I v . We claim that x ∈ (I T )v , that is, x(I T )−1 ⊆ T . Let y ∈ (I T )−1 and S ∈ O. Since S is semistar t-compatible, one has x ∈ (I S)v and therefore x(I S)−1 ⊆ S. One also has y I ⊆ T ⊆ S and therefore y ∈ (I S)−1 , whence x y ∈ S. Since this holds for all S ∈ O, one has x y ∈ T . Since this holds for all y ∈ (I T )−1 , one has x(I T )−1 ⊆ T , as claimed.  Proposition 3.5.7. Let R be a ring and  a set of prime ideals of R. The overring  p∈ R[p] of R is a semistar t-compatible extension of R. The following is the analogue of Proposition 3.3.11 for large localization. Proposition 3.5.8. Let R be a ring with total quotient ring K and let p be a prime ideal of R. Then p ∈ sK(K /R) if and only if [p]R[p] ∈ t-Spec(R[p] ). Moreover, if p ∈ sK(K /R), then p ∈ t-Spec(R)reg . In other words, one has sK(K /R) = {p ∈ Spec(R) : [p]R[p] ∈ t-Spec(R[p] )} ⊆ t-Spec(R)reg . Proof. Suppose that p ∈ sK(K /R). Then pRp is a regular t-prime of Rp by Proposition 3.3.11. Moreover, Rp is flat over R[p] . Therefore [p]R[p] is a t-prime of R[p] by Proposition 3.3.2. Alternatively, imitating the proof of Proposition 3.3.11, we show directly that H v ⊆ [p]R[p] for every finitely generated subideal H of [p]R[p] , so that [p]R[p] is t-closed and therefore t-prime. Let H be any finitely generated subideal of [p]R[p] . We may write H = (y1 , . . . , yn )R[p] , where there exists u ∈ R − p such that

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xi = uyi ∈ p for all i. Let I = (x1 , . . . , xn )R, so that I is a finitely generated subideal of p with I R[p] = u H . Since p ∈ sK(K /R), there exists a conductor ideal J = (1, a/b)−1 for some a, b ∈ R with b regular such that I ⊆ J ⊆ p. Extending to R[p] , we have [I ]R[p] ⊆ [J ]R[p] = ((1, a/b)R[p] )−1 ⊆ [p]R[p] , and applying v-closure on R[p] we get (I R[p] )v = ([I ]R[p] )v ⊆ (((1, a/b)R[p] )−1 )v = ((1, a/b)R[p] )−1 and therefore (I R[p] )v ⊆ [p]R[p] . Thus u H v ⊆ (u H )v = (I R[p] )v ⊆ [p]R[p] . But u ∈ R[p] − [p]R[p] and [p]R[p] is prime, and therefore we have H v ⊆ [p]R[p] , as desired. Conversely, suppose that [p]R[p] is t-prime in R[p] . Let I be a finitely generated subideal of p. Since I R[p] ⊆ [p]R[p] is finitely generated, we have (I R[p] )v ⊆ [p]R[p] . Since then (I R[p] )v  R[p] , one has [I −1 ]R[p] = (I R[p] )−1  R[p] , so there exists / R. Write x ∈ [I −1 ]R[p] − R[p] . Choose w ∈ R − p so that wx ∈ I −1 , so that wx ∈ wx = a/b with a, b ∈ R and b regular. Then a I ⊆ b R since wx I ⊆ R. Thus I ⊆ (b R : R a R). Moreover, if (b R : R a R) Ę p, then ua R ⊆ b R, whence uwx ∈ R, for some u ∈ R − p, and therefore x ∈ R[p] , which is a contradiction. Therefore I ⊆ (b R : R a R) ⊆ p. Finally, since I was an arbitrary finitely generated subideal of p, we have p ∈ sK(K /R). Thus, we have shown that sK(K /R) = {p ∈ Spec(R) : [p]R[p] ∈ t-Spec(R[p] )}. Moreover, by Proposition 3.3.11 or by Lemma 3.5.5, every ideal lying in this set is t-prime.  Recall that a ring R is said to be v-coherent if I −1 is (strictly) v-finite for every finitely generated regular (fractional) ideal I of R. The next proposition shows that the regular t-primes of a v-coherent ring R coincide with the strong Krull primes of T (R)/R. Lemma 3.5.9. Let R be a ring and p a prime ideal of R. If I ∈ K(R) is v-closed and I and I −1 are v-finite, then [I ]R[p] = (I R[p] )v = ([I ]R[p] )v is v-closed in R[p] . Proof. Let H ∈ K(R) be finitely generated with I = H v . Then, by Proposition 3.5.3(14), we have [I −1 ]R[p] = [H −1 ]R[p] = (H R[p] )−1 ⊇ (H v R[p] )−1 = (I R[p] )−1 ⊇ [I −1 ]R[p] , and therefore [I −1 ]R[p] = (I R[p] )−1 = ([I ]R[p] )−1 . Therefore, if I −1 is also v-finite, then it follows that ([I ]R[p] )v = (([I ]R[p] )−1 )−1 = ([I −1 ]R[p] )−1 = [(I −1 )−1 ]R[p] = [I ]R[p] . Therefore ([I ]R[p] )v = (I R[p] )v = [I ]R[p] is v-closed in R[p] .



Proposition 3.5.10 (cf. [189, Proposition 1.4]). Let R be a v-coherent ring. Then, p is a regular t-prime of R if and only if [p]R[p] is a (regular) t-prime of R[p] . Equivalently, one has sK(K /R) = {p ∈ Spec(R) : [p]R[p] ∈ t-Spec(R[p] )} = t-Spec(R)reg .

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Proof. Let p be a regular t-prime ideal of R, and let H = (a1 , . . . , an )R[p] be a finitely generated regular ideal of R[p] contained in [p]R[p] . We may choose a u ∈ R − p such that uai ∈ R for all i, so that I = (ua1 , . . . , uan ) ⊆ p is a finitely generated ideal of R with I R[p] = u H . Since p is t-closed we have I v ⊆ p and therefore [I v ]R[p] ⊆ [p]R[p] . Now, since I is finitely generated and regular, by hypothesis, (I v )−1 = I −1 is v-finite, as of course is I v . Therefore, by the lemma, we have (I v R[p] )v = [I v ]R[p] and therefore u H v ⊆ (u H )v = (I R[p] )v ⊆ (I v R[p] )v = [I v ]R[p] ⊆ [p]R[p] . But then, since u ∈ R[p] − [p]R[p] and [p]R[p] is prime, it follows that H v ⊆ [p]R[p] . Since this holds for all finitely generated regular ideals H of R[p] contained in [p]R[p] ,  it follows that [p]R[p] is t-closed, hence t-prime. Since a PVMR is equivalently a v-coherent v-Prüfer ring, we have the following. Corollary 3.5.11. Let R be PVMR with total quotient ring K . One has sK(K /R) = {p ∈ Spec(R) : [p]R[p] ∈ t-Spec(R[p] )} = t-Spec(R)reg . Since every Mori ring is v-coherent and H, by Propositions 3.5.10, 3.3.24, and 3.3.18 we also have the following. Corollary 3.5.12. Let R be a Mori ring with total quotient ring K . One has t-Max(R)reg ⊆ Ass(K /R) = wAss(K /R) = sK(K /R) = t-Spec(R)reg . Remark 3.5.13. A prime p of an integral domain D is said to be well behaved, or t-localizing, if pDp is a t-prime of Dp . A domain D is said to be well behaved (with respect to t-primes) if every t-prime of D is well behaved [189]. In the comments after [189, Proposition 1.4], Zafrullah shows that any v-coherent domain is well behaved. We may extend these notions to rings with zerodivisors in the obvious way: a prime p of a ring R is well behaved if [p]R[p] is a t-prime of R[p] , and a ring R is well behaved with respect to regular t-primes if every regular t-prime of R is well behaved. Proposition 3.5.10 then states that any v-coherent ring is well behaved with respect to regular t-primes. Next, we wish to prove the following theorem. Theorem 3.5.14. Let R be a ring and  a set of prime ideals of R. If R is Mori (resp., integrally closed, aPVMR, Krull, Prüfer, Dedekind, r-GCD, an r-UFR, r-Bézout, an r-PIR), then so is p∈ R[p] . Remark 3.5.15. (1) Let p be a prime ideal of a domain D. If D is a v-domain, then Dp need not be a v-domain, and if D is completely integrally closed, then Dp need not be completely integrally closed [68, p. 161–162].

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(2) Let p be a regular prime ideal of a ring R. It is well-known that Rp need not be Mori (resp., Prüfer) if R is Mori (resp., Prüfer) [134, Example 3.3] [120]. To prove Theorem 3.5.14, we need a few subsidiary results. Lemma 3.5.16. Let  R be a ring and  a setof prime ideals of R.For any divisorial ideal I of the ring p∈ R[p] , one has I = p∈ [I ∩ R]R[p] = p∈ [I ]R[p] .  Proof. inclusion  Let T = p∈ R[p] . The equality [I ∩ R]R[p] = [I ]R[p] and the I ⊆ p∈ [I ∩ R]R[p] are clear. To prove the reverse inclusion, let x ∈ p∈ [I ∩ R]R[p] . Then for each p ∈  there exists u p ∈ R − p such that xu p ∈ I ∩ R. Let y ∈ (T : K I ), where K = T (R). Then, for all p ∈ , one has xu p y ∈ T ⊆ R[p] , so that xu p yvp ∈ R for some vp ∈ R − p, so that x y ∈ R[p] . Therefore x y ∈ T . Since  this holds for all y ∈ (T : K I ), one has x(T : K I ) ⊆ T , whence x ∈ I v = I . The following result follows from [134, Theorem 2.13], but we provide a proof here. Proposition  3.5.17. Let R be a ring and  a set of prime ideals of R. If R is Mori, then so is p∈ R[p] .  Proof. Let T = p∈ R[p] , and let I1 ⊆ I2 ⊆ I3 ⊆ · · · be any chain of regular divisorial ideals of T . Let Jk = Ik ∩ R for all k. Since T is a semistar t-compatible overring of R by Proposition 3.5.7, and since each Ik is regular and t-closed in T , J1 ⊆ J2 ⊆ J3 ⊆ · · · each Jk is regular and t-closed in R. Therefore the sequence  stabilizes. Therefore, since by the lemma, one has Ik = p∈ [Jk ]R[p] for all k, the  sequence I1 ⊆ I2 ⊆ I3 ⊆ · · · stabilizes as well. Therefore T is Mori. By Exercises 2.2.8 and 3.3.14, one has the following. Proposition 3.5.18. Any t-linked overring of a PVMR is a PVMR, and any overring of a Prüfer ring is Prüfer. The following proposition generalizes [7, Theorem 4.4]. Proposition 3.5.19. Let R be a PVMR, and let S be any t-linked overring of R. The induced group homomorphisms Invt (R) −→ Invt (S) and Clt (R) −→ Clt (S) are surjective. Consequently, if R has trivial t-class group, then S has trivial t-class group. Proof. Let J ∈ Invt (S). Then J ∈ Freg (S) is strictly t-finite, so there exist elements x1 , x2 , . . . , xn of T (R), at least one of which is regular, such that J = ((x1 , x2 , . . . , xn )S)t . Let I = (x1 , x2 , . . . , xn )R. Then I is a finitely generated regular fractional ideal of R such that (I S)t = J . Since R is a PVMR, it follows that I is t-invertible, whence I t ∈ Invt (R) is also t-invertible. Moreover, since S is a t-linked overring of R and I ∈ Freg (R) is t-invertible, by Proposition 3.3.2, one has (I t S)t = (I S)t = J , and therefore I t −→ J under the group homomorphism Invt (R) −→ Invt (S). The proposition follows. 

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Note that, by [2, p. 33], there exists a local integrally closed Mori domain D containing a nonzero element f such that the group homomorphism Clt (D) −→ Clt (D f ) is not surjective. By the proposition above, no such domain D can be a PVMD. Corollary 3.5.20. Any t-linked overring of an r-GCD ring is r-GCD, and any overring of an r-Bézout ring is r-Bézout. The proof of the following proposition is left as an exercise. Proposition 3.5.21.  Let R be a ring and  a set of prime ideals of R. If R is integrally closed, then so is p∈ R[p] . One now obtains Theorem 3.5.14 by combining Propositions 3.5.7, 3.5.17, 3.5.18, 3.5.19, and 3.5.21.  A subintersection of an integral domain D is an overring of D of the form p∈ Dp for some set  of prime ideals of D. It is known that if D is a PVMD, then every t-linked overring of D is a subintersection of D [114, Theorem 3.8]. Let us say that a subintersection of a ring R is an overring of R of the form p∈ R[p] for some set  of prime ideals of R. We have shown that any subintersection of a ring R is a semistar t-compatible extension of R, and if R is Mori (resp., integrally closed, a PVMR, Krull, Prüfer, Dedekind, r-GCD, an r-UFR, r-Bézout, and an r-PIR), then so is any subintersection of R. Open Problem 3.5.22. Is every semistar t-compatible overring of a PVMR a subintersection of R? Is every t-linked overring of a PVMR a subintersection of R? Finally, we note the following result, which provides several useful criteria for an overring to be flat. Theorem 3.5.23 (cf. [93, Proposition 10]). Let S be an overring of a ring R. The following conditions are equivalent. (1) (2) (3) (4) (5) (6) (7)

S is flat over R. (R : R J )S = S for every finitely generated R-submodule J of S. (R : R x)S = S for all x ∈ S. For every prime ideal p of R, one has pS = S or S ⊆ R[p] . For every regular prime ideal p of R, one has pS = S or S ⊆ R[p] . For every prime ideal q of S, one has S[q] = R[q∩R] . For every regular maximal ideal q of S, one has S[q] = R[q∩R] .

Proof. (1) ⇒ (2). If K is the total quotient ring of R, then (R : R J )S = ((R : K J ) ∩ R)S = (R : K J )S = (S : K S J S) ⊇ S by Propositions 2.2.10 and 2.2.11. (2) ⇒ (3). Clear. (3) ⇒ (1). To show that S is flat it suffices to show  that the canonical map I ⊗ R S −→ S is injective for any ideal I of R. Let x = h i ⊗ si be in the kernel, where  h i ∈ I and si ∈ S for all i. Let J = i (R : R si ). Then J S = S. Moreover, x J = 0

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   since x y = h i ⊗ (si y) = h i si y ⊗ 1 = 0 ⊗ 1 = 0 for all y ∈ J . Therefore x S = x J S = 0, whence x = 0. (3) ⇒ (4). Suppose that (4) does not hold, so there exists a prime ideal p of R with pS = S and S Ę R[p] . Let x ∈ S − R[p] . Then (R : R x) ⊆ p, and therefore (R : R x)S ⊆ pS  S. Therefore (3) does not hold. (4) ⇒ (6). Let q be any prime ideal of S, and let p = q ∩ R. Then R[p] ⊆ S[q] . To prove the reverse inclusion, let x ∈ S[q] be given. Choose b ∈ S − q with b1 = bx ∈ S. Since pS ⊆ q  S, by hypothesis, one has S ⊆ R[p] . Therefore b, b1 ∈ R[p] , so there exist a, a1 ∈ R − p with ab ∈ R and a1 b1 ∈ R. Since a, b ∈ S − q, one has ab ∈ R ∩ (S − q) = R − p. Therefore a1 ab ∈ R − p, and since (a1 ab)x = a1 ab1 = a(a1 b1 ) ∈ R, we conclude that x ∈ R[p] . (6) ⇒ (7). Clear. (7) ⇒ (5). Let p be any regular prime ideal of R. Suppose that pS = S. Then there is a (regular) maximal ideal m of S such that pS ⊆ m, so that p ⊆ m ∩ R. Then S ⊆ S[m] = R[m∩R] ⊆ R[p] . (5) ⇒ (3). Suppose that (3) does not hold. Then (R : R x)S ⊆ q for some x ∈ S and some maximal ideal q of S. Let p = q ∩ R. Since (R : R x) ⊆ q ∩ R = p and (R : R x) is a regular ideal of R, the ideal p is a regular prime ideal of R. Since / R[p] . Since x ∈ S, then, one has S Ę R[p] . Moreover, one (R : R x) ⊆ p, one has x ∈ has pS ⊆ q  S, so pS = S. Therefore (5) does not hold. 

3.6 Valuations and Valuation Pairs In this section, we generalize the notion of a valuation domain to rings with zerodivisors. Definition 3.6.1. A paravaluation on a ring T is a function v : T −→  ∪ {∞}, where  is a totally ordered abelian group (where  is written additively, and where one sets γ + ∞ = ∞ + γ = ∞ = ∞ + ∞ and γ < ∞ for all γ ∈ ), satisfying the following three conditions for all x, y ∈ T : (1) v(0) = ∞ and v(1) = 0. (2) v(x y) = v(x) + v(y). (3) v(x + y)  min{v(x), v(y)}. For any paravaluation v on T , let Tv = {x ∈ T : v(x)  0}, pv = {x ∈ T : v(x) > 0}, v −1 (∞) = {x ∈ T : v(x) = ∞}. A (Manis) valuation on T is a paravaluation v : T −→  ∪ {∞} on T such that v is surjective, or equivalently such that  = v(T ) − {∞}. The totally ordered group

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 = v(T ) − {∞} of a valuation v : T −→  ∪ {∞} is called the value group of v. A valuation v : T −→  ∪ {∞} is discrete rank one if  is isomorphic to Z. Convention 3.6.2. Unfortunately, there is now potential conflict between using the symbol “v” for a paravaluation versus “v” for the v-closure operation. The intended meaning should be clear in any given context. In general, if v : T −→  ∪ {∞} is a paravaluation on T , then v(T ) − {∞} is a submonoid of . The corestriction v : T −→ v(T ) of v to its image v(T ) is a valuation on T if and only if v(T ) − {∞} is a subgroup of , if and only if for every x ∈ T with v(x) = ∞ there exists a y ∈ T such that v(y) = −v(x), if and only if for every x ∈ T − v −1 (∞) there exists a y ∈ T such that x y ∈ Tv − pv . In general, v(T × ) is a subgroup of ; therefore, if T is a field, then the corestriction of a paravaluation v on T to its image v(T ) = v(T × ) ∪ {∞} is automatically a valuation on T . For this reason, if T is a field, then the distinction between a paravaluation and a valuation is not an important one. Proposition 3.6.3. Let v be a paravaluation on a ring T . (1) (2) (3) (4)

v(a + b) = min{v(a), v(b)} for all a, b ∈ T with v(a) = v(b). Tv is a subring of T and is integrally closed in T . pv is a prime ideal of Tv . v −1 (∞) is a non-T -regular prime ideal of Tv (called the prime (of v) at infinity) contained in pv , and v −1 (∞) is also a prime ideal of T .

Proof. The proofs are all trivial, except for the proof that Tv is integrally closed in T . Suppose that x ∈ T − Tv is integral over Tv . Write x n + an−1 x n−1 + · · · + a1 x + a0 = 0 with ai ∈ Tv for all i. Since v(x) < 0 and v(−ai )  0 for all i, one has v(x n ) < v(x i )  v(−ai x i ) for all i < n, so that v(x n ) < v(−an−1 x n−1 − · · · − a1 x − a0 ),  contradicting the fact that x n = −an−1 x n−1 − · · · − a1 x − a0 . Definition 3.6.4. Let T be a ring. A valuation pair of T is a pair (R, p), where R is a subring of T and p is a prime ideal of R such that for every x ∈ T − R there exists y ∈ p such that x y ∈ R − p. The following theorem justifies the term “valuation pair.” Theorem 3.6.5 ([141, Proposition 1]). Let R be a subring of a ring T and p a prime ideal of R. The following conditions are equivalent. (1) (R, p) is a valuation pair of T . (2) There exists a valuation v : T −→  ∪ {∞} on T such that R = Tv and p = pv . (3) If S is a subring of T containing R and q is a prime ideal of S with q ∩ R = p, then S = R. In other words, R is the only subring of T containing R and a prime ideal lying over p.

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Proof. The implications (2) ⇒ (1) ⇒ (3) are trivial. To prove (3) ⇒ (1), we use the “lying over” theorem [17, Theorem 5.10]: if R ⊆ S is an integral extension, then every prime ideal of R lies under some prime ideal of S. For the case at hand, let S be the integral closure of R in T . Then S has a prime ideal lying over p, and so by condition (3) one must have S = R. Therefore R is integrally closed in T . Let x ∈ T − R, and let B = R[x] and J = pB. Then J is an ideal of B such that J ∩ R ⊇ p. If J ∩ R = p, then J is contained in a prime ideal q of B such that q ∩ R = p, contradicting (3). Therefore J ∩ R  p. Let b ∈ (J ∩ R) − p, and write b = p0 + p1 x + · · · + pn x n , where pi ∈ p for all i and n is minimal. Multiplying by pnn−1 shows that x pn is integral over R, and hence in R. If x pn ∈ p, then b = p0 + p1 x + · · · + ( pn−1 + x pn )x n−1 , contradicting the minimality of n. Therefore y = pn satisfies x y ∈ R − p. Therefore (1) holds. Finally, we prove (1) ⇒ (2). For all x, y ∈ T , write x ∼ y if (p :T x) = (p :T y). Then ∼ is an equivalence relation on T . Let v(x) denote the equivalence class of x, so that v(0) = {x ∈ T : (p :T x) = T } = (p :T T ) and v(1) = R − p. Let G = {v(x) : x ∈ T } − {v(0)}. For v(x), v(y) ∈ G, let v(x) + v(y) = v(x y), which defines a (well-defined) associative binary operation on G with identity v(1). From (1), one can deduce that for all x ∈ T − v(0) there is a y ∈ T − v(0) such that x y ∈ R − p. Thus every element of G has an inverse with respect to +, and therefore G is an abelian group. For all v(x), v(y) ∈ G v(x)  v(y) if (p :T x) ⊆ (p :T y). This defines a partial ordering on G so that G is a totally ordered abelian group. Finally, let v(0) = ∞. Under these definitions, G is a totally ordered abelian group,  v : T −→ G ∪ {∞} is a valuation, R = Tv , and p = pv . Remark 3.6.6. Let R be a subring of a ring T and p be a prime ideal of R. By [117, Theorem 1], the pair (R, p) is a valuation pair of T if and only if there exists an algebraically closed field L and a ring homomorphism R −→ L, with kernel p, that has no extension to a ring homomorphism R  −→ L with R  a subring of T properly containing R. Definition 3.6.7. Two valuations v : T −→  ∪ {∞} and v  : T −→   ∪ {∞} on a ring T are equivalent if there exists an isomorphism ϕ :  −→   of totally ordered abelian groups such that ϕ(v(x)) = v  (x) for all x ∈ T , where ϕ(∞) = ∞. Corollary 3.6.8. Let (R, p) be a valuation pair of a ring T , and let v : T −→  ∪ {∞} be a valuation on T such that R = Tv and p = pv . (1) (2) (3) (4) (5)

R is integrally closed in T . R = (p :T p). v −1 (∞) = (p :T T ). v(x)  v(y) if and only if (p :T x) ⊆ (p :T y), for all x, y ∈ T . v(x) < v(y) if and only if there is a z ∈ T such that zx ∈ R − p and zy ∈ p, for all x, y ∈ T . (6) A valuation v  on T is equivalent to v if and only if pv = pv , in which case one also has Tv = Tv . (7) One has R = T if and only if the value group  of v is trivial, if and only p = v −1 (∞).

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(8) If R = T , then p = {x ∈ R : x y ∈ R for some y ∈ T − R} and v −1 (∞) = (R : R T ), and thus p and v −1 (∞) are uniquely determined by R and T . Definition 3.6.9. (1) A ring R is a valuation ring of T if (R, p) is a valuation pair of T for some prime ideal p of R, or equivalently if R = Tv for some valuation v : T −→  ∪ {∞} on T . (2) A ring R is a valuation ring if R is a valuation ring of T (R). (3) If R is a valuation ring, then the totally ordered group G defined in the proof of Theorem 3.6.5, where T = T (R), is called the value group of R. (By Corollary 3.6.12 below, it is uniquely determined by R.) For example, an integral domain is a valuation ring if and only if it is a valuation domain in the sense of Definition 3.1.1. Definition 3.6.10. Let T be a ring. (1) A valuation v : T −→  ∪ {∞} on T is trivial if the value group  of v is trivial. (2) A valuation pair (R, p) of T is proper if R = T . (3) A valuation ring R of T is proper if R = T . Corollary 3.6.11. Let T be a ring. (1) A valuation v on T is trivial if and only if T = Tv , if and only if pv = v −1 (∞). (2) If p is any prime ideal of T , then (T, p) is a valuation pair of T ; in fact, the map v : T −→ {0, ∞} given by v(x) = 0 if x ∈ / p and v(x) = ∞ if x ∈ p is a trivial valuation on T with T = Tv and p = pv = v −1 (∞). (3) If p is a non-regular prime ideal of a ring R, then R[p] = T (R) and therefore (R[p] , [p]R[p] ) is a valuation pair of T (R). Corollary 3.6.12. Let T be a ring. A ring R is a valuation ring of T if and only R = T or there exists a unique prime ideal p of R such that (R, p) is a valuation pair of T . Corollary 3.6.13. Let T be a ring. There is a one-to-one correspondence between the set of all valuation pairs of T and the set of all equivalence classes of valuations on T . Moreover, there are one-to-one correspondences between the set of all proper valuation rings of T , the set of all proper valuation pairs of T , and the set of all equivalence classes of nontrivial valuations on T . Proofs of the remaining results in this section are left as exercises. Proposition 3.6.14 (cf. [117, p. 428]). Let R be a ring with total quotient ring K . (1) If (R, p) is a valuation pair of K , then R[p] = R and [p]R[p] = p. (2) If (R, p) is a valuation pair of K , then for any prime ideal q of R containing (R : R K ) one has R[q] = R if and only if q ⊇ p. (3) (K , p) is a valuation pair of K for every prime ideal p of K .

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(4) If R = K , then there exists at most one prime ideal p of R such that (R, p) is a valuation pair of K . Definition 3.6.15. Let T be a ring and v : T −→  ∪ {∞} a valuation on T , and let R = Tv . (1) For any R-submodule I of T , let Iv = {x ∈ T : v(x)  v(y) for some y ∈ I }, which is an R-submodule of T . Let (−)v denote the self-map I −→ Iv of Mod R (T ). (2) I is closed under v if Iv = I , or equivalently if x ∈ I for all x ∈ T such that v(x)  v(y) for some y ∈ I . (3) Let v = {(x)v : x ∈ T } − {(0)v }. Definition 3.6.16. Let R be a subring of a ring T . A T -semistar operation on R is a nucleus  on the ordered monoid Mod R (T ), that is, it is a closure operation  on the poset Mod R (T ) such that I  J  ⊆ (I J ) for all I, J ∈ Mod R (T ). A T -semistar operation  on R is unital if R  = R. In particular, a semistar operation on a ring R is equivalently a unital T (R)semistar operation on R. Definition 3.6.17. LetR be a subring of a ring T . A T -semistar operation  on R is of finite type if I  = {J  : J ∈ Mod R (T ) and J ⊆ I is finitely generated} for all I ∈ Mod R (T ). Proposition 3.6.18. Let R be a subring of a ring T . (1) If  is any T -semistar operation  on R, then the operation t : I −→



{J  : J ∈ Mod R (T ) and J ⊆ I is finitely generated}

on Mod R (T ) is the largest finite type T -semistar operation  on R that is less than or equal to . (2) The operation vT : I −→ (I −1T )−1T on Mod R (T ) is the largest unital T -semistar operation on R. (3) tT = (vT )t is the largest finite type unital T -semistar operation on R. −1T −1T Definition 3.6.19. Let R be a subring of a ring T . Then vT : I −→ (I ) denotes the largest unital T -semistar operation on R, and tT = (vT )t : I −→ {J vT : J ∈ Mod R (T ) and J ⊆ I is finitely generated} denotes the largest finite type unital T -semistar operation on R.

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It is easy to check that if v is a valuation on a ring T , then (−)v is a finite type unital T -semistar operation on Tv . Moreover, the T -semistar operation (−)v is valuative in the following sense. Definition 3.6.20. Let R be a subring of a ring T . A T -semistar operation  on R is valuative if the following three conditions hold: (1) For all x, y ∈ T , either (x) ⊆ (y) or (y) ⊆ (x) . exists a y ∈ T such that (x y) = R  . (2) For all x ∈ T , if x ∈ / (0) , then there  (3) For all I ∈ Mod R (T ), one has I = {(x) : x ∈ I }. Note that conditions (1) and (2) hold if and only if  = {(x) : x ∈ T } − {(0) } is a totally ordered group, ordered by ⊇, under the operation (I, J ) −→ (I J ) of -multiplication. Moreover, if conditions (1) and (2) hold, then condition (3) holds if and only if  is of finite type. More generally, if  is a T -semistar operation on R satisfying conditions (1) and (2), then t is the largest valuative T -semistar operation that is less than or equal to . Proposition 3.6.21. Let T be a ring. (1) Let v : T −→  ∪ {∞} be a valuation on T , and let R = Tv . (a) Iv for any I ∈ Mod R (T ) is the smallest element of Mod R (T ) containing I that is closed under v. (b) (−)v : Mod R (T ) −→ Mod R (T ) is a valuative finite type unital T -semistar operation on R. (c) v is a totally ordered abelian group under the operation ((x)v , (y)v ) −→ (x y)v . (d) The map γv : v ∪ {∞} −→  ∪ {∞} acting by (x)v −→ v(x) for all x ∈ T , where ∞ = (0)v , restricts to a well-defined isomorphism v −→  of totally ordered abelian groups. (e) The map  v : T −→ v ∪ {∞} acting by x −→ (x)v is a valuation on T and v , and therefore v is equivalent to  v. one has v = γv ◦  (2) If v and v  are valuations on T , then v and v  are equivalent if and only if v = v . (−)v = (−)v , if and only if  (3) Let R be a subring of T , and let  be a valuative T -semistar operation on R. Let  = {(x) : x ∈ T } − {∞}, where ∞ = (0) . Then the map v : T −→  ∩ {∞} acting by x −→ (x) is a valuation on T such that R  = Tv , (−)v = , and v =  . (4) Let R be a subring of T . The association [v] −→ (−)v , with inverse  −→ [v ], provides a one-to-one correspondence between the set of all equivalence classes [v] of valuations v on T with R ⊆ Tv and the set of all valuative T -semistar operations  on R. (5) There are one-to-one correspondences between the set of all valuation pairs of T , the set of all equivalence classes of valuations on T , and the set of all valuative unital T -semistar operations on the valuation rings of T . (6) R is a valuation ring of T if and only if there exists a valuative unital T -semistar operation on R.

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(7) If R is a proper valuation ring of T , then there exists a unique valuative unital T -semistar operation on R. (8) Suppose that T is a total quotient ring. There are one-to-one correspondences between the set of all valuation pairs of T , the set of all equivalence classes of valuations on T , and the set of all valuative semistar operations on the valuation rings of T . (9) R is a valuation ring if and only if there exists a valuative semistar operation on R. Theorem 3.6.22. Let v : T −→  ∪ {∞} be a valuation on a ring T , let R = Tv , and let I ∈ Mod R (T ). (1) Suppose that R = T . Then I ⊆ v −1 (∞) if and only if I −1T = T , if and only if I vT = v −1 (∞), if and only if I tT = v −1 (∞). (2) If I is finitely generated, then α = min{v(x) : x ∈ I } ∈  ∪ {∞} exists, and if also I Ę v −1 (∞) (which holds if I is T -regular), then α ∈ . (3) Suppose that α = inf{v(x) : x ∈ I } ∈  exists. Then there exist y, z ∈ T such that α = v(y) and −α = v(z), and then one has I −1T = {x ∈ T : v(x)  −α} = (z)v and I vT = Iv = {x ∈ T : v(x)  α} = (y)v . (4) Suppose that R = T . Then I tT = T for all I ∈ Mod R (T ). (5) Suppose that R = T . Then tT = (−)v , that is, I tT = Iv for all I ∈ Mod R (T ). Thus, an R-submodule of T is tT -closed if and only if it is closed under v. Moreover, tT is the unique valuative unital T -semistar operation on R. (6) The set of all R-submodules of T that are closed under v is totally ordered. (7) The set of all tT -closed R-submodules of T is totally ordered. (8) A prime ideal q of R is closed under v if and only if v −1 (∞) ⊆ q ⊆ pv . (9) Suppose that R = T . Then a prime ideal q of R is tT -closed if and only if q is closed under v, if and only if v −1 (∞) ⊆ q ⊆ pv . (10) R × = v −1 (0) ∩ T × . Corollary 3.6.23. Let R be a valuation ring with total quotient ring K = R, and let v : K −→  ∪ {∞} be any valuation on K such that R = K v . (1) t = (−)v , that is, I t = Iv for all I ∈ Mod R (T ). Thus, I ∈ K(R) is t-closed if and only if it is closed under v. (2) t is the unique valuative semistar operation on R. (3) I t = x∈I (x)t for all I ∈ K(R). (4) (0)t = v −1 (∞) = (R : K K ) is a prime ideal of R. (5) The set K(R)t of all t-closed Kaplansky fractional ideals of R is totally ordered. (6) A prime ideal q of R is t-prime if and only if q is closed under v, if and only if v −1 (∞) ⊆ q ⊆ pv . (7) t-Spec(R) = {q ∈ Spec(R) : v −1 (∞) ⊆ q ⊆ pv } is totally ordered. (8) t-Max(R) = {pv }. (9) R × = v −1 (0) ∩ K reg = v −1 (0)reg . (10) I ∈ K(R) is (strictly) t-finite if and only if I t = (x)t for some x ∈ I (or x ∈ K ). (11) I ∈ K(R) is t-invertible if and only if I is (strictly) t-finite and I Ę (R : K K ).

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(12) Invt (R) = Invt (R) = {(x)t : x ∈ K − (R : K K )} = {(x)t : x ∈ K and (x)t is regular}. (13) Clt (R) = Clt (R) is trivial if and only if (x)t is principal for all x ∈ K such that (x)t is regular, if and only if R is t-Marot. Corollary 3.6.24. Every valuation ring is a PVMR. Let  be a semistar operation on a ring R. The ring R is said to be -local if it has a unique -maximal ideal. Thus, any valuation ring that is not a total quotient ring is t-local. Another corollary of the theorem is that one can characterize the valuation rings via the t-operation. Corollary 3.6.25. Let R be a subring of a ring T . The following conditions are equivalent. (1) (2) (3) (4)

R is a valuation ring of T . There exists a valuative unital T -semistar operation on R. The T -semistar operation tT is valuative. For all x, y ∈ T , one has (x)tT ⊆ (y)tT or (y)tT ⊆ (x)tT , and for all x ∈ T − (R :T T ) there exists a y ∈ T such that (x y)tT = R. (5) The set of all tT -closed R-submodules of T is totally ordered, and every finitely generated I ∈ Mod R (T ) that is not contained in (R :T T ) is tT -invertible. Corollary 3.6.26. Let R be ring with total quotient ring K . The following conditions are equivalent. (1) (2) (3) (4)

R is a valuation ring. There exists a valuative semistar operation on R. The t-operation on R is valuative. For all x, y ∈ K , one has (x)t ⊆ (y)t or (y)t ⊆ (x)t , and for all x ∈ K − (R : K K ) there exists a y ∈ K such that (x y)t = R. (5) K(R)t is totally ordered, and every finitely generated I ∈ K(R) that is not contained in (R : R K ) is t-invertible. (6) The set of all t-closed ideals of R is totally ordered, and every finitely generated ideal I of R that is not contained in (R : R K ) is t-invertible. Proposition 3.6.27. Let v : T −→  ∪ {∞} be a valuation on a ring T . (1) aγ = {x ∈ T : v(x)  γ} is a Tv -submodule of T that is closed under v, for all γ ∈ . (2) (aγ )−1T = a−γ and (aγ )vT = aγ for all γ ∈ . T = aγ+δ and (aγ (aγ )−1T )tT = R for all γ, δ ∈ . (3) (aγ aδ )t (4) I vT = {aγ : γ ∈ , I ⊆ aγ } for all I ∈ K(R). Proposition 3.6.28. Let R be a valuation ring with total quotient ring K , and let v : K −→  ∪ {∞} be any valuation on K such that R = K v .

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(1) aγ = {x ∈ K : v(x)  γ} is an R-submodule of K that is closed under v, for all γ ∈ . (2) (aγ )−1 = a−γ and (aγ )v = aγ for all γ ∈ . t (3) (aγ aδ ) = aγ+δ and (aγ (aγ )−1 )t = R for all γ, δ ∈ . v (4) I = {aγ : γ ∈ , I ⊆ aγ } for all I ∈ K(R). (5) The map  −→ Invt (R) acting by γ −→ aγ is an anti-isomorphism of ordered abelian groups that induces a group isomorphism /v(K × ) −→ Clt (R). Proposition 3.6.29. Let R be a valuation ring with total quotient ring K = R, let v : K −→  ∪ {∞} be any valuation on K such that R = K v , and let p = pv . (1) The following conditions are equivalent for all x, y ∈ K with v(x) = ∞. (a) v(x) = v(y). (b) (x)t = (y)t . (c) (x)w = (y)w . (d) [x]R[p] = [y]R[p] (e) There exist u, v ∈ R − p such that ux = vy. (2) I t = I v = {x ∈ K : v(x) = v(y) for some y ∈ I } for all I ∈ K(R). (3) I w = {(I : K a) : a ∈ R − p} = [I ]R[p] for all I ∈ K(R). (4) I t = I w for all regular I ∈ K(R), and therefore t  w.

3.7 Krull Rings, Dedekind Rings, and r-UFRs In this section, we use the tools developed in Sections 3.3, 3.5, and 3.6 to generalize the characterizations of the Krull domains and Dedekind domains in Section 3.4 to characterizations of the Krull rings and Dedekind rings. It behooves us to determine the regular t-maximal ideals of any Krull ring as we did in Proposition 3.4.5 for the Krull domains. Definition 3.7.1. If R is a ring, then X 1 (R) denotes the set of all height one primes 1 (R) denotes the set of all primes of R that are minimal among the of R and X reg regular primes of R. The regular height ht reg p of a prime ideal p of a ring R is the supremum of the number of primes in any chain (rather than the length of the chain) of regular primes contained in p. The primes of regular height zero are the non-regular primes, 1 (R) is the set of all primes of R of regular height one. Since every prime and X reg contains a minimal prime and all minimal primes are non-regular, the regular height of a prime p is less than or equal to the usual height of p. More generally, one has

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ht reg p  ht p  htreg p + dim T (R)p . Thus equalities hold if dim T (R)p = 0, that is, if all non-regular primes contained in p are minimal. In particular, the notions agree if R is an integral domain, in which case 1 1 (R), and moreover X 1 (R)reg ⊆ X reg (R) for any ring R. The regular X 1 (R) = X reg dimension dimreg R of a ring R is the supremum of the regular heights of the primes of R. Thus dimreg R  dim R  dimreg R + dim T (R), and equalities hold if dim T (R) = 0, that is, if all non-regular primes of R are minimal. The rings of regular dimension zero are the total quotient rings. Consequently, every zero dimensional ring is of regular dimension zero, hence a total quotient ring. However, there exist rings ofarbitrary regular dimension that are infinite dimenn sional. For example, the ring ∞ n=1 (Z/ p Z) is a ring of regular dimension zero that is infinite dimensional [82, Example 4]. In Theorem 3.7.29 we show that a Dedekind ring is equivalently a Krull ring of regular dimension at most one. 1 (R) ⊆ wAss(K /R) for any ring R with total quotient Lemma 3.7.2. One has X reg ring K . 1 (R), then p is minimal over any (and therefore some) regular princiProof. If p ∈ X reg pal ideal contained in it, and every regular principal ideal of R is a conductor ideal. 

Regular t-height and regular t-dimension are defined analogously to how regular height and regular dimension were defined earlier, replacing regular primes with regular t-primes. A t-prime p of a ring R is of regular t-height zero if it is not regular. A t-prime p of a ring R is of regular t-height one if it is minimal among the regular 1 (R) ⊆ t-Spec(R)reg and every regular prime contains a t-primes of R. Since X reg regular t-prime, the primes of regular height one and the t-primes of t-regular height one coincide. Regular t-dimension zero means that there are no regular t-primes, which is equivalent to being a total quotient ring. A ring of regular t-dimension at most one is a ring in which every regular t-prime ideal is t-maximal. Consequently, by Lemma 3.7.2 and Corollaries 3.3.12 and 3.3.19, we have the following. Proposition 3.7.3. Let R be a ring with total quotient ring K . The following conditions are equivalent. (1) (2) (3) (4) (5)

R is of regular t-dimension at most one. t-Max(R)reg = t-Spec(R)reg . 1 (R) = t-Spec(R)reg . X reg 1 X reg (R) = t-Max(R)reg . 1 X reg (R) = t-Max(R)reg = wAss(K /R) = sK(K /R) = t-Spec(R)reg .

Moreover, R is an H ring of regular t-dimension at most one if and only if 1 (R) = t-Max(R)reg = Ass(K /R) = wAss(K /R) = sK(K /R) = t-Spec(R)reg . X reg

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219

By Exercise 3.3.17, every t-invertible t-prime of a ring is t-maximal. It follows that every Krull ring is of regular t-dimension at most one. A more direct proof of this fact is provided below. Proposition 3.7.4. Let R be a Krull ring. Then R is of regular t-dimension at most one; in other words, every regular t-prime ideal of R is t-maximal. Proof. Let p be a regular t-prime ideal. Suppose to obtain a contradiction that q is a regular t-prime ideal of R such that p  q. Let J = q−1 p, which is an ideal of R with J q ⊆ p. Since p is prime and q Ę p, one has q−1 p = J ⊆ p. Since p is regular and thus t-invertible, one therefore has q−1 ⊆ R and thus (q−1 q)t ⊆ qt = q, which contradicts the fact that q is also t-invertible.  Since every Krull ring is an H ring, Propositions 3.7.3 and 3.7.4 yield the following. Corollary 3.7.5. Let R be a Krull ring with total quotient ring K . One has 1 (R) = t-Max(R)reg = Ass(K /R) = wAss(K /R) = sK(K /R) = t-Spec(R)reg . X reg

To summarize several prior results, we note the following. Proposition 3.7.6. Let R be a ring with total quotient ring K . (1) One has 1 X reg (R) ∪ Ass(K /R) ⊆ wAss(K /R) ⊆ sK(K /R) ⊆ t-Spec(R)reg

Moreover, t-Max(R)reg is the set of all maximal elements of the poset 1 (R) is the set of all minimal elements of the poset t-Spec(R)reg , while X reg reg t-Spec(R) . (2) R is an H ring if and only if t-Max(R)reg ⊆ Ass(K /R) ⊆ wAss(K /R) ⊆ sK(K /R) ⊆ t-Spec(R)reg . (3) If R is Mori, then R is a H ring and t-Max(R)reg ⊆ Ass(K /R) = wAss(K /R) = sK(K /R) = t-Spec(R)reg . (4) R is of regular t-dimension at most one if and only if 1 (R) = t-Max(R)reg = wAss(K /R) = sK(K /R) = t-Spec(R)reg . X reg

(5) R is an H ring of regular t-dimension at most one if and only if 1 (R) = t-Max(R)reg = Ass(K /R) = wAss(K /R) = sK(K /R) X reg

= t-Spec(R)reg .

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(6) If R is Krull, then R is a Mori ring of regular t-dimension at most one and 1 (R) = t-Max(R)reg = Ass(K /R) = wAss(K /R) = sK(K /R) X reg

= t-Spec(R)reg . Along with the t-primes, another tool used to study the Krull rings is that of a discrete rank one valuation ring, defined below. Definition 3.7.7. (1) A valuation pair (R, p) of a ring T is discrete rank one if the value group of some (or any) valuation v on T such that R = Tv and p = pv is isomorphic to Z. (2) A valuation ring R is discrete rank one if (R, p) is a discrete rank one valuation pair of T (R) for some prime ideal p of R, or equivalently if the value group of R is isomorphic to Z. (3) A valuation ring R is discrete rank zero if R is a total quotient ring, or equivalently if the value group of R is trivial. (4) A valuation ring R is discrete rank at most one if R is discrete rank zero or discrete rank one. Example 3.7.8. A total quotient ring is equivalently a discrete rank zero valuation ring. By Corollary 3.1.11, a DVR is equivalently a discrete rank at most one valuation ring that is a domain. We wish to prove the following characterization of the Krull rings. Theorem 3.7.9 ([118, 143]). Let R be a ring with total quotient ring K . The following conditions are equivalent. (1) R is a Krull ring. (2) R is a completely integrally closed Mori ring. (3) R is of finite t-character and (R[p] , [p]R[p] ) is a discrete rank one valuation pair of K for every regular t-maximal (or every regular t-prime) ideal p of R. (4) There exists a family {(Rλ , pλ )}λ∈ of discrete rank one valuation pairs of K  such that R = {Rλ : λ ∈ } and for every regular x ∈ R one has x ∈ pλ for only finitely many λ ∈ . (5) There exists a family{(Rλ , pλ )}λ∈ of discrete rank at most one valuation pairs of K such that R = {Rλ : λ ∈ } and for every regular x ∈ R one has x ∈ pλ for only finitely many λ ∈ . }λ∈ of discrete rank at most one valuations vλ : K −→ (6) There exists a family {vλ Z ∪ {∞} such that R = {Rvλ : λ ∈ }, where Rvλ = {x ∈ K : vλ (x)  0} and for every regular x ∈ R one has vλ (x) = 0 for all but finitely many λ ∈ . Remark 3.7.10. Huckaba introduced Krull rings in 1976, defining them as rings satisfying condition (6) of the theorem [106]. In 1980, R. E. Kennedy proved them completely integrally closed and Mori [118], and 2 years later Matsuda proved the

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converse, namely, that a completely integrally closed Mori ring is Krull [143, Theorem 5]. Condition (3) of the theorem provides a canonical representation of a Krull ring as an intersection of discrete rank one valuation rings. It is well-known that every nonzero ideal of a Dedekind domain factors as a product of maximal ideals, uniquely up to reordering the factors (which generalizes the fact that any PID is a UFD). This generalizes to any t -Dedekind ring as follows. Proposition 3.7.11. Let  be an r-finite type semistar operation on a ring R, and suppose that R is -Dedekind. Then every regular -closed ideal of R can be written as a -product of regular -maximal ideals of R, uniquely up to reordering the factors. In other words, for every regular -closed ideal I of R, there exists a unique finite set {p1 , . . . , pk } of distinct regular -maximal ideals pi of R and unique positive integers e1 , . . . , ek such that I = (pe11 · · · pekk ) . Moreover, every regular -prime of R is -maximal, that is, -Max(R)reg =  Spec(R)reg . Proof. First we prove existence. Suppose to obtain a contradiction that there exists a regular -closed ideal of R that cannot be written as a -product of regular -maximal ideals of R. Since R is -Noetherian, by Zorn’s lemma the set of all such ideals of R has a maximal element I . The ideal I , being necessarily a proper -closed ideal of R, is contained in some -maximal ideal p of R. Let J = (I p−1 ) , which is a closed ideal of R containing I . If J = I , then since I is -invertible one has p = R, which is a contradiction. Therefore I  J . By maximality of I , then, J is a -product (p1 · · · pk ) of regular -maximal ideals of R, whence, so is I = (pp1 · · · pk ) . This is our desired contradiction. Next, we prove uniqueness. Suppose that (p1 · · · pk ) = (q1 · · · ql ) , where the pi and q j are -maximal. Clearly, if k = 0 then l = 0 and there is nothing to prove. Suppose that k, l  1. Then p1 ⊇ (q1 · · · ql ) ⊇ q1 · · · ql , so p1 ⊇ qi for some i. Since qi is -maximal, it follows that p1 = qi . Then since both are -invertible we can cancel p1 and qi from the equation (p1 · · · pk ) = (q1 · · · ql ) and proceed by induction. Finally, if p is a regular -prime, then, writing p = (p1 · · · pk ) for -maximal ideals pi , we see that p ⊇ pi for some i, whence p = pi since pi is -maximal.  Corollary 3.7.12. Let  be an r-finite type semistar operation on a ring R. Then R is -Dedekind if and only if the monoid Freg (R) of all -closed regular fractional ideals of R is a group (or equivalently is equal to its group Inv (R) of invertible elements), in which case it is the free abelian group on the set -Max(R)reg = -Spec(R)reg of all regular -maximal (or regular -prime) ideals of R. Corollary 3.7.13. A ring R is Dedekind if and only if the monoid Freg (R) of all regular fractional ideals of R is a group (or equivalently is equal to the group Inv(R) of invertible ideals of R), in which case it is the free abelian group on the set Max(R)reg = Spec(R)reg of all regular maximal (or regular prime) ideals of R. Definition 3.7.14. Let R be a Krull ring. For any I ∈ K(R) and any positive integer n, we let I −n = (I n )−1 . By Corollary 3.7.12, for any regular fractional ideal I of R there exists a unique finite set {p1 , . . . , pk } of distinct regular t-maximal ideals pi

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of R and unique nonzero (positive or negative) integers e1 , . . . , ek such that I t = (pe11 · · · pekk )t . We let ordp (I ) = ei if p = pi for some i, and ordp (I ) = 0 otherwise. Equivalently, ordp (I ) is the largest nonnegative integer e such that I ⊆ (pe )t . If instead R is -Dedekind for some r-finite type semistar operation , then R is a Krull ring and v  t  , so there is no harm in assuming R is Krull rather than -Dedekind in the definition above (since then t-Max(R)reg = -max(R)reg and one has I t = (pe11 · · · pekk )t if and only if I  = (pe11 · · · pekk ) ). Lemma 3.7.15. Let R be a Krull ring, and let p be a regular t-maximal (or regular t-prime) ideal of R and I and J regular fractional ideals of R. (1) (2) (3) (4) (5) (6)

ordp (I t ) = ordp (I ). If I ⊆ J , then ordp (I )  ordp (J ). ordp (I J ) = ordp (I ) + ordp (J ). ordp (I + J ) = min{ordp (I ), ordp (J )}. ordp (I ∩ J ) = max{ordp (I ), ordp (J )}. ordp (I ) = sup{ordp (I + pn ) : n ∈ Z}.

Proof. Statements (1)–(3) are clear. By multiplying by a suitable regular element of R we may assume that I and J are ideals of R, and then by t-multiplying by a suitable power of p we may assume that ordp (I ) = 0, that is, p Ğ I . Since p Ğ I + J , we have ordp (I + J ) = 0. This proves (4), from which (6) immediately follows. Finally, statement (5) follows from statements (3) and (4) and the equation ((I + J )(I ∩  J ))t = (I J )t , which holds by Theorem 2.11.9. Definition 3.7.16. Let R be a Krull ring, and let p be a regular t-maximal ideal of R. For any fractional ideal I of R, we let vp (I ) = sup{ordp (I + pn ) : n ∈ Z}. For any a ∈ T (R), we set vp (a) = vp (a R) ∈ Z ∪ {∞}. As a consequence of the previous lemma, one has vp (I ) = ordp (I ) for all regular fractional ideals I of R. Proposition 3.7.17. Let  be an r-finite type semistar operation on a ring R, and suppose that R is -Dedekind. Let p be a regular -maximal (or regular -prime) ideal of R and I and J fractional ideals of R. (1) vp (I ) = ordp (I ) if I is regular. (2) The sequence ordp (I + pn ) of elements of Z ∪ {∞} is nondecreasing as n → ∞, and therefore vp (I ) = limn→∞ ordp (I + pn ). (3) vp (I  ) = vp (I ). (4) If I ⊆ J , then vp (I )  vp (J ). (5) vp (I J ) = vp (I ) + vp (J ). (6) vp (I + J ) = min{vp (I ), vp (J )}.

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vp (I ∩ J )  max{vp (I ), vp (J )}, with equality if I and J are regular. If I is an ideal of R, then vp (I ) = sup{n ∈ Z : I ⊆ (pn ) }. vp (I ) = min{vp (a) : a ∈ I }, and there exists a c ∈ I such that vp (c) = vp (I ). The map vp : T (R) −→ Z ∪ {∞} acting by vp : a −→ vp (a R) for all a ∈ T (R) is a discrete rank one valuation on T (R). (11) One has T (R)vp = R[p] and pvp = [p]R[p] . (7) (8) (9) (10)

Proof. Statements (1)–(3) are clear, and then statements (4), (6), and (7) follow from Lemma 3.7.15. To prove (5), we may assume without loss of generality that I and J are ideals of R. In that case, for any positive integer n, one has (I + p2n )(J + p2n ) ⊆ I J + p2n ⊆ (I + pn )(J + pn ), and therefore ordp (I + p2n ) + ordp (J + p2n )  ordp (I J + p2n )  ordp (I + pn ) + ordp (J + pn ),

so taking limits we see that vp (I ) + vp (J )  vp (I J )  vp (I ) + vp (J ), whence equalities hold. To prove (8), suppose that I is an ideal of R. Suppose that I ⊆ (pn ) . Then I + pn ⊆ (pn ) , so that vp (I )  ordp (I + pn )  ordp (pn ) = n. Conversely, suppose that vp (I )  n. Then ordp (I + pk )  n for some k. We may then write (I + pk ) = (pn J ) for some ideal J of R, and then we have I ⊆ (I + pk + pn ) = (pn J + pn ) = (pn ) . To prove (9), we may assume that I is an ideal of R. Let n = vp (I ). First, assume that n < ∞. By (8) we have I Ę (pn+1 ) . Choose c ∈ I − (pn+1 ) . Then c R ⊆ I ⊆ (pn ) but c R Ę (pn+1 ) , so that vp (c) = n, again by (8). On the other hand, if n = ∞, then for all a ∈ I one has a R ⊆ I ⊆ (pk ) for all k and therefore v(a) = ∞. Thus, we have shown in either case that there exists a c ∈ I such that vp (c) = vp (I ). Now, let a ∈ I be arbitrary. Observe that a R ⊆ I , and therefore vp (I )  v(a), for all a ∈ I , so vp (I )  inf{vp (a) : a ∈ I }. But also c ∈ I , so that vp (I ) = vp (c)  inf{vp (a) : a ∈ I }. This proves (9). Next we prove (10). Let a, b ∈ K . Then (a + b)R ⊆ a R + b R and therefore vp (a + b)  vp (a R + b R) = min{vp (a), vp (b)}. Likewise, vp (ab) = vp (a Rb R) = vp (a) + vp (b). Moreover, vp is surjective by (9) since for all n there exists a c ∈ pn such that vp (c) = vp (pn ) = n. Therefore vp is a discrete rank one valuation on T (R). Finally, we prove (11). Note first that, by (8), for all y ∈ R one has vp (y) = 0 if and only if y ∈ R − p. Let x = a/b ∈ T (R) with a, b ∈ R and b regular. If x ∈ R[p] , then x y ∈ R for some y ∈ R − p and therefore vp (x) = vp (x y)  0. Likewise, if x ∈ [p]R[p] , then x y ∈ p for some y ∈ R − p and therefore vp (x) = vp (x y) > 0. To prove the converse, for each -maximal ideal q = p we may choose cq ∈ R with

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vp (cq ) = 0 and vq (cq ) = 1. Indeed, cq = aq + bq satisfies this condition, where aq is any element of q2 − p and bq is any element of pq − (q2 ) , since then vp (aq ) = 0 and vp (bq ) > 0, and vq (aq ) > 1 and vq (bq ) = 1. Now, suppose that vp (x)  0, so that  v (b) vp (a)  vp (b). Let y = q =p,b∈q cqq . Then y ∈ R − p and ypvp (b) ⊆ (b R) = b R and so x y R = ab−1 y R ⊆ ap−vp (b) ⊆ ap−vp (a) ⊆ R, whence x y ∈ R. Therefore x ∈ R[p] . Similarly, if vp (x) > 0, then vp (a)  vp (b) + 1 and so x y R ⊆ ap−vp (b) ⊆ ap−vp (a) p ⊆ p and therefore x ∈ [p]R[p] .  Lemma 3.7.18. Let R be a ring with total quotient ring K . Suppose there exists a family  {vλ }λ∈ of discrete rank one valuations vλ : K −→ Z ∪ {∞} such that R = {Rvλ : λ ∈ }, where Rvλ = {x ∈ K : vλ (x)  0}. Let vλ (I ) = min{vλ (x) : x ∈ I } for all I ∈ F(R) and all λ. One has the following. (1) vλ (I ) = min{vλ (x) : x ∈ I } ∈ Z ∪ {∞} is well-defined for all I ∈ F(R). (2) Let I, J ∈ F(R) with I regular and divisorial. Then J ⊆ I if and only if vλ (J )  vλ (I ) for all λ. (3) Suppose further that, for every regular x ∈ R one has vλ (x) = 0 for all but finitely many λ ∈ . If J ∈ F(R) is regular, then vλ (J ) = 0 for all but finitely many λ. In particular, any regular ideal J of R is contained in at most  λ: vλ (J ) =0 (1 + vλ (J )) < ∞ divisorial ideals of R, and therefore R is Mori. Proof. Note that vλ (J ) is well-defined for all J ∈ F(R) since and c J ⊆ R for some regular element c of R (which must satisfy v(c) < ∞) and then vλ (x)  −v(c) for all x ∈ J . Suppose that I is regular and vλ (J )  vλ (I ) for all λ, and let x ∈ J . Then vλ (x I −1 ) = vλ (x) + vλ (I −1 )  vλ (I ) + vλ (I −1 ) = vλ (I I −1 )  0 for all λ, and therefore x I −1 ⊆ R, so x ∈ (I −1 )−1 = I . Therefore J ⊆ I . The converse is obvious. Now, suppose the hypothesis in statement (3), and let J ∈ F(R) be regular. We must show that vλ (J ) = 0 for all but finitely many λ. Let c ∈ R with c J ⊆ R and c regular. Since vλ (c) = 0 for all but finitely many λ and vλ (c J ) = vλ (c) + vλ (J ), we may assume without loss of generality that J ⊆ R. Let a be a regular element of J . If vλ (J ) = 0, then vλ (x) > 0 for all x ∈ J , and, in particular, vλ (a) > 0. But by hypothesis there exist only finitely many such λ. Therefore, vλ (J ) = 0 for all but finitely many λ. Moreover, if J is contained in some regular divisorial ideal I , then vλ (I ) ∈ {0, 1, 2, . . . , vλ (J )} for all J , and therefore by statement (2) there are  at most λ: vλ (J ) =0 (1 + vλ (J )) < ∞ such ideals I . We are now ready to prove Theorem 3.7.9. Proof of Theorem 3.7.9. Conditions (1) and (2) are equivalent by Corollary 2.6.20. To see that (1) implies (3), apply statements (10) and (11) of Proposition 3.7.17 if p ∈ t-Spec(R) is regular and note that K = R[p] if p is not regular. Moreover,  since R = p∈t-Max(R)reg R[p] for any ring R and [p]R[p] ∩ R = p for any prime p, condition (3) implies condition (4). The implications (4) ⇒ (5) ⇔ (6) are obvious. Suppose that condition (6) holds. Let x ∈ K and c a regular element of R such that cx n ∈ R for all n. Then vλ (x)  − n1 vλ (c). Thus vλ (x)  supn {− n1 vλ (c)} = 0 for all

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λ, so that x ∈ R. Thus R is completely integrally closed. Moreover, R is Mori by Lemma 3.7.18. Therefore (6) ⇒ (2).  Definition 3.7.19. Let p be a prime ideal of a ring R. For any nonnegative integer n, the ideal p(n) = pn Rp ∩ R = [pn ]R[p] ∩ R of R is called the nth symbolic power of p. Proposition 3.7.20. Let  be an r-finite type semistar operation on a ring R, and suppose that R is -Dedekind. Let I, J ∈ F(R), and let p ∈ -Max(R)reg . (1) vp (I −1 ) + vp (I )  0, with equality if I is regular. (2) If I is regular, then vq (J )  vq (I ) for all q ∈ -Max(R)reg if and only if J ⊆ I  . (3) If I is an ideal, then vp (I ) = 0 if and only if I Ę p. Consequently, vp (q) = 0 if q = p, for all q ∈ -Max(R).  (4) If I is regular and J = p pvp (I ) , then vq (J ) = vq (I ) for all q ∈ -Max(R)reg and consequently I  = J  . (5) [I ]R[p] ⊆ {x ∈ T (R) : vp (x)  vp (I )}, with equality if I is regular. (6) I  = p∈-Max(R)reg [I ]R[p] = {x ∈ T (R) : vp (x)  vp (I ) for all p ∈ -Max (R)reg } if I is regular. vp (I ) ]R[p] if I is regular. (7) [I  ]R [p] = [I ]R[p] = [p  vp (I ) ]R[p] if I is regular. (8) I = p∈-Max(R)reg [p  (9) I  = p∈-Max(R)reg p(vp (I )) if I is a regular ideal. (10) If I is regular, then the following conditions are equivalent. (a) I is divisorial. (b) I is -closed.  (c) I = p∈-Max(R)reg [I ]R[p] .  (d) I = p∈-Max(R)reg [pvp (I ) ]R[p] . Proof. Throughout the proof we use Proposition 3.7.17, especially the fact that vp : T (R) −→ Z ∪ {∞} is a valuation and vp (I ) = min{vp (a) : a ∈ I }. Statement (1) is clear. Statement (2) follows from Lemma 3.7.18 and the fact that vq (I  ) = vq (I ). Statement (3) follows from the definition of vp (I ), and (4) follows from (2) and (3). To prove (5), we may suppose without loss of generality that I is an ideal of R. Let x ∈ [I ]R[p] , so that there exists y ∈ R − p such that x y ∈ I . Then vp (x) = vp (x y)  vp (I ). Conversely, suppose that I is regular, and let x ∈ T (R) with vp (x)  vp (I ). Let a ∈ I with vp (a) = vp (I ). Since I is regular, vp (I ) < ∞. Therefore there exists a b ∈ T (R) such that vp (b) = −vp (a), so that vp (ab) = 0, whence ab ∈ R[p] − [p]R[p] . Thus we have uab ∈ R − p for some u ∈ R − p. Moreover, vp (bx) = vp (x) + vp (b) = vp (x) − vp (a) = vp (x) − vp (I )  0, so that bx ∈ R[p] , whence wbx ∈ R for some w ∈ R − p. Since a ∈ I and u, wbx ∈ R, one has uwabx ∈ I . Therefore y = wuab ∈ R − p satisfies yx ∈ I , so x ∈ [I ]R[p] . This proves (5). Next, (6) follows immediately from (2) and (5), and (7) follows from (5) since vp (I ) = vp (pvp (I ) ). Moreover, (8), (9), and (10) follow immediately from (6) and (7). 

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 Recall that if  is a set then the free abelian group Z() = λ∈ Z is a (partially) ordered group under the coordinatewise partial ordering induced from that on Z. From Proposition 3.7.20, we deduce the following generalization of Theorem 3.4.9 to rings with zerodivisors. Corollary 3.7.21. Let R be a Krull ring. (1) A regular fractional ideal of R is divisorial if and only if it is of the form I (n) =



1 [pn p ]R[p] = {x ∈ T (R) : vp (x)  n p for all p ∈ X reg (R)}

1 (R) p∈X reg

(X reg (R)) 1 (R) ∈ Z for some n = (n p )p∈X reg . Moreover, n p = vp (I (n)) for all 1 p ∈ X reg (R). (2) The map n −→ I (n) defines an order-reversing isomorphism from the ordered 1 (free abelian) group Z(X reg (R)) onto the group Freg (R)v = Invv (R) of divisorial (v-invertible) regular fractional ideals of R under v-multiplication. The inverse 1 (R) . map acts by I −→ (vp (I ))p∈X reg 1

1 (3) p = I (ep ) for all p ∈ X reg (R), where ep ∈ Z(X reg (R)) is the elementary unit vector with nonzero p-th coordinate. In particular, the group Freg (R)v under 1 (R). v-multiplication is the free abelian group on the set X reg  (4) A regular divisorial ideal I of R is contained in exactly 1 (R) p∈X reg (1 + vp (I )) < ∞ many regular divisorial ideals of R. 1

Remark3.7.22. For any Krull ring R, the function d R : I(R)reg −→ Z>0 given by I −→ p∈X reg 1 (R) (1 + vp (I )) is analogous to (and is equal to in the case R = Z if one identifies I(Z)reg with Z>0 ) the well-known divisor function d : Z>0 −→ Z>0 , where d(n) for any positive integer n is the number of positive integer divisors of n. Moreover, one has d R (I J ) = d R (I )d R (J ) for all I, J ∈ I(R)reg such that (I + J )v = R, which generalizes the fact that d(mn) = d(m)d(n) for all m, n ∈ Z>0 with gcd(m, n) = 1. The following result provides three more equivalent characterizations of the Krull rings. Proposition 3.7.23. Let R be a ring. The following conditions are equivalent. (1) R is a Krull ring. (2) R[p] is a Krull ring for every p ∈ sK(T (R)/R) and every regular element of R lies in at most finitely many p ∈ sK(T (R)/R). (3) R[p] is a Krull ring for every p ∈ wAss(T (R)/R) and every regular element of R liesin at most finitely many p ∈ wAss(T (R)/R). 1 1 (R)}, R[p] is a Krull ring for every p ∈ X reg (R), and every (4) R = {R[p] : p ∈ X reg 1 regular element of R lies in at most finitely many p ∈ X reg (R).

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Proof. It is clear that condition (1) implies conditions (2), (3), and (4). Suppose that (2) holds. Let p ∈ sK(T (R)/R), so that R  = R[p] is Krull. Since p = [p]R[p] is a t   prime of R  by Proposition 3.5.4, it follows that (R[p  ] , [p] R[p ] ) = (R[p] , [p]R[p] ) is a   discrete rank one valuation pair of T (R ) = T (R). Moreover, one has R = {R[p] : p ∈ sK(T (R)/R)}, and by hypothesis every regular element of R lies in at most finitely many p ∈ sK(T (R)/R). Therefore R is Krull by Theorem 3.7.9. Therefore (2) ⇒ (1). The proofs that (3) ⇒ (1) and (4) ⇒ (1) are similar.  Open Problem 3.7.24. If R is of finite t-character and R[p] is a Krull ring for every t-prime p of R, then does it follow that R is a Krull ring? The following result shows that a valuation ring is Krull if and only if it is discrete rank at most one. Proposition 3.7.25. Let R be a ring. The following conditions are equivalent. (1) (2) (3) (4) (5)

R is a Krull valuation ring. R is a Krull ring with at most one (regular) t-maximal ideal. R is a discrete rank at most one valuation ring. The ordered monoid Freg (R)t is cyclic (hence trivial or isomorphic to Z). The group Invt (R) is trivial or infinite cyclic and generated by a regular prime ideal of R. (6) R is a total quotient ring or there exists a regular prime ideal p of R such that for every regular element a of R one has (a) = (pn )t for some nonnegative integer n. (7) R is a Mori valuation ring. (8) R is a valuation ring and for every regular ideal I of R one has I t = (a)t for some a ∈ R. Proof. We may suppose without loss of generality that R is not a total quotient ring. One has (1) ⇒ (2) by Corollary 3.6.23 and (2) ⇒ (3) ⇒ (1) by Theorem 3.7.9. Suppose that (4) holds. Since Freg (R)t is a group, every regular fractional ideal is t-invertible, and therefore R is Krull. Then, by Corollary 3.7.12, the group Freg (R)t is cyclic if and only if R has at most one t-maximal ideal. Therefore (4) ⇒ (2). Likewise, by Corollary 3.7.12, one has (2) ⇒ (4), and clearly (4) ⇒ (5) ⇒ (6). Next, we show that (6) ⇒ (3), which then implies that conditions (1)–(6) are equivalent. Suppose that (6) holds. It follows that p is t-invertible. Moreover, we claim that p is v-closed, and hence t-closed. Indeed, since p ⊇ pv p−1 p and p is prime, one has p ⊇ pv or p ⊇ p−1 p, but the latter condition would imply that pt = R since p is t-invertible, which in turn would imply that Invt (R) is trivial, and therefore R is a total quotient ring. Therefore p is t-invertible and t-closed. Now, for all a ∈ R, let v(a) = sup{n ∈ Z0 : a ∈ (pn )t }. If a ∈ R is regular and, say, (a) = (pn )t for some nonnegative integer n, then clearly v(a) = n < ∞. One can then show that the map v : R −→ Z ∪ {∞} extends uniquely to a paravaluation v : T (R) −→ Z ∪ {∞} on T (R). (See Exercise 3.7.7.) Now, since p = (p2 )t , there exists an element π of p such that v(π) = 1. Let a0 be any regular element of p, so that v(a0−1 ) = −n for some

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positive integer n. Then v(π n−1 a0−1 ) = −1. Therefore v is surjective, whence v is a valuation on T (R). Finally, if v(a/b)  0, where a, b ∈ R and b is regular, then v(a)  v(b) and (b) = (pv(b) )t , whence a ∈ (pv(b) )t = (b) and therefore a/b ∈ R. Thus R = T (R)v . Therefore R is a discrete rank one valuation ring. Thus we have (6) ⇒ (3), as claimed. Finally, one has (1) ⇔ (7) ⇔ (8) by Corollaries 3.6.23(10) and 3.6.24.  Because of Propositions 3.7.23 and 3.7.25, the Krull valuation rings play the same role for the Krull rings as the DVRs do for the Krull domains (and Dedekind domains). Corollary 3.7.26. The ideal class group Cl(R) of a Dedekind valuation ring R, or more generally the t-class group Clt (R) of any Krull valuation ring (or discrete rank at most one valuation ring) R, is finite cyclic. Proof. Let R be a Krull valuation ring. We may suppose without loss of generality that R is not a total quotient ring. Then R has a unique regular t-maximal ideal p and the group Invt (R) is infinite cyclic and generated by p. Let a be any regular non-unit of R. Since (a) ∈ Invt (R), one has (a) = (pn )t for some positive integer n, and so  the generator p has finite order in Clt (R). Example 3.7.27 below shows that a Krull valuation ring need not be a Dedekind valuation ring and a Dedekind valuation ring need not be an r-PIR valuation ring. For domains, or more generally for Marot rings, all three conditions are equivalent: see Corollary 3.10.21. Example 3.7.27. (1) [107, Section 27 Example 7] provides an example of a discrete rank one valuation pair (R, p) of the total quotient ring of a ring R such that R is not Prüfer and p is not a maximal ideal of R and, in particular, such that R is a Krull valuation ring that is not Dedekind. (2) The ring R of Example 2.7.13 is a Dedekind valuation ring that has nontrivial (t-)class group and is therefore not an r-UFR (nor an r-PIR). Remark 3.7.28. (1) Although a Krull valuation ring need not be an r-UFR, it must be a t-UFR, where the t-UFRs are defined as in Remark 2.9.30. (See also Exercise 3.7.13.) It follows that a Krull valuation ring is equivalently a t-UFR with at most one (regular) t-maximal ideal. Similarly, by Corollary 3.8.6 of the next section, a valuation ring is equivalently a PVMR with at most one (regular) t-maximal ideal. (2) Thus far, we have characterized the PVMR valuation rings and the Krull valuation rings. The valuation rings that are r-PIR, Dedekind, r-Bézout, Prüfer, r-UFR, and r-GCD, respectively, are characterized in Section 3.10.

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Next, as an application of our various equivalent characterizations of the Krull rings, we provide several equivalent characterizations of the Dedekind rings. First note that, if p is a maximal ideal of a ring R, then p(n) = pn for all nonnegative k integers n. Moreover, if p1 , p2 , . . . , pk are distinct maximal ideals of R, then i=1 piei = k ei i=1 pi for any nonnegative integers e1 , e2 , . . . , ek . Theorem 3.7.29. Let R be a ring. The following conditions are equivalent. (1) (2) (3) (4) (5) (6) (7)

R is a Dedekind ring. R is a Krull ring of regular dimension at most one. R is a Krull ring in which every regular t-maximal ideal is maximal. R is a Krull ring in which every regular maximal ideal is t-closed. R is an integrally closed r-Noetherian ring of regular dimension at most one. R is a TD Krull ring. R is of finite character and (R[p] , [p]R[p] ) is a discrete rank one valuation pair of T (R) for every regular maximal ideal (or every regular prime ideal) p of R.

Proof. That (1) ⇒ (2) follows from Corollary 3.7.13, and it is clear that (2) ⇒ (3) ⇔ (4). Suppose that (3) holds. It follows from Proposition3.7.20(9) and the observations made prior to the statement of the theorem that I = p∈t-Max(R)reg pvp (I ) for all regular ideals I of R, and therefore by Proposition 3.7.20(4) every regular ideal I of R is v-closed. In particular, R is a Krull divisorial ring and is therefore Dedekind. Therefore statements (1)–(4) are equivalent. Moreover, one has (1) ⇒ (5) ⇒ (2) by Corollary 2.6.26. Next, by Theorem 3.7.9, one has (1) ⇒ (6) ⇒ (7). Finally, we show that (7) ⇒ (4). Suppose that (7) holds. Then R is a Krull ring by Theorem 3.7.9. Moreover, if p is a regular maximal ideal of R, then by Corollary 3.6.23(8) the prime ideal [p]R[p] is a t-maximal ideal of R[p] , whence p is t-closed by Proposition 3.5.8.  Corollary 3.7.30. Let D be an integral domain. The following conditions are equivalent. (1) (2) (3) (4) (5) (6) (7)

D is a Dedekind domain. D is a Krull domain of dimension at most one. D is a Krull domain in which every nonzero t-maximal ideal is maximal. D is a Krull domain in which every nonzero maximal ideal is t-closed. D is an integrally closed Noetherian domain of dimension at most one. D is a TD Krull domain. D is of finite character and Dp is a DVR for every maximal ideal p of D. Assuming Proposition 3.10.16 of Section 3.10, we can deduce the following.

Proposition 3.7.31. Let R be a ring with total quotient ring K . The following conditions are equivalent. (1) R is Dedekind. (2) R is of finite character and R[p] is a Dedekind valuation ring for every regular maximal ideal (or every prime ideal) p of R.

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(3) R is of finite character and R[p] is Dedekind for every regular maximal ideal (or every prime ideal) p of R. Proof. It is clear that (1) ⇒ (2) ⇒ (3). Suppose that (3) holds. We first show that R is Krull. If q  p are regular primes of R, then [q]R[p]  [p]R[p] are regular primes of the Dedekind ring R[p] , contradicting the fact that any Dedekind ring has regular dimension at most one. Therefore R also has regular dimension at most one, whence 1 (R). Consequently, condition (4) of by Proposition 3.7.23 holds, Max(R)reg = X reg whence R is Krull. Finally, since R[p] is Prüfer for all regular primes p of R, it follows from Proposition 3.10.16 of Section 3.10 that R is Prüfer. Therefore, since R is both Krull and Prüfer, we deduce that R is Dedekind, and therefore (3) ⇒ (1).  In particular, the Dedekind valuation rings play the same role for the Dedekind rings as the DVRs do for the Dedekind domains. Finally, let us say that a ring R is almost Dedekind if R[p] is Dedekind for every prime ideal p of R. Assuming Proposition 3.10.16 of Section 3.10, we have the following. Proposition 3.7.32. Let R be a ring. The following conditions are equivalent. (1) R is almost Dedekind. (2) R[p] is a Dedekind valuation ring for every regular maximal (or every prime) ideal p of R. (3) R is Prüfer and R[p] is Mori (or r-Noetherian) for every regular maximal (or every prime) ideal p of R. Corollary 3.7.33. Let R be a ring. The following conditions are equivalent. (1) (2) (3) (4)

R is Dedekind. R is almost Dedekind and of finite character. R is almost Dedekind and r-Noetherian. R is Prüfer and r-Noetherian.

3.8 PVMRs, Prüfer Rings, and -Prüfer Rings In this section, we characterize the Prüfer rings, PVMRs, and t -Prüfer rings for any semistar operation . The following theorem characterizes the Prüfer rings. Theorem 3.8.1 ([93, Theorem 13] [130, Theorem 10.18]). Let R be a ring with total quotient ring K . The following conditions are equivalent. (1) R is Prüfer. (2) (a, b) is invertible for all a, b ∈ R with a regular. (3) (R[p] , [p]R[p] ) is a valuation pair of K for every prime ideal p of R.

3.8 PVMRs, Prüfer Rings, and -Prüfer Rings

(4) (5) (6) (7) (8) (9) (10) (11)

231

(R[p] , [p]R[p] ) is a valuation pair of K for every regular maximal ideal p of R. Every overring of R is flat over R. Every overring of R is integrally closed. R is integrally closed and for all a, b ∈ R with a regular one has (a, b)2 = (a 2 , b2 ). R is integrally closed and for all a, b ∈ R with a regular one has (a, b)n = (a n , bn ) for some integer n > 1. R is integrally closed and for all a, b ∈ R with a regular one has a n−1 b ∈ (a n , bn ) for some integer n > 1. If I J = I H for I, J, H ∈ K(R) with I finitely generated and regular, then J = H. If I J = I H for regular ideals I, J, H of R with I finitely generated, then J = H.

Proof. (1) ⇒ (2). Clear. (2) ⇒ (3). Let x ∈ K − R[p] , where p is a prime ideal of R. Choose a regular element c of R such that cx ∈ R. By (2), then, the ideal (c, cx) of R is invertible, so there exists an I ∈ K(R) such that (c, cx)I = R. Now, cI x ⊆ R and cI ⊆ R, and x∈ / R[p] , so we must have cI ⊆ p. Therefore, if also cI x ⊆ p, then R = (c, cx)I = cI + cI x ⊆ p, which is a contradiction. So we must have cI x Ę p. Therefore there exists r ∈ I such that cr x ∈ / p. It follows that cr ∈ p and cr x ∈ R − p, so that cr ∈ [p]R[p] and cr x ∈ R[p] − [p]R[p] . Thus, we have shown that for an arbitrary x ∈ K − R[p] there exists an element a = cr of [p]R[p] such that ax ∈ R[p] − [p]R[p] , that is, (R[p] , [p]R[p] ) is a valuation pair of K . (3) ⇒ (4). Clear. (4) ⇒ (5). Let S be an overring of R, and let m be any regular maximal ideal of S. Let p = m ∩ R and q a (regular) maximal ideal of R containing p. Then R[q] ⊆ R[p] ⊆ S[m] . Let x ∈ K − R[p] . Since x ∈ K − R[q] , one has x y ∈ R − q ⊆ R − p for some y ∈ q. If y ∈ / p, then x y ∈ R would imply that x ∈ R[p] , which is a contradiction. Therefore y ∈ p. Now suppose to obtain a contradiction that x ∈ S[m] . Then there exists u ∈ S − m such that ux ∈ S. But y ∈ p ⊆ m and x y ∈ R − p ⊆ S − m. Hence (ux)y ∈ m but u(x y) ∈ S − m, which is absurd. Therefore x ∈ K − S[m] . Thus we have S[m] = R[m∩R] . Since this holds for every regular maximal ideal m of S, we conclude from Theorem 3.5.23 that S is flat over R. (5) ⇒ (6). Let S be an overring of R and let T be its integral closure. We wish to show that T = S. By (5), T is flat over R, so that (R : R x)T = T for all x ∈ T by Theorem 3.5.23. Therefore T ⊆ (S : R x)T ⊆ (S : S x)T ⊆ T , whence (S : S x)T = T , for all x ∈ T , and therefore T is flat over S. However, since T is integral over S, if (S : S x) is a proper ideal of S, then (S : S x)T is a proper ideal of T , by the “lying over” theorem [17, Theorem 5.10]. Therefore (S : S x) = S for all x ∈ T , that is, S = T . (6) ⇒ (7). Let a, b ∈ R with a regular. Let x = b/a ∈ K . Since R[x 2 ] is integrally closed and x is integral over R[x 2 ], one has x ∈ R[x 2 ]. Therefore we may write x = an x 2n + an−1 x 2(n−1) + · · · + a1 x 2 + a0

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with each ai ∈ R, with an = 0, and with n minimal subject to these conditions. We claim that n = 1. Clearly n > 0. Suppose to obtain a contradiction that n > 1. Multiplying the equation above by an2n−1 we see that an x is integral over R and therefore an x ∈ R. Then, multiplying the equation above by ann−1 , we have (an x 2 )n + an−1 (an x 2 )n−1 + · · · + ann−2 a1 (an x 2 ) + (ann−1 a0 − ann−2 (an x)) = 0, so that an x 2 is integral over R and therefore an x 2 ∈ R. But then the equation x = (an x 2 + an−1 )x 2(n−1) + an−2 x 2(n−2) + · · · + a1 x 2 + a0 contradicts the minimality of n. Thus we must have n = 1. Therefore we have x = a0 + a1 x 2 , and consequently ab = a0 a 2 + a1 b2 ∈ (a 2 , b2 ). It follows, then, that (a, b)2 = (a 2 , b2 ). (7) ⇒ (8) ⇒ (9). Clear. (9) ⇒ (7). Let a, b ∈ R with a regular, so that a n−1 b ∈ (a n , bn ) for some integer n > 1. Let m > 1 be the least so that a m−1 b ∈ (a m , bm ). We may write a m−1 b = ra m + sbm with r, s ∈ R. Let x = sb/a. Since x m − s m−2 x + r s m−1 = 0 and R is integrally closed, we have x ∈ R. Then a m−1 b = ra m + xabm−1 , whence a m−2 b = ra m−1 + xbm−1 and therefore a m−2 b ∈ (a m−1 , bm−1 ). By minimality of m, then, we must have m − 1 = 1, that is, m = 2. Therefore ab ∈ (a 2 , b2 ), whence (a, b)2 = (a 2 , b2 ). (7) ⇒ (2). Let a, b ∈ R with a regular. Since ab ∈ (a 2 , b2 ) we have ab = ra 2 + 2 sb for some r, s ∈ R. Let x = sb/a, so that x 2 − x + r s = 0, whence x is integral over R. Since R is integrally closed, it follows that x ∈ R, and therefore, since ra = (1 − x)b, one has (a, b)(s, 1 − x) = (as, bs, a(1 − x), b(1 − x)) = (as, ax, a(1 − x), ra) = (a). Therefore (a, b) is invertible. (2) ⇒ (4). Done already. (4) ⇒ (1). Suppose to obtain a contradiction that (4) holds but (1) does not, so there exists a finitely generated regular ideal I of R that is not invertible. Then there exists a prime ideal p of R such that I I −1 ⊆ p and thus I ⊆ p and p is regular. We claim that, since (R[p] , [p]R[p] ) is a valuation pair of K and I ⊆ R is finitely generated and regular, there exists a y ∈ K such that y I ⊆ R[p] but y I Ę [p]R[p] . It follows from this claim that there exists a u ∈ R − p such that uy I ⊆ R but uy I Ę p, and therefore I I −1 Ę p, which is our desired contradiction. To prove our claim, write I = (a1 , . . . , an ) with each ai ∈ R, and let v : K −→  ∪ {∞} denote any valuation on K such that R[p] = K v and [p]R[p] = pv . We may assume without loss of generality that v(a1 )  v(ai ) for all i. Since I is regular, I is not contained in the prime at infinity, and therefore v(a1 ) = ∞. Since v is surjective, there is a y ∈ K such that v(y) = −v(a1 ) and therefore v(ya1 ) = 0. Hence ya1 ∈ R[p] − [p]R[p] and yai ∈ R[p] for all i > 1. Therefore y I ⊆ R[p] and y I Ę [p]R[p] , as claimed. (1) ⇒ (10) ⇒ (11). Clear. (11) ⇒ (7). Observe that (a, b)3 = (a, b)(a 2 , b2 ) for all a, b ∈ R, so if a is reg ular then (a, b)2 = (a 2 , b2 ).

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Next, using similar methods, we characterize the t -Prüfer rings for any semistar operation . Definition 3.8.2. Let  be a semistar operation on a ring R. The ring R has enough regular -prime ideals if every proper regular -closed ideal of R is contained in a -prime ideal of R (or equivalently, if every regular ideal I of R such that I  = R is contained in some -prime ideal of R). By Lemma 3.2.8, if  is of r-finite type, then R has enough regular -prime ideals. Proposition 3.8.3. Let R be a ring with total quotient ring K , and let  be a semistar operation on R. Each of the following conditions implies the next. (1) R is -Prüfer. (2) (a, b) is -invertible for all a, b ∈ R with a regular. (3) (R[p] , [p]R[p] ) is a valuation pair of K for every -prime ideal p of R. Moreover, if R has enough regular -prime ideals, then all three conditions are equivalent. Proof. Trivially one has (1) ⇒ (2). Suppose that (2) holds. Let x ∈ K − R[p] , where p is -prime. Choose a regular element c of R such that cx ∈ R. By (2), the ideal (c, cx) of R is -invertible, so there exists an I ∈ K(R) such that ((c, cx)I ) = R. Now, cI x ⊆ R and cI ⊆ R, and x ∈ / R[p] , so we must have cI ⊆ p. Therefore, if also cI x ⊆ p, then R = ((c, cx)I ) = (cI + cI x) ⊆ p = p, which is a contradiction. So we must have cI x Ę p. Therefore there exists r ∈ I such that cr x ∈ / p. It follows that cr ∈ p and cr x ∈ R − p, so that cr ∈ [p]R[p] and cr x ∈ R[p] − [p]R[p] . Thus, we have shown that for an arbitrary x ∈ K − R[p] there exists an element a = cr of [p]R[p] such that ax ∈ R[p] − [p]R[p] , that is, (R[p] , [p]R[p] ) is a valuation pair of K . Thus (2) ⇒ (3). Finally, suppose that R has enough regular -prime ideals. We show that (3) ⇒ (1). Suppose to obtain a contradiction that (3) holds but (1) does not, so there exists a finitely generated regular ideal I of R that is not -invertible. Since (I I −1 ) = R, there exists a -prime ideal p of R such that I I −1 ⊆ p and thus I ⊆ p and p is regular. As in the proof of (4) ⇒ (1) of Theorem 3.8.1, since (R[p] , [p]R[p] ) is a valuation pair of K and I ⊆ R is finitely generated and regular, there exists a y ∈ K such that y I ⊆ R[p] but y I Ę [p]R[p] . It follows that there exists a u ∈ R − p such that uy I ⊆ R and uy I Ę p, and therefore I I −1 Ę p, which is our desired contradiction.  Theorem 3.8.4. Let R be a ring with total quotient ring K , and let  be a semistar operation on R. The following conditions are equivalent. (1) (2) (3) (4)

R is t -Prüfer. (a, b) is t -invertible for all a, b ∈ R with a regular. (R[p] , [p]R[p] ) is a valuation pair of K for every t -prime ideal p of R. (R[p] , [p]R[p] ) is a valuation pair of K for every regular t -maximal ideal p of R.

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(5) If (I J )w = (I H )w for I, J, H ∈ K(R) with I finitely generated and regular, then J w = H w . (6) If I J = I H for regular ideals I, J, H of R with I finitely generated, then J w = H w . (7) R is integrally closed and for all a, b ∈ R with a regular one has ((a, b)2 )w = (a 2 , b2 )w . (8) R is integrally closed and for all a, b ∈ R with a regular one has ((a, b)n )w = (a n , bn )w for some integer n > 1. (9) R is integrally closed and for all a, b ∈ R with a regular one has a n−1 b ∈ (a n , bn )w for some integer n > 1. Proof. We may suppose without loss of generality that  = t is of finite type. By Proposition 3.8.3 conditions (1)–(4) are equivalent. One has (1) ⇒ (5) since every finitely generated regular fractional ideal I of a t -Prüfer ring is t -invertible, hence w -invertible, so I can be canceled from both sides of the equation (I J )w = (I H )w . Moreover, one clearly has (5) ⇒ (6) ⇒ (7) ⇒ (8) ⇒ (9), where (6) ⇒ (7) since (a, b)3 = (a, b)(a 2 , b2 ). Thus it remains only to show that (9) ⇒ (7) ⇒ (2). Suppose that (9) holds. Let a, b ∈ R with a regular, and let m > 1 be the least so that a m−1 b ∈ (a m , bm )w . Then a m−1 b J ⊆ (a m , bm ) for some finitely generated ideal J = (d1 , d2 , . . . , dk ) of R with J  = R. For each i, we may write a m−1 bdi = ri a m + si bm with ri , si ∈ R. Let xi = si b/a. Since xim − sim−2 di xi + r sim−1 = 0 and R is integrally closed, we have x ∈ R. Then a m−1 bdi = ri a m + xi abm−1 , whence a m−2 bdi = ri a m−1 + xi bm−1 for all i and therefore a m−2 b J ⊆ (a m−1 , bm−1 ), so that a m−2 b ∈ (a m−1 , bm−1 )w . By minimality of m, then, we must have m − 1 = 1, that is, m = 2. Therefore ab ∈ (a 2 , b2 )w , whence ((a, b)2 )w = (a 2 , b2 )w . Therefore (9) ⇒ (7). Finally, suppose that (7) holds. Let a, b ∈ R with a regular. Since ab ∈ (a 2 , b2 )w we have abdi = ri a 2 + si b2 for some ri , si , di ∈ R with J = (d1 , d2 , . . . , dk ) a finitely generated ideal of R with J  = R. Let xi = si b/a, so that xi2 − di xi + ri si = 0, whence xi is integral over R. Since R is integrally closed, it follows that xi ∈ R. Let Ii = (si , di − xi ), which is an ideal of R. Since ri a = b(di − xi ), one has (a, b)Ii = (asi , bsi , a(di − xi ), b(di − xi )) = (asi , axi , a(di − xi ), ri a) = (a)(di , xi , ri , si )

and consequently di a ∈ (a, b)Ii ⊆ (a) for all i. Let I = I1 + I2 + · · · + Ik , which is an ideal of R with J a ⊆ (a, b)I ⊆ (a) and therefore J ⊆ (a, b)I a −1 ⊆ R. Finally, we see that R = J t ⊆ ((a, b)I a −1 )t ⊆ R, so that (a, b) is t -invertible. Thus (7) ⇒ (2).  Corollary 3.8.5 ([183, Theorem 2.1]). Let R be a ring with total quotient ring K . The following conditions are equivalent. (1) R is a PVMR. (2) (a, b) is t-invertible for all a, b ∈ R with a regular.

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(3) (R[p] , [p]R[p] ) is a valuation pair of K for every t-prime ideal p of R. (4) (R[p] , [p]R[p] ) is a valuation pair of K for every regular t-maximal ideal p of R. (5) If (I J )w = (I H )w for I, J, H ∈ K(R) with I finitely generated and regular, then J w = H w . (6) If I J = I H for regular ideals I, J, H of R with I finitely generated, then J w = H w. (7) R is integrally closed and for all a, b ∈ R with a regular one has ((a, b)2 )w = (a 2 , b2 )w . (8) R is integrally closed and for all a, b ∈ R with a regular one has ((a, b)n )w = (a n , bn )w for some integer n > 1. (9) R is integrally closed and for all a, b ∈ R with a regular one has a n−1 b ∈ (a n , bn )w for some integer n > 1. Corollary 3.8.6. A valuation ring is equivalently a PVMR with at most one (regular) t-maximal ideal. Proof. If R is a PVMR with at most one regular t-maximal ideal, then R = R[p] is a valuation ring by Proposition 3.5.4 and Corollary 3.8.5. Conversely, suppose that R is a valuation ring, so that (R, p) is a valuation pair of K = T (R) for some prime ideal p of R. If K = R, then trivially R is a PVMR with no t-maximal ideals. If K = R, then, by Corollary 3.6.23(8), p is the only t-maximal ideal of R, and since (R[p] , [p]R[p] ) = (R, p) is a valuation pair of K , it follows from Corollary 3.8.5 that R is a PVMR.  Open Problem 3.8.7. Is a PVMR paravaluation ring necessarily a valuation ring? The use of the w-operation in Corollary 3.8.5 is essential: we show below that the w-operation cannot be replaced by the t-operation in statements (7)–(9) of the corollary above. The importance of the w-operation is also highlighted by the following characterization of the PVMRs. Recall that a ring R is TW if t  w (that is, if t = w on Freg (R)), and R is TD if t  d. Theorem 3.8.8. A PVMR is equivalently an integrally closed TW ring. Proof. By Corollary 2.3.31 and Proposition 2.6.5, any PVMR is an integrally closed TW ring. To prove the converse, let R be a ring, and let a, b ∈ R with a regular. From the fact that (a, b)3 = (a, b)(a 2 , b2 ), it follows immediately that (a, b)2 (a 2 , b2 )−1 ⊆ ((a, b) :T (R) (a, b)). Assuming that R is integrally closed, one has ((a, b) :T (R) (a, b)) = R by Proposition 2.3.17. Therefore (a, b)2 (a 2 , b2 )−1 ⊆ R, and so (a 2 , b2 )−1 ⊆ ((a, b)2 )−1 . But the reverse containment is obvious, and so (a 2 , b2 )−1 = ((a, b)2 )−1 , whence (a 2 , b2 )t = ((a, b)2 )t . Assuming also that R is TW, one then has  (a 2 , b2 )w = ((a, b)2 )w . Therefore, by Corollary 3.8.5, R is a PVMR. The proof of the theorem above shows that, if R is integrally closed and a, b ∈ R with a regular, then (a 2 , b2 )t = ((a, b)2 )t . It follows that the w-operation cannot be replaced by the t-operation in statements (7)–(9) of Corollary 3.8.5. Note that a ring is TW if and only if the t-operation is r-stable. Therefore, by Proposition 2.4.40, we have the following corollary.

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Corollary 3.8.9. Let R be a ring with total quotient ring K . The following conditions are equivalent. (1) R is a PVMR. (2) R is integrally closed and the t-operation is r-stable. (3) R is integrally closed, (I ∩ J )t = I t ∩ J t for all I, J ∈ Freg (R), and (I : K J )t = (I t : K J ) for I, J ∈ Freg (R) with J finitely generated. (4) R is integrally closed and (I : R J )t = (I t : R J ) for all regular ideals I and J of R with J finitely generated. We also have the following. Corollary 3.8.10. Let R be a ring and  a semistar operation on R. Then R is t -Prüfer if and only if R is integrally closed and w  t. Proof. A ring R is t -Prüfer if and only if R is a PVMR and t  t, if and only if R is integrally closed and t  w and t  t, if and only if R is integrally closed and  w  t. Corollary 3.8.11 (cf. [186, Theorem 8]). A Prüfer ring is equivalently an integrally closed TD ring. Remark 3.8.12. Let  be a semistar operation on a ring R. We say that I ∈ K(R) is -a.b. (resp. -e.a.b.) if (I J ) ⊆ (I H ) implies J  ⊆ H  for all J, H ∈ K(R) (resp., for all finitely generated J, H ∈ K(R). We say that  is a.b. (resp., e.a.b.), or that R is -a.b. (resp., -e.a.b.), if every finitely generated regular I ∈ K(R) is -a.b. (resp., -e.a.b.). These definitions generalize the corresponding notions for semistar operations on integral domains, as defined in [65], for example. (“a.b.” and “e.a.b.” stand for “arithmetisch brauchbar” and “endlich arithmetisch brauchbar,” respectively.) It is clear that any -invertible I ∈ K(R) is -a.b., and therefore if R is -Prüfer then R is -a.b. Moreover, if R is -a.b., then R is integrally closed. One can show that any finitely generated regular I ∈ K(R) is -e.a.b. if and only if it is t -a.b., and therefore R is -e.a.b. if and only if R is t -a.b., if and only if R is t -e.a.b. Moreover, Theorem 3.8.4 and its proof imply that R is t -Prüfer if and only if R is w -a.b., if and only if R is w -e.a.b. Remark 3.8.13. Let R be a ring, and let  be a semistar operation on R. (1) In Lemma 4.2.51 of Section 4.2 we show that a finitely generated regular fractional ideal I of R is t -invertible if and only if I Rp is principal (or invertible) in Rp for every t -maximal (or t -prime) ideal p of R. (2) Let p ∈ Spec R. In Corollary 4.2.52 of Section 4.2 we show that (R[p] , [p]R[p] ) is a valuation pair of T (R) if and only if I Rp is principal (or invertible) in Rp for every finitely generated regular fractional ideal I of R. (3) From either (1) or (2) it follows that R is t -Prüfer if and only if I Rp is principal (or invertible) in Rp for every finitely generated regular fractional ideal I of R and every t -maximal (or t -prime) ideal of R.

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The following definition extends the notion of an essential prime in Definition 3.1.1 to rings with zerodivisors. Definition 3.8.14. A prime ideal p of a ring R is essential if (R[p] , [p]R[p] ) is a valuation pair of T (R). We let Ess(R) denote the set of all essential primes of R. Note that, by Corollary 3.6.11(3), the non-regular prime ideals of any ring are all essential. Example 3.8.15. If D is an integral domain, then a prime p of D is essential if and only if Dp is a valuation domain, if and only if Dp is Prüfer. Proposition 3.8.16. Let R be a ring with total quotient ring K . One has Ess(R)reg ⊆ sK(K /R) ⊆ t-Spec(R)reg . Moreover, equalities hold if and only if R is a PVMR. Proof. Let p be a regular essential prime of R, so that (R[p] , [p]R[p] ) is a valuation pair of K and R[p] = K . Then, by Corollary 3.6.23(6), the ideal [p]R[p] of R[p] is t-closed, and therefore p ∈ sK(K /R). Moreover, by Corollary 3.8.5, R is a PVMR if and only if every regular t-prime of R is essential.  By Proposition 3.8.3, we have the following. Proposition 3.8.17. Let R be a ring and  a semistar operation on R. If  is of r-finite type, or more generally if R has enough regular -prime ideals, then R is -Prüfer if and only if every regular -prime ideal of R is essential. Corollary 3.8.18. A PVMR is equivalently a ring in which every regular t-maximal (or t-prime) ideal is essential. A Prüfer ring is equivalently a ring in which every regular maximal (or prime) ideal is essential. The following definition generalizes the notion of an essential domain (Definition 3.2.2) to rings with zerodivisors.  Definition 3.8.19. if R = {R[p] : p ∈ Ess(R)} (or equiva A ring R is essential lently if R = {R[p] : p ∈ Ess(R)reg }). A P-domain is an integral domain D such that every weak Bourbaki associated prime of T (D)/D is essential, that is, such that Dp is a valuation domain for every weak Bourbaki associated prime of T (D)/D [150]. This notion also extends to rings with zerodivisors in the obvious way: let us say that a P-ring is a ring R such that every weak Bourbaki associated prime of T (R)/R is essential. We then have the following implications. PVMR ⇒ P-ring ⇒ essential ⇒ integrally closed.

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It is known that none of the implications above is reversible, even for integral domains, and a domain D is a P-domain if and only if every localization of D at a multiplicative subset is an essential domain [150]. In Chapters 1–3, we have been occupied with the study of various ring-theoretic properties. A major problem we have largely neglected is to determine how such properties relate under various types of ring extensions. For example, one is generally interested in how various ring-theoretic properties ascend to and descend from various rings of polynomials. In the exercises, this is pursued, in particular, for the ring R[X] for any set X of indeterminates and for the ring R + X T (R)[X ], where R is a ring of various types. For example, it is known that R[X ] is a Krull ring if and only if R is a finite direct product Krull domains (see Exercise 3.7.16), and R[X ] is a PVMR if and only if R is a PVMR and T (R) is von Neumann regular [135]. Another type of result one is interested in is how various ring-theoretic properties ascend from a ring to its integral closure in a finite or integral extension of its total quotient ring. The following theorem is a prototypical example of this kind of result. Theorem 3.8.20 (cf. [93, Proposition 14]). Let R be a Prüfer ring with total quotient ring K , let L be any total quotient ring that is an integral extension of K , and let S be the integral closure of R in L. Then S is a Prüfer ring. Proof. It is clear that L = S K and therefore L is the total quotient ring of S. Let q be any prime ideal of S, and let p = q ∩ R. We show that (S[q] , [q]S[q] ) is a valuation pair of L. Let x ∈ L − S[q] . Since x is integral over K , we may write x n + a1 x n−1 + · · · + an = 0 with ai ∈ K for all i. If one has ai ∈ R[p] for all i, then x is integral over R[p] and therefore over S[q] ⊇ R[p] ; however, S[q] is integrally closed, so it follows / R[p] for some i. Let from this that x ∈ S[q] , which is a contradiction. Therefore ai ∈ v be the valuation on R corresponding to p. Then v takes a minimum value on ai , and that value must be negative. After multiplying by a suitable element b0 of R[p] , we see / [p]R[p] that b0 x n + b1 x n−1 + · · · + bn = 0, where bi = b0 ai ∈ R[p] for all i and bi ∈ for some i. Since [q]S[q] ∩ R[p] = [p]R[p] , it follows that bi ∈ S[q] for all i and bi ∈ / [q]S[q] for some i. We therefore may assume that the equation b0 x n + b1 x n−1 + · · · + / [q]S[q] for bn = 0 is an equation of least degree such that bi ∈ S[q] for all i and bi ∈ some i. Now, S[q] is integrally closed and b0 x is integral over S[q] , so that b0 x ∈ S[q] . We claim that b0 ∈ [q]S[q] . Suppose not. Then b0 u ∈ S − q for some u ∈ S − q. Since b0 x ∈ S[q] , we also have b0 xv ∈ S for some v ∈ S − q. Therefore (b0 uv)x = / S[q] . Therefore since v ∈ (b0 xv)u ∈ S and b0 uv ∈ S, so b0 uv ∈ q since also x ∈ S − q, it follows that b0 u ∈ q, which is a contradiction. Therefore b0 ∈ [q]S[q] . Now, since the equation (b0 x + b1 )x n−1 + b2 x n−2 + · · · + bn = 0 is of degree less than n in x and b0 x + b1 ∈ S[q] , all of its coefficients must lie in [q]S[q] , and therefore b0 x + / [q]S[q] b1 , b2 , b3 , . . . , bn ∈ [q]S[q] . By choice of the bi , one must therefore have b0 ∈ or b1 ∈ / [q]S[q] , and therefore b1 ∈ / [q]S[q] . Therefore, since also b0 x + b1 ∈ [q]S[q] , / [q]S[q] . Therefore, we have shown that, for every x ∈ L − S[q] , it follows that b0 x ∈ there exists a b0 ∈ [q]S[q] such that b0 x ∈ S[q] − [q]S[q] . Thus, (S[q] , [q]S[q] ) is a valuation pair of L = T (S), for every prime ideal q of S, and therefore S is Prüfer. 

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Open Problem 3.8.21. Find analogues of Theorem 3.8.20 for the Dedekind rings, Krull rings, and PVMRs.

3.9 Integrally Closed Rings, -Prüfer Rings, and Content Ideals In this section, we derive some consequences of the results of the previous section and the well-known Dedekind–Mertens lemma, which we state and prove here. Definition 3.9.1. Let R be a ring and T a (commutative) R-algebra. The T -content cT ( f ) of a polynomial f ∈ T [X ] is the R-submodule of T generated by the coefficients of f . If the R-algebra T is understood in a given context, e.g., if T is the total quotient ring of R, then we may write c( f ) or c R ( f ) for cT ( f ). If f, g ∈ T [X ], then trivially one has cT ( f )cT (g) ⊇ cT ( f g). The reverse inclusion does not hold in general. However, the Dedekind–Mertens lemma shows that a weaker version of the reverse inclusion does hold. Theorem 3.9.2 (Dedekind–Mertens lemma). Let R be a ring and T an R-algebra, and let f, g be polynomials in T [X ], and let n = deg g. Then cT ( f )n+1 cT (g) = cT ( f )n cT ( f g). Proof. Since c( f )c(g) ⊇ c( f g), the inclusion c( f )n+1 c(g) ⊇ c( f )n c( f g) is trivial, so it suffices to prove the reverse inclusion. We prove the inclusion by induction on n. Let f (X ) = f 0 + f 1 X + . . . + f m X m and g(X ) = g0 + g1 X + . . . + gn X n be arbitrary polynomials in T [X ]. The inclusion is trivial in the case n = 0. Let n > 0. Let f j = 0 for j < 0 and for j > m, and let G n be the R-submodule of T generated by g0 , g1 , . . . , gn−1 . By the inductive hypothesis, we have c( f )n G n ⊆ c( f )n−1 c( f (g − gn X n )) ⊆ c( f )n−1 (c( f g) + c( f )gn ). Moreover, since c(g) = G n + gn R, we have c( f )n c(g) = c( f )n (G n + gn R) = c( f )n G n + c( f )n gn . Therefore, combining this equality with the previous inclusion, we have c( f )n c(g) ⊆ c( f )n−1 (c( f g) + c( f )gn ) + c( f )n gn = c( f )n−1 c( f g) + c( f )n gn . This implies the containment

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f i c( f )n c(g) ⊆ c( f )n c( f g) + c( f )n f i gn for all i. In addition, it is clear that f i gn ∈ c( f g) + ( f i+1 , f i+2 , . . . , f m )c(g), whence f i c( f )n c(g) ⊆ c( f )n c( f g) + ( f i+1 , f i+2 , . . . , f m )c( f )n c(g), for all i. From this last inclusion, we infer that f i c( f )n c(g) ⊆ c( f )n c( f g) successively for each i = m, m − 1, . . . , 0, and therefore c( f )n+1 c(g) ⊆ c( f )n c( f g).  As a corollary, note that, if f ∈ T [X ] and cT ( f ) is T -invertible, then cT ( f )cT (g) = cT ( f g) for all g ∈ T [X ]. Example 3.9.3. Let R be a ring and a, b ∈ R with a regular. Applying the Dedekind– Mertens lemma to f = a X + b and g = a X − b and noting that c( f g) = (a 2 , b2 ) and c( f )c(g) = (a, b)2 , we see that (a, b)3 = (a, b)(a 2 , b2 ). This equation was important for establishing the results of the previous section. Remark 3.9.4. Obviously, if the equation cT ( f )n+1 cT (g) = cT ( f )n cT ( f g) holds for some positive integer n = N , then the same equation holds for all n  N . Given a polynomial g ∈ T [X ], the smallest n for which the equation holds for all f ∈ T [X ] may be much smaller than deg g. The smallest positive integer n such that cT ( f )n cT (g) = cT ( f )n−1 cT ( f g) for all polynomials f ∈ T [X ] is called the Dedekind–Mertens number μT (g) of g. The Dedekind–Mertens lemma says that this number is bounded above by 1 + deg g. Much work has been done to find sharper bounds in various circumstances (e.g., [37, 88, 99, 131, 136]). In the extreme case, a polynomial g ∈ R[X ] is said to be Gaussian if its Dedekind– Mertens number is 1, that is, if c R ( f g) = c R ( f )c R (g) for all f ∈ R[X ]. By the Dedekind–Mertens lemma, if g ∈ R[X ] and c R (g) is invertible, then g is Gaussian. A problem associated with Kaplansky was to determine whether or not every Gaussian polynomial with regular content has invertible content. Subsequent work in the 1990s and 2000s eventually yielded a positive answer to this question when T. G. Lucas proved in 2005 that a polynomial g ∈ R[X ] such that c R (g) has no nonzero annihilators (or equivalently such that g is regular in R[X ]) is Gaussian if and only if c R (g) is “Q 0 -invertible,” if and only if c R (g) is locally principal [131, 136]. This yields a positive answer to the question because a regular ideal is Q 0 -invertible if and only if it is invertible. A ring R is said to be Gaussian if every polynomial in R[X ] is Gaussian, that is, if c R ( f g) = c R ( f )c R (g) for all f, g ∈ R[X ]. For example, an integral domain is Gaussian if and only if it is Prüfer. More generally, as shown in Corollary 3.9.10 below, a ring R is Prüfer if and only if every polynomial g ∈ R[X ] such that c R (g) is regular is Gaussian. (The theorem follows easily from Lucas’ 2005 result but was proved earlier and has a simpler proof.) It follows that every Gaussian ring is Prüfer.

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However, for rings with zerodivisors the converse is false, and in fact there exist non-Gaussian total quotient (hence Prüfer) rings: see [86, Example 2.4] or Exercise 3.9.9. In the exercises for Sections 3.2, 3.8, and 3.9, definitions for the terms listed in italics below are provided, and it is shown that, for any ring R, each of the following conditions implies the next. (1) (2) (3) (4) (5) (6)

R is semihereditary. R is of weak global dimension at most one. R is arithmetical. R is Gaussian. R is locally Prüfer. R is Prüfer.

The implications (1) ⇒ (2) ⇒ (3) ⇒ (4) ⇒ (5) ⇒ (6) are well-known: see [87], for example, where it is also shown that none of these implications is reversible. For integral domains R, however, all six of the conditions are equivalent. The following result characterizes the integrally closed rings and is a consequence of the Dedekind–Mertens lemma and Proposition 2.3.17. Theorem 3.9.5 (cf. [168, Lemme 1]). Let R be ring with total quotient ring K . The following conditions are equivalent. (1) R is integrally closed. (2) c R ( f g)v = (c R ( f )c R (g))v for all f, g in R[X ] (or equivalently in K [X ]) with c R ( f ) regular. (3) c R ( f g)v = (c R ( f )c R (g))v for all f, g in R[X ] (or equivalently in K [X ]) with c R ( f ) and c R (g) regular. Proof. That (2) ⇒ (3) is clear. Suppose that (3) holds. Let x ∈ K be integral over R, so that x is a root of a monic polynomial f ∈ R[X ]. One then has f = (X − x)g for some monic g ∈ K [X ]. Then R = c( f ) = c( f )v = (c(X − x)c(g))v = ((1, x)c(g))v ⊇ (1, x)v , so x ∈ R. Therefore R is integrally closed. Thus (3) ⇒ (1). Finally, suppose that (1) holds, that is, that R is integrally closed. Let f, g ∈ R[X ] with c( f ) regular. By the Dedekind–Mertens lemma, one has c( f )n+1 c(g) = c( f )n c( f g), where n = deg g. Therefore c( f )c(g)c( f g)−1 ⊆ (c( f )n : K c( f )n ) = R, where the last equality holds by Proposition 2.3.17, and so c( f g)−1 ⊆(c( f )c(g))−1 . But also c( f g) ⊆ c( f )c(g), so c( f g)−1 ⊇ (c( f )c(g))−1 , whence c( f g)−1 =  (c( f )c(g))−1 and therefore c( f g)v = (c( f )c(g))v . Therefore (1) ⇒ (2). Note that the v-operation in the theorem above can be replaced with the t-operation since content ideals are finitely generated. The proof of the theorem above applied to f = a X + b and g = a X − b inspired the proof of Theorem 3.8.8, which required showing that (a 2 , b2 )t = ((a, b)2 )t if R is integrally closed. Next, recall that a GCD domain D is equivalently a t-Bézout domain. In a GCD domain D, the ideal (S D)v is principal, generated by gcd(S), for any finite subset S

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of D; it is common to call the gcd d of the coefficients of a polynomial f ∈ D[X ] its content, which is to be distinguished from its D-content, c D ( f ). Of course, the two are related by the equation (d) = c D ( f )v . By the previous theorem, then, we have the following. Corollary 3.9.6. Let D be a GCD domain and f, g ∈ D[X ]. Then the gcd of the coefficients of f g is the gcd of the coefficients of f times the gcd of the coefficients of g. The following result characterizes domains in which a similar content equation holds for ideals. Theorem 3.9.7 (Gauss–Gilmer–Tsang theorem). An integral domain D is Prüfer if and only if c D ( f g) = c D ( f )c D (g) for all f, g in D[X ]. The Gauss–Gilmer–Tsang theorem generalizes as follows. Theorem 3.9.8 (cf. [6, Corollary 1.2]). Let R be a ring with total quotient ring K and  a semistar operation on R. The following conditions are equivalent. (1) R is t -Prüfer. (2) c R ( f g)w = (c R ( f )c R (g))w for all f, g in R[X ] (or equivalently in K [X ]) with c R ( f ) regular. (3) c R ( f g)w = (c R ( f )c R (g))w for all f, g in R[X ] (or equivalently in K [X ]) with c R ( f ) and c R (g) regular. Proof. We may suppose without loss of generality that  = t is of finite type. If R is -Prüfer and c R ( f ) is regular, then c R ( f ), being also finitely generated, is w -invertible, and therefore we may cancel c R ( f )n from both sides of the Dedekind– Mertens content equation to obtain the equation c R ( f g)w = (c R ( f )c R (g))w . Thus (1) ⇒ (2) ⇒ (3). Suppose that (3) holds. Since w  v, one has c R ( f g)v = (c R ( f )c R (g))v for all f, g in R[X ] with c R ( f ) and c R (g) regular, so Theorem 3.9.5 implies that R is integrally closed. Now, let a, b ∈ R with a regular. Applying the hypothesized content formula to f = a X + b and g = a X − b, we get (a 2 , b2 )w = ((a, b)2 )w . Therefore (3) ⇒ (1) by Theorem 3.8.4.  Corollary 3.9.9. Let R be a ring with total quotient ring K . The following conditions are equivalent. (1) R is a PVMR. (2) c R ( f g)w = (c R ( f )c R (g))w for all f, g in R[X ] (or equivalently in K [X ]) with c R ( f ) regular. (3) c R ( f g)w = (c R ( f )c R (g))w for all f, g in R[X ] (or equivalently in K [X ]) with c R ( f ) and c R (g) regular. Corollary 3.9.10. Let R be a ring with total quotient ring K . The following conditions are equivalent.

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(1) R is Prüfer. (2) c R ( f g) = c R ( f )c R (g) for all f, g in R[X ] (or equivalently K [X ]) with c R ( f ) regular. (3) c R ( f g) = c R ( f )c R (g) for all f, g in R[X ] (or equivalently K [X ]) with c R ( f ) and c R (g) regular. Combining Theorem 3.9.5 and Corollary 3.9.9 yields another proof that any integrally closed TW ring is a PVMR. Remark 3.9.11. In Remark 3.8.12, we defined the notions of an a.b. and an e.a.b. semistar operation. Let us say that a ring R is -DM if c R ( f g) = (c R ( f )c R (g)) for all f, g ∈ R[X ] with c R ( f ) regular. By the Dedekind–Mertens lemma, if R is e.a.b., then R is -DM. However, the converse does not hold. Indeed, by [79, Theorem 34.6], a domain R is v-e.a.b. if and only if R is a v-domain, while R is v-DM if and only if R is integrally closed. One has the following implications. -Prüfer ⇒ -a.b. ⇒ -e.a.b. ⇒ -DM ⇒ integrally closed. However, none of these implications is reversible, even for domains. Indeed, [65, Example 16] provides an example of a -e.a.b. domain that is not -a.b., and it is known that for domains one has v-Prüfer ⇔ v-a.b. ⇔ t-a.b. ⇔ v-e.a.b. ⇔ t-e.a.b, so that t-a.b. does not imply t-Prüfer.

3.10 Valuation Rings In Section 3.7 we characterized the Krull valuation rings, and in Section 3.8 we showed that a valuation ring is equivalently a PVMR with at most one t-maximal ideal. In this section we show that a valuation ring is equivalently a generalized t-Prüfer ring for which the set of all regular t-closed ideals is totally ordered, and we characterize the valuation rings that are Prüfer, Dedekind, r-Bézout, r-PIR, r-GCD, and r-UFR, respectively. First, we note the following result, which shows that, in a precise sense, valuation pairs are ubiquitous. Proposition 3.10.1. If R is a subring of a ring T and p is a prime ideal of R, then there exists a valuation pair (V, q) of T such that V contains R and q lies over p. To prove the proposition, we need the following definition. Definition 3.10.2. Let T be a ring, and let S(T ) denote the set of all pairs (R, p) such that R is a subring of T and p is a prime ideal of R. For all (R, p) and (S, q)

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in S(T ), we write (R, p)  (S, q) and say that the pair (S, q) dominates the pair (R, p) if S contains R and q lies over p. The relation  defined above is clearly a partial ordering on S(T ). By Theorem 3.6.5, we have the following. Proposition 3.10.3. Let T be a ring. A pair (R, p) is a valuation pair of T if and only if (R, p) is a maximal element of S(T ), that is, if (R, p) is dominated only by (R, p). One obtains Proposition 3.10.1 from Proposition 3.10.3 using a simple application of Zorn’s lemma. Compare the following definition with Definition 3.6.9. Definition 3.10.4. Let R be a subring of a ring T . We say that R is a paravaluation ring of T if R = Tv for some paravaluation v of T . We say that R is a paravaluation ring if R is a paravaluation ring of T (R). The following theorem shows that paravaluation rings are useful for studying integral closure. Theorem 3.10.5 ([107, Theorem 9.1]). Let R be a subring of a ring T . The integral closure of R in T is equal to the intersection of the paravaluation rings of T containing R. Proof. Let S be the integral closure of R in T . Since each paravaluation ring of T is integrally closed in T , the ring S is contained in the intersection of all such paravaluation rings containing R. Let x be any element of T that is not in S. We must show that x ∈ / V for some paravaluation ring V of T containing R. Let I = {a ∈ T : ax n = 0 for some n  1}, which is an ideal of T . Since nilpotent elements of T are integral over R, one has x ∈ / I . Let T = T /I , and let x be the image of x in T . Clearly, x is regular in T . It follows from the fact that x is not integral over R that x is not integral over R = (R + I )/I (which is the image of R in T ). Since x ∈ R[x −1 ] would lead to an equation of integrality of x over R, one has x ∈ / R[x −1 ], −1 −1 −1 and therefore x R[x ] is a proper ideal of R[x ]. Let p be a prime ideal of R[x −1 ] containing x −1 . Since R[x −1 ] is a proper subring of T [x −1 ], by Proposition 3.10.1 there exists a valuation pair (V , q) of T [x −1 ] dominating (R[x −1 ], p), say, with associated valuation v and value group . Let v 0 be the restriction of v to T . Then v 0 is a paravaluation on the ring T . Define v0 : T −→  ∪ {∞} via v0 (x) = v 0 (x) for all x ∈ T . Then v0 is a paravaluation on T , and V = Tv0 is a paravaluation ring of T containing R with x ∈ / V (since x −1 ∈ p implies v0 (x) = −v(x −1 ) < 0).  Corollary 3.10.6. Let R be a subring of a ring T . Then R is integrally closed in T if and only if R is equal to the intersection of the paravaluation rings of T containing R. J. Gräter in 1989 provided the first example of an integrally closed ring R that is properly contained in the intersection of the valuation rings of T (R) containing R [90]. However, we show later in this section that equality holds at least if R is Marot, since in that case every paravaluation ring T (R) containing R is in fact a valuation ring.

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Proposition 3.10.7. Let R be a ring. The following conditions are equivalent. (1) (2) (3) (4)

For all x, y ∈ R one has (x)t ⊆ (y)t or (y)t ⊆ (x)t . For all x, y ∈ T (R) one has (x)t ⊆ (y)t or (y)t ⊆ (x)t . The set of all t-closed ideals of R is totally ordered. The set of all t-closed Kaplansky fractional ideals of R is totally ordered.

Moreover, R is a valuation ring if and only if R is a generalized t-Prüfer ring (or PVMR) satisfying any of the conditions above, if and only if R is a generalized t-Prüfer ring (or PVMR) and the set of all regular t-closed ideals of R is totally ordered. Proof. It is easy to verify that conditions (1)–(4) are equivalent. If R is a valuation ring, then by Corollary 3.6.26 the set of all t-closed ideals of R is totally ordered and R is a PVMR. Conversely, suppose that R is a generalized t-Prüfer ring and the set of all regular t-closed ideals of R is totally ordered. Then one has (I + J )(I ∩ J ) = I J for all regular t-closed ideals I and J of R, whence R is a PVMR by Theorem 2.11.9. Moreover, R has at most one regular t-maximal ideal, and therefore R is a valuation ring by Corollary 3.8.6.  The following result follows from Corollary 3.8.6. Proposition 3.10.8. One has the following. (1) A valuation ring (resp., Krull valuation ring, r-GCD valuation ring, r-UFR valuation ring) is equivalently a PVMR (resp., Krull ring, r-GCD ring, r-UFR) with at most one (regular) t-maximal ideal. (2) A Prüfer valuation ring (resp., Dedekind valuation ring, r-Bézout valuation ring, r-PIR valuation ring) is equivalently a Prüfer ring (resp., Dedekind ring, r-Bézout ring, r-PIR) with at most one regular maximal ideal. Example 3.10.9. (1) Any valuation domain that is not a DVR is an r-Bézout valuation ring that is not an r-PIR, and hence not Krull. (2) Example 2.7.13 is an example of a Dedekind ring with a unique regular maximal ideal, hence a Dedekind valuation ring, with nontrivial ideal class group, and it is therefore neither an r-PIR nor r-GCD. Another example of such a valuation ring is provided by [108, Section 27, Example 10]. Unlike the situation with integral domains, a valuation ring with zerodivisors need not be Prüfer. Example 3.10.10. Contrary to [22, Theorem 2.3] and [108, Theorem 6.5], a valuation ring with a unique regular maximal ideal need not be Prüfer. In fact, a local valuation ring with regular maximal ideal need not be Prüfer. The following example is due to T. G. Lucas (private communication). Let D = k[X, Y ](X,Y ) , where k is a field, and let P = X 1 (D) − {Y D} be the set of all height one primes of D other than Y D.

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 Let R = D(+)M, where M = p∈P T (D/p). Since D is local with maximal ideal (X, Y ), the ring R is local with (regular) maximal ideal (X, Y )(+)M. Now, the total quotient ring of R is T (R) = D[1/Y ](+)M. Let v be the restriction of the Y -adic valuation on k(X, Y ) to D[1/Y ]. Then D = { f ∈ D[1/Y ] : v( f )  0} and X D = { f ∈ D[1/Y ] : v( f ) > 0}. Moreover, v extends to a valuation v  on T (R) = D[1/Y ](+)M so that v  ((a, m)) = v(a) for all a ∈ D[1/Y ] and all m ∈ M, and one has R = T (R)v . Thus R is a local valuation ring. But R is not Prüfer since ((X, 0), (Y, 0)) is a regular ideal of R that is not principal, and hence not invertible. Note that R reg = {(cY n , m) : c ∈ D × , n ∈ Z0 , m ∈ M}. It follows that R is an r-UFR, and (Y, 0) is the unique regular prime element of R up to associates. It also follows that every regularly generated ideal of R is principal. Therefore R is a local weak PIR r-UFR valuation ring that is not Prüfer. Conjecture 3.10.11. There exist examples of the following: (1) (2) (3) (4)

a Prüfer valuation ring that is neither Krull nor r-GCD; a Krull valuation ring that is neither r-GCD nor Prüfer; an r-GCD valuation ring that is neither Prüfer nor Krull; a valuation ring that is not Krull, r-GCD, or Prüfer. The following result characterizes the Prüfer valuation rings.

Proposition 3.10.12. Let R be a ring. The following conditions are equivalent. (1) (2) (3) (4) (5) (6) (7)

R is a Prüfer valuation ring. R is a TD valuation ring. R is a DW valuation ring. R is quasi-Marot and the set of all regular ideals of R is totally ordered. R is a Prüfer ring with at most one regular maximal ideal. R is a valuation ring and every regular maximal ideal of R is t-closed. R is either a total quotient ring or a valuation ring whose unique t-maximal ideal is the unique regular maximal of R.

Proof. One has (1) ⇔ (2) ⇔ (3) since any valuation ring is a PVMR and is therefore TW. By Corollary 3.6.23(5), the set of all t-closed ideals of any valuation ring is totally ordered, and therefore (2) ⇒ (4). Suppose that (4) holds. Then (I + J )(I ∩ J ) = I J for all regular ideals I and J of R, whence R is Prüfer by Theorem 2.11.9. Moreover, R has at most one regular maximal ideal. Therefore (4) ⇒ (5). Finally, one has (5) ⇔ (1) by Proposition 3.10.8 and (5) ⇔ (6) ⇔ (7) by Theorem 3.8.1 and Corollary 3.6.23(8).  Definition 3.10.13. A Prüfer valuation pair (of K ) is a valuation pair (R, p) of K = T (R) such that R is Prüfer. Corollary 3.10.14 ([107, Lemma 6.4]). Let R be a ring with total quotient ring K and p a prime ideal of R. The following conditions are equivalent.

3.10 Valuation Rings

(1) (2) (3) (4)

247

(R, p) is a Prüfer valuation pair of K = R. (R, p) is a valuation pair of K and p is the unique regular maximal ideal of R. R is Prüfer and p is the unique regular maximal ideal of R. R is Prüfer, p is regular, and R = R[p] .

Corollary 3.10.15. Let R be a ring with total quotient ring K and p a prime ideal of R. The following conditions are equivalent. (1) (R[p] , [p]R[p] ) is a Prüfer valuation pair of K = R[p] . (2) (R[p] , [p]R[p] ) is a valuation pair of K and [p]R[p] is the unique regular maximal ideal of R[p] . (3) R[p] is Prüfer and [p]R[p] is the unique regular maximal ideal of R[p] . (4) R[p] is Prüfer and [p]R[p] (or equivalently, p) is regular. (5) R[p] is Prüfer and K = R[p] . The condition that R[p] is Prüfer is generally stronger than the condition that (R[p] , [p]R[p] ) is a valuation pair of K . However, for Marot rings R, they are equivalent. (See Corollary 3.10.20.) By Proposition 3.5.18, Corollary 3.10.15, and Theorem 3.8.1, we have the following. Proposition 3.10.16. Let R be a ring with total quotient ring K and p a prime ideal of R. The following conditions are equivalent. (1) (2) (3) (4) (5)

R is Prüfer. R[p] is a Prüfer valuation ring for every prime ideal p of R. R[p] is a Prüfer valuation ring for every regular maximal ideal p of R. R[p] is Prüfer for every prime ideal p of R. R[p] is Prüfer for every regular maximal ideal p of R.

Remark 3.10.17. Let R be a ring. The following definitions are taken from [120]. (1) R is locally Prüfer if Rp is Prüfer for every prime ideal p of R. (2) R is maximally Prüfer if Rp is Prüfer for every maximal ideal p of R. (3) R is regular locally Prüfer if Rp is Prüfer for every regular prime ideal p of R (or equivalently for every regular maximal ideal p of R). One has the following implications. locally Pr¨ufer ⇒ maximally Pr¨ufer ⇒ regular locally Pr¨ufer ⇒ Pr¨ufer. However, none of these implications is reversible [120]. Thus Proposition 3.10.16 shows that the Prüfer condition behaves much more nicely with respect to large localization than it does with respect to ordinary localization. For integral domains, and more generally for Marot rings, the five conditions in the proposition below are equivalent, but for arbitrary rings they are all inequivalent.

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Proposition 3.10.18 (cf. [107, Theorem 7.7]). Let R be a ring. Each of the following conditions implies the next. (1) (2) (3) (4) (5)

R is an r-Bézout valuation ring; equivalently, R is a Marot valuation ring. R is a Prüfer valuation ring. R is a valuation ring. R is a paravaluation ring. For all regular x ∈ T (R), one has x ∈ R or x −1 ∈ R; equivalently, the set of all regularly generated ideals of R is totally ordered.

Moreover, if R is Marot, then the five conditions above are equivalent. Proof. The equivalences of the two conditions in statement (1) and of the two conditions in statement (5) are left as an exercise, and the implications (1) ⇒ (2) ⇒ (3) ⇒ (4) ⇒ (5) are clear. We assume R is Marot and prove (5) ⇒ (1). We may assume without loss of generality that R = T (R). Suppose that (5) holds. It follows that the set of all regular ideals of R is totally ordered, and therefore R is a Prüfer valuation ring. Therefore, since R is Marot, condition (1) holds.  Since every overring of a Marot ring is Marot, we also obtain from previous results the following corollaries. Corollary 3.10.19. If R is a ring, then the integral closure of R is contained in the intersection of the valuation rings of T (R) containing R. Moreover, equality holds if R is Marot. Corollary 3.10.20. Let R be a Marot ring with total quotient ring K and p a prime ideal of R. Then (R[p] , [p]R[p] ) is a valuation pair of K if and only if (R[p] , [p]R[p] ) is a Prüfer valuation pair of K , if and only if R[p] is Prüfer. We also note the following. Corollary 3.10.21. Let R be a ring. Each of the following conditions implies the next. (1) R is an r-PIR and a valuation ring; equivalently, R is a Marot Dedekind valuation ring. (2) R is a Dedekind valuation ring. (3) R is a Krull valuation ring; equivalently, R is a discrete rank at most one valuation ring. (4) R is a Krull paravaluation ring. (5) R is Krull and the set of all regularly generated ideals of R is totally ordered. Moreover, if R is Marot, then the five conditions above are equivalent. Note that a DVR is equivalently a domain that satisfies any of the five conditions of the corollary above. By Example 3.7.27, conditions (1), (2), and (3) of the corollary are inequivalent. Thus, the notion of a DVR generalizes to rings with zerodivisors in at least three inequivalent ways. We suspect that conditions (3), (4), and (5) of the corollary are also inequivalent, but we are unable to verify this.

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249

Open Problem 3.10.22. Are conditions (3) and (4) of Corollary 3.10.21 equivalent? Are conditions (4) and (5) of Corollary 3.10.21 equivalent? The following proposition characterizes the Dedekind valuation rings. Proposition 3.10.23. Let R be a ring with total quotient ring K = R. The following conditions are equivalent. (1) R is a Dedekind valuation ring. (2) R is a Mori Prüfer valuation ring. (3) R is completely integrally closed and has a unique regular maximal ideal p, and p is invertible. (4) Freg (R) = {. . . , (p2 )−1 , p−1 , R, p, p2 , . . .} for some invertible maximal ideal p of R. (5) Freg (R) is isomorphic to Z as an ordered monoid. (6) R is a Krull ring with a unique regular prime ideal. (7) (R, p) is a discrete rank one valuation pair of K , where p is the unique regular maximal ideal of R. (8) R is a Prüfer discrete rank one valuation ring. (9) R is a TD Krull valuation ring. (10) R is a valuation ring and for every regular ideal I of R one has I = (a)t for some a ∈ R. Proof. The implications (2) ⇔ (1) ⇒ (3) are clear. Suppose that (3) holds. Then Freg (R) ⊇ {. . . , (p2 )−1 , p−1 , R, p, p2 , . . .}. Suppose to obtain a contradiction that the inclusion is proper. Let I denote the set of all proper regular ideals I of R that are not of the formpn for any n. Then I is nonempty. Let C be any chain of ideals in I. Then I = C is a proper regular ideal of R. If I = pn for some n, then, since pn is finitely generated, one has I ⊆ J for some J ∈ C, so I = J , which is a contradiction. Therefore I = pn for all n, so I ∈ I. Thus every chain in I has an upper bound in I. Therefore, by Zorn’s lemma, I has a maximal element H . Now, H is a proper regular ideal of R, so H ⊆ p and therefore H ⊆ H p−1 ⊆ R. Since R is completely integrally closed, one has (H : K H ) = R by Proposition 2.3.17. If H p−1 = H , then, p−1 ⊆ (H : K H ) = R , which contradicts the invertibility of p. Thus the ideal H p−1 properly contains H , so by maximality of H one has H p−1 = pn for some n and thus H = pn+1 , which is also a contradiction. Thus (3) ⇒ (4). Now, clearly (4) ⇒ (5). Conversely, if (5) holds, then there exists an invertible ideal I of R that generates the group Freg (R), and therefore every proper regular ideal of R is a power of I and thus is contained in I , so that I is maximal, whence condition (4) holds. Therefore (4) ⇔ (5). Next, (5) ⇒ (6) by Corollary 3.7.12, and (6) ⇒ (7) by Theorem 3.7.9. One also has (7) ⇔ (8) by Corollary 3.10.14. Moreover, if (8) holds, then by Proposition 3.7.25 R is Krull and Prüfer and therefore Dedekind. Therefore (8) ⇒ (1). Finally, one has (1) ⇔ (9) ⇔ (10) by Proposition 3.7.25.  The next two results characterize the valuation rings that are r-UFRs and r-PIRs, respectively.

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3 Semistar Operations on Commutative Rings: Local Methods

Proposition 3.10.24. Let R be a ring with total quotient ring K = R. The following conditions are equivalent. (1) (2) (3) (4)

(5) (6) (7) (8) (9) (10) (11) (12)

R is an r-UFR and a valuation ring. R is an r-UFR with exactly one t-maximal ideal. R is an r-UFR with exactly one regular prime element up to associates. There is a regular prime element π of R such that for every regular element r of R one has r = uπ n for a unique unit u of R and a unique nonnegative integer n. Prin(R) = {. . . , (π −2 ), (π −1 ), (1), (π), (π 2 ), . . .} for some regular prime element π of R. R is a discrete rank one valuation ring whose unique t-maximal ideal is principal. R is a Krull ring with a unique t-maximal ideal p, and p is principal. R is a Krull valuation ring with trivial t-class group. R is a t-Marot Krull valuation ring.  n R = R[p] for some principal prime ideal p of R such that ∞ n=0 p is not regular. (R, p) is a discrete rank one valuation pair of K , where p is the unique tmaximal ideal of R, and p is principal. R is a valuation ring and for every regular ideal I of R one has I t = (a) for some a ∈ R.

Proposition 3.10.25. Let R be a ring with total quotient ring K = R. The following conditions are equivalent. (1) (2) (3) (4) (5) (6) (7) (8) (9)

R is an r-PIR and a valuation ring. R is an r-PIR with exactly one regular prime element up to associates. R is an r-PIR with exactly one regular maximal ideal. R is a Krull ring with a unique regular prime ideal p, and p is principal. R is a Marot Krull valuation ring. R is a Marot Dedekind valuation ring. R is a Dedekind valuation ring with trivial ideal class group. R is a Dedekind valuation ring whose unique regular maximal ideal is principal. Freg (R)={. . . , (π −2 ), (π −1 ), (1), (π), (π 2 ), . . .} for some regular element π of R. (10) (R, p) is a discrete rank one valuation pair of K , where p is the unique regular maximal ideal of R, and p is principal. Given the results in Sections 3.6 through 3.10, we can distinguish eight classes of valuation rings as characterized as in the table below.

3.10 Valuation Rings

Valuation ring R r-PIR valuation

Dedekind valuation

r-Bézout valuation

Prüfer valuation

r-UFR valuation

Krull valuation

r-GCD valuation

valuation

251

Equivalent characterizations r-PIR with at most one regular prime ideal r-UFR with at most one regular prime ideal Marot discrete rank at most one valuation Dedekind valuation with Cl(R) = 0 valuation and ∀I ∈ Ireg (R) ∃a ∈ R : I = (a) Dedekind with at most one regular prime ideal Krull with at most one regular prime ideal Prüfer discrete rank at most one valuation valuation and ∀I ∈ Ireg (R) ∃a ∈ R : I = (a)t r-Bézout with at most one regular prime ideal Marot valuation TD t-Marot valuation valuation and ∀ f.g. I ∈ Ireg (R) ∃a ∈ R : I = (a) Prüfer with at most one regular maximal ideal quasi-Marot, and the set of all regular ideals of R is totally ordered valuation with all regular maximal ideals t-closed TD valuation valuation and ∀ f.g. I ∈ Ireg (R) ∃a ∈ R : I = (a)t r-UFR with at most one (regular) t-maximal ideal t-Marot Mori valuation t-Marot discrete rank at most one valuation valuation and ∀I ∈ Ireg (R) ∃a ∈ R : I t = (a) Krull with at most one (regular) t-maximal ideal Mori valuation discrete rank at most one valuation valuation and ∀I ∈ Ireg (R) ∃a ∈ R : I t = (a)t r-GCD with at most one (regular) t-maximal ideal valuation with Clt (R) = 0 t-Marot valuation t-Marot and the set of all regular t-closed ideals of R is totally ordered valuation and ∀ f.g. I ∈ Ireg (R) ∃a ∈ R : I t = (a) PVMR with at most one (regular) t-maximal ideal generalized t-Prüfer and the set of all regular t-closed ideals of R is totally ordered the set of all t-closed ideals of R is totally ordered, and every f.g. ideal I Ę (R : R K ) is t-invertible ∀x, y ∈ K ((x)t ⊆ (y)t or (y)t ⊆ (x)t ), and ∀x ∈ K − (R : R K ) ∃y ∈ K ((x y)t = R)

On the next page is the smallest full and complete implication lattice that, among all PVMRs, includes the r-PID, Dedekind, r-Bézout, Prüfer, r-UFR, Krull, r-GCD, and CIC conditions along with the divisorial, w-divisorial, TD, TV, DW, and TW conditions. The lattice forms a four dimensional cube, with axes corresponding to the CIC, TV, TD, and trivial t-class group properties.

Krull

CIC PVMR

r-UFR

PVMR

r-GCD

CIC Prüfer

TV PVMR

CIC r-Bézout

IC divisorial

TV r-GCD

CIC r-GCD

Dedekind

r-PIR

Prüfer

divisorial r-Bézout

r-Bézout

252 3 Semistar Operations on Commutative Rings: Local Methods

3.10 Valuation Rings

253

We conjecture that for valuation rings with or without zerodivisors, the given implication lattice is complete; in other words, we conjecture that all 32 arrows in the lattice are irreversible for the valuation rings. If this is correct, then the implication lattice divides the class of all valuation rings into 16 disjoint nonempty subclasses. This would be in stark contrast to the situation for integral domains. Indeed, for the valuation domains, or more generally for the Marot valuation rings, the lattice collapses to the following. Dedekind valuation ring

CIC valuation ring

divisorial valuation ring

valuation ring

Open Problem 3.10.26. Which of the 32 arrows in the implication lattice on the previous page are irreversible for the valuation rings? Remark 3.10.27. Let R be a valuation ring with valuation v and value group . (1) By Exercise 3.6.15, the ring R is completely integrally closed if and only if t-Spec(R) = {v −1 (∞), pv }, if and only if |t-Spec(R)|  2, and if and only if R is of regular t-dimension at most one. (2) By Exercise 3.6.18, the ring R is TV if and only if R is H, if and only if pv is divisorial, and if and only if pv = (a)t for some a ∈ R, and R is divisorial if and only if R is TV and Prüfer, if and only if it is H and Prüfer. (3) By Exercise 3.6.23, the ring R is t-Marot if and only if R has trivial t-class group, if and only if v maps the regular elements of R onto the nonnegative elements of , and R is Marot if and only if R is Prüfer and has trivial class group, if and only if R is Prüfer and v maps the regular elements of R onto the nonnegative elements of .

3.11 Spectral Semistar Operations and -Prüfer-Like Conditions This section develops some further characterizations of the t -Prüfer rings and contrasts them with some subtly inequivalent characterizations of various related classes of rings. Our main tools are two generalizations of the notion of a spectral semistar operation on an integral domain [62, Section 4] to rings with zerodivisors (both of which turn out to be more or less equivalent for Marot rings).

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3 Semistar Operations on Commutative Rings: Local Methods

Let R be a ring and O a collection of overrings of R whose intersection is R.  Recall that O denotes the semistar operation on R given by I O = {I S : S ∈ O} for all I ∈ K(R). Definition3.11.1. Let R be a ring and  ⊆ Spec R a set of prime ideals of R such that R = {R[p] : p ∈ }. (1) Let I [] =

 {[I ]R[p] : p ∈ }

for all I ∈ K(R). The operation [] : I −→ I [] is a semistar operation on R. (2) Let  = O , where O = {R[p] : p ∈ }, that is, let  denote the semistar operation on R such that  {I R[p] : p ∈ } I  = for all I ∈ K(R). Definition 3.11.2. A semistar operation   on a ring R is spectral if  = [] for some set  ⊆ Spec R such that R = {R[p] : p ∈ }. Theorem 3.11.3. Let R be a ring and  ⊆ Spec R a set of prime ideals of R such that R = {R[p] : p ∈ } (which holds if and only if for every x ∈ T (R) − R there exists a p ∈  such that (R : R x) ⊆ p). One has the following.  (1) [] : I −→ {[I ]R[p] : p ∈ } is a stable semistar operation on R. (2) Every proper [] -closed ideal of R is contained in some prime in . (3) [] -Spec(R) = {q ∈ Spec(R) : q ⊆ p for some p ∈ }. (4) [] -Max(R) is the set of all maximal elements of . (5) R is [] -Prüfer if and only if (R[p] , [p]R[p] ) is a valuation pair of T (R) for every p ∈ , if and only if  ⊆ Ess(R). Proof. It is easy to check that [] is a semistar operation on R. Let I, J ∈ K(R). Then (I ∩ J )[] =

  {[I ∩ J ]R[p] : p ∈ } = {[I ]R[p] ∩ [J ]R[p] : p ∈ } = I [] ∩ J [] .

Moreover, if J is finitely generated, then (I : K J )[] =



{[(I : K J )]R[p] : p ∈ } =



{([I ]R[p] : K J ) : p ∈ } = (I [] : K J ).

Therefore [] is stable. Moreover, if I is a proper [] -closed ideal of R, then there exists an a ∈ R and a p ∈  such that a ∈ / [I ]R[p] and therefore I ⊆ (I : R a) ⊆ p. This proves statements (1) and (2).  Next, let q ∈ Spec(R). Suppose that q ⊆ p for some p ∈ . Then q[] = {[q]R[r] : r ∈ } ⊆ [q]R[p] ∩ R = q and therefore q is [] -prime. Conversely, if q is [] -prime, then, by statement (2), q is contained in some prime in . This proves statements

3.11 Spectral Semistar Operations and -Prüfer-Like Conditions

255

(3) and (4). Finally, statement (5) follows from statements (2)–(4) and Proposition 3.8.3.   Recall that a ring R is essential if R = {R[p] : p ∈ Ess(R)}. Corollary 3.11.4. A ring R is essential if and only if R is -Prüfer for some spectral semistar operation  on R. Moreover, any essential ring R is [Ess(R)] -Prüfer and v-Prüfer. The following result provides equivalent characterizations of the spectral semistar operations. Theorem 3.11.5 (cf. [161, Theorem 22]). Let  be a semistar operation on a ring R. The following conditions are equivalent. (1)  is spectral. (2)  is stable and every proper -closed ideal of R is contained in some -prime of R.  (3)  = [-Spec(R)] , that is, I  = {[I ]R[p] : p ∈ -Spec(R)} for all I ∈ K(R). Proof. That (1) ⇒ (2) follows from (1)–(3) of Theorem 3.11.3, and clearly (3) ⇒ (1). To prove (2) ⇒ (3), suppose that  is stable and every proper  -closed ideal of R is contained in some -prime of R. First, we claim that I  = {[I  ]R[p] : p ∈ -Spec(R)} for all I ∈K(R). To prove the claim, it suffices to assume that I is closed and show that {[I ]R[p] : p ∈ -Spec(R)} ⊆ I . Suppose x ∈ K lies in the given intersection, where K is the total quotient ring of R. For each p ∈ -Spec(R), we can write bp x = ap with ap ∈ I and bp ∈ R − p. Then, since bp ∈ (I : R x) − p, we have (I : R x) Ę p, for all p ∈ -Spec(R). Since (I : R x) is -closed, it follows that (I : R x) = R and  therefore x ∈ I . Now, since R = {R[p] : p ∈ -Spec(R)}, by Theorem 3.11.3 it follows that c = [-Spec(R)] is a stable semistar operation on R. By our claim proved above, we have (I  )c = I  for all I ∈ K(R) and therefore c  . Now, if J  = R, where J ∈ I(R), then we claim that J c = R. For suppose that J c = R. Then there exists x ∈ R − [J ]R[p] for some -prime ideal p of R. Then (J : R x) ⊆ p, so R = (R : R x) = (J  : R x) = (J : R x) ⊆ p, which is a contradiction. Finally, let I, J ∈ K(R) with J finitely generated and J  = R. Then since c J = R, we have (I : K J ) ⊆ (I : K J )c ⊆ (I c : K J ) = (I c : K J c ) = (I c : K R) = I c . Taking the union over all such J , we see that I  ⊆ I c . Therefore   c, so equality holds.  Corollary 3.11.6. Let  be a semistar operation on a ring R. One has the following. (1) w -Spec(R) = {q ∈ Spec(R) : q ⊆ p for some p ∈ t -Max(R)} = {q ∈ Spec(R) : qt = R}. (2) w -Max(R) = t -Max(R).  (3) w = [t -Max(R)] , that is, I w = {[I ]R[p] : p ∈ t -Max(R)} for all I ∈ K(R).

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3 Semistar Operations on Commutative Rings: Local Methods

Thus, for example, one has d = [Max(R)] = [Spec(R)]  [Max(R)reg ] = [Spec(R)reg ] , and therefore also d = Max(R) = Spec(R)  Max(R)reg = Spec(R)reg . By the theorem and Proposition 2.5.16, one has the following. Corollary 3.11.7. Let  be a semistar operation on a ring R. The following conditions are equivalent. (1) (2) (3) (4)

R is t -Prüfer. R is w -Prüfer R is [t -Max(R)] -Prüfer. R is [t -Max(R)reg ] -Prüfer.

For example, one has w = [t-Max(R)] , and R is a PVMR if and only if R is wPrüfer, if and only if R is [t-Max(R)reg ] -Prüfer. Note that   [] , and therefore if R is  -Prüfer then R is [] -Prüfer. Let us say that a prime ideal p of a ring R is strongly essential if R[p] is Prüfer (or equivalently if p is non-regular or satisfies the equivalent conditions of Corollary 3.10.15). We let SEss(R) denote the set of all strongly essential primes of R. By Corollary 3.10.15 and Proposition 3.10.18, one has the following. Proposition 3.11.8. For any ring one has SEss(R) ⊆ Ess(R), and equality holds if R is Marot. Lemma 3.11.9. Let R be a ring and p a prime ideal of R. If p is not regular, or if [p]R[p] is the unique regular maximal ideal of R[p] , or if R[p] is Prüfer, then R[p] is flat over R. Proof. If p is not regular, then R[p] = T (R) is flat over R. Suppose that P = [p]R[p] is the unique regular maximal ideal of S = R[p] , so that p = P ∩ R and S[P] = R[P∩R] . Then S[q] = R[q∩R] for every regular maximal ideal q of S, so that S is flat over R by Theorem 3.5.23. Finally, if S = R[p] is Prüfer, then by Corollary 3.10.15 either p  is not regular or else [p]R[p] is the unique regular maximal ideal of S. By Lemma 3.11.9 and Theorem 3.2.3, one can characterize the  -Prüfer rings as follows.  Theorem 3.11.10. Let R be a ring and  a subset of Spec R such that R = {R[p] : p ∈ }. (1) If R[p] is flat over R for all p ∈ , then the semistar operation  on R is stable. (2) If I ∈ K(R) is  -invertible, then I R[p] is invertible in R[p] for all p ∈ ; moreover, the converse holds if I is finitely generated and R[p] is flat over R for all p ∈ . (3) R is  -Prüfer if and only if R[p] is Prüfer for all p ∈ , if and only if  ⊆ SEss(R). Compare statement (3) of the theorem with statement (5) of Theorem 3.11.3. An obvious corollary of Theorems 3.11.3 and 3.11.10 is that, if  is a semistar operation on a Marot ring R, then R is [] -Prüfer if and only if R is  -Prüfer. In fact, we can show the stronger result that under these conditions one has []   .

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257

Lemma 3.11.11. Let R be a ring, p a prime ideal of R, and J a regularly generated ideal of R. Then [J ]R[p] = R[p] if and only if J R[p] = R[p] , if and only if J Ę p. Proof. Suppose that [J ]R[p] = R[p] , or equivalently, that J Ę p. Since J is regularly generated, there exists a regular x ∈ J − p. Since 1 = x −1 x ∈ R and x ∈ R − p, one has x −1 ∈ R[p] and therefore 1 = x x −1 ∈ J R[p] , whence J R[p] = R[p] . The converse  is clear since J R[p] ⊆ [J ]R[p] . Definition 3.11.12. For any prime ideal p of a ring R, let R(p) = {x ∈ T (R) : ux ∈ R for some regular u ∈ R − p}. Lemma 3.11.13. Let R be a ring and p a prime ideal of R. (1) (2) (3) (4)

Let U = (R − p)reg . Then R(p) = U −1 R is a flat overring of R contained in R[p] . pR(p) is a prime ideal of R(p) lying over p in R and under [p]R[p] in R[p] . R[p] = (R(p) )[pR(p) ] . R = {R(p) : p ∈ Max(R)reg }.

Lemma 3.11.14 (cf. [107, Theorem 7.6]). Let R be a Marot ring and p a prime ideal of R. Then R[p] = R(p) and [p]R[p] = pR(p) . In particular, R[p] is flat over R. Moreover, if p is regular, then [p]R[p] is the unique regular maximal ideal of R[p] . Proof. Certainly R(p) ⊆ R[p] . Let x ∈ R[p] , so that there exists a y ∈ R − p such that yx ∈ R. Since x ∈ T (R), there exists a regular u ∈ R such that ux ∈ R. The ideal I = (u, y) of R is regular and satisfies I x ⊆ R and I Ę p. Since R is Marot, I is regularly generated, so there must be a regular v ∈ I − p, so that vx ∈ I x ⊆ R. Therefore x ∈ R(p) . It follows that R[p] = R(p) . Suppose, now, that p is regular. Let I be a proper regular ideal of R(p) . Then I = H R(p) , where H = I ∩ R is a regular ideal of R not meeting U = (R − p)reg . It follows that every regular element of H lies in p, and therefore H ⊆ p since R is Marot. Therefore I = H R(p) ⊆ pR(p) . Therefore every proper regular ideal of R(p) is contained in pR(p) , so that pR(p) is the unique regular maximal ideal of R(p) . Finally, one has pR(p) ⊆ [p]R[p] and [p]R[p] is a proper ideal of R(p) , whence  pR(p) = [p]R[p] since pR(p) is maximal. Proposition 3.11.15. Let R be a Marot ring and  a subset of Spec R such that  R = {R[p] : p ∈ }. Then I [] = I  for all regular I ∈ K(R). Consequently, one has []   , and therefore R is [] -Prüfer if and only if R is  -Prüfer. Proof. For any regular ideal I of R, one has I [] = R if and only if [I ]R[p] = R for all p ∈ , and likewise I  = R if and only if I R[p] = R for all p ∈ . Therefore by Lemma 3.11.11, one has I [] = R if and only I  = R for all regular ideals I of R. Moreover, by Lemma 3.11.11, the overring R[p] of R is flat over R for every prime ideal p of R, and therefore by Theorems 3.11.3 and 3.11.10 both [] and  are stable. It follows that for all regular I ∈ K(R), one has

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 {(I : K J ) : J ∈ Ireg (R), J [] = R}  = {(I : K J ) : J ∈ Ireg (R), J  = R}

I [] =

= I  The proposition follows.



Corollary 3.11.16. Let  be a semistar operation on a Marot ring R. Then   w  t -Max(R)  t -Max(R)reg and in fact I w = {I R[p] : p ∈ t -Max(R)} = {I R[p] : p ∈ t -Max(R)reg } for all regular I ∈ K(R). Let  be a semistar operation on a ring R. We say that R is -large-locally Prüfer if R[p] is Prüfer for every -prime ideal p of R. By the lemma above, Corollaries 3.11.6 and 3.11.7, Theorem 3.11.10, Corollary 3.11.16, and the fact that t -Max(R)  [t -Max(R)] , we have the following. Proposition 3.11.17. Let  be an r-finite type semistar operation on a ring R. The following conditions are equivalent. (1) (2) (3) (4)

R is -large-locally Prüfer. R[p] is Prüfer for every regular -maximal ideal p of R. R is -Spec(R) -Prüfer. R is -Max(R)reg -Prüfer.

Moreover, each of the conditions above implies that R is -Prüfer, and the converse also holds if R is Marot. Corollary 3.11.18. Let R be a ring. The following conditions are equivalent. (1) (2) (3) (4)

R is t-large-locally Prüfer. R[p] is Prüfer for every regular t-maximal ideal p of R. R is t-Spec(R) -Prüfer. R is t-Max(R)reg -Prüfer.

Moreover, each of the conditions above implies that R is a PVMR, and the converse also holds if R is Marot. Note that any valuation ring is a PVMR, but it is t-large-locally Prüfer if and only if it is Prüfer, since if (R, p) is a valuation pair of T (R) = R then p is the only regular t-maximal ideal of R and R = R[p] . Thus, any non-Prüfer valuation ring is a PVMR that is not t-large-locally Prüfer. Proposition 3.11.19. Let R be a ring with total quotient ring K . Every strongly essential prime of R is essential, and therefore SEss(R)reg ⊆ Ess(R)reg ⊆ sK(K /R) ⊆ t-Spec(R)reg . Moreover, equalities hold if and only if R is t-large-locally Prüfer, if and only if R is a PVMR in which every essential prime ideal is strongly essential.

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 Let us say that a ring R is strongly essential if R = {R[p] : p ∈ SEss(R)}. Clearly, any strongly essential ring is essential. By Theorem 3.11.10, a strongly essential ring is  -Prüfer and therefore v-Prüfer, where  = SEss(R). Also, every PVMR is essential, while every t-large-locally Prüfer ring is strongly essential. Thus we have the following implications. Prüfer

t-large-locally Prüfer

strongly essential

PVMR

essential

v-Prüfer

Of course, if R is Marot, then t-large-locally Prüfer is equivalent to PVMR and strongly essential is equivalent to essential. Open Problem 3.11.20. Does there exist an essential ring (or a valuation ring) that is not strongly essential? Does there exist a strongly essential PVMR that is not t-large-locally Prüfer? Does there exist a strongly essential valuation ring that is not Prüfer? Notable special cases of Theorem 3.11.3 are where  is t-Max(R), sK(T (R)/R), or wAss(T (R)/R). We say that a ring R is a K-ring if every strong Krull prime of T (R)/R is essential, and R is a K-domain if R is a domain that is a K-ring. We also say that R is a P-ring if every weak Bourbaki associated prime of T (R)/R is essential. A domain that is a P-ring is said to be a P-domain [150, p. 2]. A K-ring (resp., P-ring) is equivalently a ring R that is [] -Prüfer, where  = sK(T (R)/R) (resp.,  = wAss(T (R)/R)). Proposition 3.11.21. Let R be a ring. Then R is a K-ring if and only if R[p] is a PVMR for every p ∈ sK(T (R)/R). Likewise, R is a P-ring if and only if R[p] is a PVMR for every p ∈ wAss(T (R)/R). Proof. We prove the first statement. The second is proved in like fashion. Suppose that R[p] is a PVMR for every p ∈ sK(T (R)/R). Let p ∈ sK(T (R)/R). Then the ideal P = [p]R[p] of S = R[p] is t-prime. Therefore, since S is a PVMR, it follows that (S[P] , [P]S[P] ) = (R[p] , [p]R[p] ) is a valuation pair of T (S) = T (R). Therefore R is a K-ring. Conversely, if R is a K-ring and p ∈ sK(T (R)/R), then (R[p] , [p]R[p] )  is a valuation pair of T (R), and therefore R[p] is a PVMR by Corollary 3.8.6. Let us say that a ring R is an almost PVMR if R[p] is a PVMR for every (regular) prime ideal p of R. Let us also say that a ring R is a near PVMR if R[p] is a PVMR for every regular t-maximal (or every t-prime) ideal p of R. By Theorem 3.5.14, any PVMR is an almost PVMR. Likewise, by Proposition 3.11.21, any near PVMR is a K-ring. Therefore we have the following implications. PVMR ⇒ almost PVMR ⇒ near PVMR ⇒ K-ring ⇒ P-ring ⇒ essential ⇒ v-Prufer. ¨

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Since any v-coherent v-Prüfer ring is (equivalently) a PVMR, the implications above are reversible for the v-coherent rings. Likewise, since any H v-Prüfer ring is a PVMR, the implications above are also reversible for the H rings. It is known that a v-Prüfer domain is not necessarily essential, an essential domain is not necessarily a P-domain, and an almost PVMR that is a domain is not necessarily a PVMD [68, 150]. (For example, by [150, Example 2.1], an almost PVMR need not be a PVMR and in fact there exists a domain D such that Dp is a GCD domain, hence a PVMD, for every prime ideal p of D and yet D is not a PVMD.) Thus, the first arrow above and the last two arrows are irreversible, but the other three arrows are a mystery: according to [68], it is unknown whether or not there exists a P-domain that is not an almost PVMR. Thus we have the following. Open Problem 3.11.22. Does there exist a P-domain that is not an almost PVMR? Does there exist a P-domain that is not a near PVMR? Does there exist a P-domain that is not a K-domain? Does there exist a K-domain that is not an almost PVMR? Does there exist a K-domain that is not near PVMR? Does there exist a near PVMR that is not an almost PVMR? We expect that most or all of the questions in Open Problems 3.11.20 and 3.11.22 have positive answers. At any rate, it is clear there are several classes of rings between the PVMRs and the essential rings that differ from both in subtle ways. On account of Theorem 3.11.3, the K-ring and P-ring conditions are quite natural. By contrast, however, the t-large-locally Prüfer, almost PVMR, and near PVMR conditions are interesting only insofar as they provide further insight into the various subtleties involved with PVMR-like conditions. Such subtleties, as we have witnessed, are not nearly as pronounced for the Dedekind, Krull, and Prüfer rings as they are for the PVMRs. The following result summarizes how the Dedekind, Prüfer, Krull, and PVMR properties behave with respect to large localization. Proposition 3.11.23. Let R be a ring. One has the following. (1) R is Dedekind if and only if R[p] is Dedekind (or a Dedekind valuation ring) for all p ∈ Max(R)reg and every regular element of R lies in only finitely many p ∈ Max(R)reg . (2) R is Prüfer if and only if R[p] is Prüfer (or a Prüfer valuation ring) for all p ∈ Max(R)reg . (3) The following conditions are equivalent. (a) R is Krull.  1 (R)}, R[p] is a Krull (or a Krull valuation ring) for (b) R = {R[p] : p ∈ X reg 1 every p ∈ X reg (R), and every regular element of R lies in at most finitely 1 many p ∈ X reg (R). (c) R[p] is Krull (or a Krull valuation ring) for all p ∈ wAss(T (R)/R) and every regular element of R lies in only finitely many p ∈ wAss(T (R)/R). (d) R[p] is Krull (or a Krull valuation ring) for all p ∈ sK(T (R)/R) and every regular element of R lies in only finitely many p ∈ sK(T (R)/R).

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261

(e) R is v-coherent, R[p] is Krull (or a Krull valuation ring) for all p ∈ t-Max(R)reg , and every regular element of R lies in only finitely many p ∈ t-Max(R)reg . (4) R is a PVMR if and only if R is v-coherent and R[p] is a PVMR (or a valuation ring) for all p ∈ t-Max(R)reg . Finally, we prove that the poset t-Spec(R)reg of all regular t-prime ideals of a PVMR R, or more generally the poset Ess(R)reg of all regular essential primes of a ring R, is a tree. A poset S is said to be a tree if for all x ∈ S the subset {y ∈ S : y  x} of S is totally ordered, or equivalently if no two incomparable elements of S have an upper bound in S. (Parting from more standard terminology, we do not require that the subset {y ∈ S : y  x} be well ordered.) Theorem 3.11.24. Let R be a ring. One has the following. (1) For any prime ideal p of R, the association q −→ [q]R[p] provides an isomorphism from the poset of all prime ideals (resp., regular prime ideals) q of R contained in p to the poset of all prime ideals (resp., regular prime ideals) of R[p] contained in [p]R[p] . (2) Any prime ideal of R contained in an essential prime of R is essential. (3) If R is a PVMR, then any regular prime ideal of R contained in a t-prime ideal of R is t-prime. (4) If (R, p) is a valuation pair for some regular prime p of R, then t-Spec(R)reg = {q ∈ Spec(R)reg : q ⊆ p} is totally ordered. (5) If p is a regular essential prime of R, then t-Spec(R[p] )reg = {Q ∈ Spec(R[p] )reg : Q ⊆ [p]R[p] } is totally ordered. (6) The poset Ess(R)reg is a tree, and in fact for all p ∈ Ess(R)reg the poset {q ∈ Spec(R)reg : q ⊆ p} = {q ∈ Ess(R)reg : q ⊆ p} is totally ordered. (7) If R is a PVMR, then the poset t-Spec(R)reg is a tree, and in fact for all p ∈ t-Spec(R)reg the poset {q ∈ Spec(R)reg : q ⊆ p} = {q ∈ t-Spec(R)reg : q ⊆ p} is totally ordered. (8) If R is Prüfer, then the poset Spec(R)reg is a tree. (9) If R is a K-ring, then the poset sK(T (R)/R) is a tree, and in fact for all p ∈ sK(T (R)/R) the poset {q ∈ Spec(R)reg : q ⊆ p} = {q ∈ sK(T (R)/R) : q ⊆ p} is totally ordered. (10) If R is a P-ring, then the poset wAss(T (R)/R) is a tree. Proof. Statement (1) follows from Proposition 3.5.3(19), statement (2) is easily verified, and (3) follows from (2) and Proposition 3.3.19. Statement (4) follows from (3) and Corollaries 3.6.23(7) and 3.6.24, and (5) follows from (4). Finally, statement (6) follows from (1) and (5), and (7)–(10) follow from (6).

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Exercises Section 3.1 1. Prove the equivalence of statements (1)–(4) of Theorem 3.1.3. 2. Let v be a valuation on a field K . Verify that K v is a subring of K and that pv is the unique maximal ideal of K v . 3. Let v : K −→  ∪ {∞} be a valuation on a field K , and let x1 , x2 , . . . , xn ∈ K . Show that if the v(xi ) are all distinct, then v(x1 + x2 + · · · + xn ) = min{v(xi )}. 4. Verify the unproved assertions in the proof of Proposition 3.1.7. 5. Prove Corollaries 3.1.9 and 3.1.11. 6. Show that, if V is a valuation domain, then V /p and Vp are valuation domains for every prime ideal p of V . 7. Let V be a valuation domain. Express the Krull dimension of V in terms of | Spec V |. 8. Show that the maximal ideal of a valuation domain V is principal if and only if the value group of V has a least element greater than 0. 9. Let K be an ordered field. An element x of K is said to be finite if x lies between two integers in K , that is, if m  x  n for some m, n ∈ Z; otherwise, x is said to be infinite. Show that the set D of all finite elements of K is a valuation domain. If x ∈ K is finite but x −1 is infinite, then x is said to be infinitesimal. Show that the maximal ideal of D is equal to the set of all infinitesimals of D. 10. Let v : K −→  ∪ {∞} be a valuation on a field K . Prove that v can be extended uniquely to a valuation v  : K (X ) −→  ∪ {∞} on the field K (X ) of rational functions in X such that v  (c0 + c1 X + · · · + cn X n ) = min{v(ci )} for all c0 , c1 , . . . , cn ∈ K . 11. Let D be a valuation domain with quotient field K , and let  be the totally ordered abelian group Prin(D) of nonzero principal fractional ideals of D, ordered by ⊇. Note that the group Prin(D) is written multiplicatively, not additively, and its “0” is the ideal D = (1). Show that the map v : K −→ Prin(D) ∪ {∞} acting by x −→ x D, where ∞ is the ideal (0), is a valuation on K with D = K v . 12. Let V be a valuation domain with maximal ideal m and V  an overring of V . a) Prove that V  is a valuation domain. Let its maximal ideal be denoted m . b) Prove that m ⊆ m, with equality holding if and only if V = V  . c) Prove that m is a prime ideal of V and V  = Vm . 13. Prove directly, without using Theorem 3.1.3, that every valuation domain is integrally closed. 14. Prove that any nonzero principal prime ideal of a valuation domain V must be equal to the maximal ideal of V . 15. Let D be a UFD with quotient field K = D, and let π be a nonzero prime element of D. a) Show that there exists a unique normalized discrete rank one valuation vπ : K −→ Z ∪ {∞} on K . The valuation vπ is called the π-adic valuation on K .

Exercises

263

b) Describe all π-adic valuations explicitly for the PIDs Z and k[X ] for any field k. c) Show that the valuation ring associated to the π-adic valuation on K is equal to the localization D(π) of D at the prime ideal (π). d) Let P be a complete  set of pairwise non-associate primes of D. Show that a is associate to  π∈P π vπ (a) for all nonzero a ∈ D. e) Show that D = π∈P D(π) . f) Show that every discrete rank one valuation v : K −→ Z ∪ {∞} on K has the form v = nvπ for some positive integer n. 16. A subset F of a totally ordered abelian group  is said to be a filter in  if F is a nonempty proper subset of  + = {x ∈  : x > 0} such that β ∈ F for all β ∈  such that β  α for some α ∈ F. a) Let α ∈  + . Show that the set {β ∈  + : β  α} is a filter in . Such filters are said to be principal. b) Let v : K −→  ∪ {∞} be a surjective valuation on a field K , and let D = K v . Show that the correspondence I −→ F = v(I − {0}) defines a bijection between the set of nonzero ideals I of D and the set of filters F in , where the inverse map is the association F −→ I = v −1 (F) ∪ {0}. c) In the notation of part (b), show that an ideal I of D is principal if and only if the filter F = v(I − {0}) is principal, and the ideal I is prime if and only if the complement  + − F of F in  + is closed under addition. 17. In this exercise we show that for every field k and every totally ordered abelian group  there is a valuation domain V with residue field k and value group . Let k be a field and  a totally ordered abelian written additively.  group, γ γ k X with a) Let k[] denote the k-vector space g∈  basisγ {X : γ ∈ }. Elements of k[] are formal linear combinations g∈ cγ X , where cγ ∈ k for all γ ∈  and all but finitely many of them are equal to 0. Prove that there is a unique k-algebra structure on k[] such that X γ X δ = X γ+δ for all γ, δ ∈ .  b) For all f = γ∈ cγ X γ , define deg( f ) for f = 0 to be the largest γ ∈  such that cγ = 0, and set deg(0) = ∞. Show that the function deg : k[] −→  ∪ {∞} is well-defined and satisfies the following conditions for all f, g ∈ k[]: (1) deg( f ) = ∞ if and only if f = 0, (2) deg( f g) = deg( f ) + deg(g), and (3) deg( f + g)  min{deg( f ), deg(g)}. c) From part (b), conclude that k[] is an integral domain. Its quotient field is denoted k(). Elements of k() are formal quotients f /g, where f, g ∈ k[] and g = 0.

264

18.

19.

20.

21.

3 Semistar Operations on Commutative Rings: Local Methods

d) Show that the map deg extends uniquely to a valuation deg : k() −→  ∪ {∞} on the field k(). Describe the associated valuation ring, denoted k()deg e) Show that the valuation ring k()deg has value group isomorphic to  and residue field isomorphic to k. f) Show that if  = Z is the totally ordered abelian group Z, then k() is isomorphic to the field k(X ) of rational functions. Describe the associated valuation ring of k(X ). g) Show that if  is the totally ordered (free) abelian group Zn , ordered lexicographically, then k() is isomorphic to the field k(X 1 , X 2 , . . . , X n ). Describe the associated valuation ring of k(X 1 , X 2 , . . . , X n ). A subset X of a poset S is convex if for all s ∈ S with x  s  x  for some x, x  ∈ X one has s ∈ X . Let v : K −→  ∪ {∞} be a surjective valuation on a field K , and let D = K v . Prove the following. a) v(m) is equal to  + = {x ∈  : x > 0}, where m is the maximal ideal of D. b) An ideal I of D is principal if and only if v(I ) contains a least element. c) Let I be a subset of D. Then I is a prime ideal of D if and only if v(I − {0}) is the complement in  + of a convex subgroup of . d) The poset Spec D ordered by ⊆ is order isomorphic to the poset of all convex subgroups of  ordered by ⊇. e) The set of all convex subgroups of  is totally ordered by inclusion. f) D is one dimensional if and only if D is not a field and  has no nontrivial proper convex subgroups. g) The Krull dimension of D is equal to the supremum of the lengths of the chains of convex subgroups of . A totally ordered abelian group  is said to be Archimedean if for all γ, δ ∈  with γ > 0 there exists a positive integer n such that nγ  δ. (A famous theorem of Hölder states that a totally ordered abelian group is Archimedean if and only if it has no nontrivial proper convex subgroups, if and only if it is order isomorphic to a subgroup of R.) Let V be a valuation domain that is not a field. Prove the following conditions equivalent. 1) V is one dimensional. 2) The value group of V is Archimedean. 3) V is completely integrally closed. (Hint: For the proof of (1) ⇔ (2), note that, for any nonzero elements x, y of a valuation domain V , one has nv(x) < v(y) for all positive integers n if √ and only if x ∈ / (y), if and only if there exists a prime ideal of V containing y but not x.) Let V be a valuation domain with quotient field K = V . Prove the following. a) There are at most two star operations on V , namely, d and v. b) One has K(V ) = F(V ) ∪ {K }. c) There are at most two semistar operations on V , namely, d and v. Prove Proposition 3.1.14.

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265

22. Prove Proposition 3.1.13. Use the proposition to show that a domain is a Dedekind domain if and only if it is Noetherian, integrally closed, and at most one dimensional. 23. A valuation v : K −→  ∪ {∞} on a field K is said to be trivial if v(x) = 0 for all nonzero x ∈ K . Prove the following. a) Let  be any totally ordered abelian group. Then  is Z-torsion-free, that is, if na = 0 for some a ∈  and some integer n, then either n = 0 or a = 0. b) Any valuation on a finite field is trivial. c) For every prime p, every valuation on the algebraic closure of F p is trivial. Section 3.2 1. Let  be a finite type semistar operation on an integral domain D. Prove that the following conditions are equivalent for all I ∈ Freg (D). 1) I is -invertible. 2) I is strictly -finite and I Dp is invertible in Dp for every p ∈ -Max(D). 3) I is strictly -finite and I Dp is principal in Dp for every p ∈ -Max(D). 2. Using Exercise 3.1.19, show that any one dimensional Prüfer domain is completely integrally closed. 3. Show that any prime ideal of an integral domain D contained in an essential prime of D is essential. 4. Let p be a t-prime ideal of a PVMD D. Show that (pn )t is equal to the nth symbolic power p(n) = pn Dp ∩ D of p for any positive integer n. 5. Show that, for all t-prime ideals p and q of a PVMD D, one has (p + q)t = D if p and q are incomparable (that is, if neither ideal is contained in the other). 6. Let  be a semistar operation on an integral domain D, let p be a w -prime ideal of D, and let I be an ideal of D. Prove the following. a) If I Dp ∩ D = I , then I is w -closed. b) (pn )w is equal to the nth symbolic power p(n) = pn Dp ∩ D of p for any positive integer n. c) If p is w -prime and maximal, then pn is w -closed for all n. 7. Let  be a semistar operation on an integral domain D. a) Show that the following conditions are equivalent for any prime ideal p of D. 1) p is w -prime. 2) p is contained in a t -closed ideal of D. 3) p is contained in a t -maximal ideal of D. 4) p is contained in a t -prime ideal of D. 5) pt = D. b) Show that w -Spec(D) = {p ∈ Spec(D) : p ⊆ q for some q ∈ t -Spec(D)}. c) Explain how part (b) generalizes the fact that w -Max(D) = t -Max(D). d) Find an example of a w-prime ideal that is not t-prime. 8. [148]. Let D be an integral domain that is not a field. Prove the following conditions equivalent. 1) D is a DW domain, that is, w  d.

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10.

11.

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3 Semistar Operations on Commutative Rings: Local Methods

2) The only finite type stable semistar operation on D is the trivial semistar operation. 3) Every maximal ideal of D is w-closed. 4) Every maximal ideal of D is t-closed. 5) Max(D) = t-Max(D). 6) If I ∈ Freg (D) is finitely generated and I t = D, then I = D. Let D be an integral domain. Prove the following conditions equivalent. 1) D is Prüfer. 2) I ∩ (J + K )=I ∩ J + I ∩ K for all (finitely generated) I, J, K ∈ Freg (D). 3) I + (J ∩ K ) = (I + J ) ∩ (I + K ) for all (finitely generated) I, J, K ∈ Freg (D). 4) I (J ∩ K ) = I J ∩ I K for all (finitely generated) I, J, K ∈ Freg (D). Let D be an integral domain. Using Exercise 2.2.14, prove the following conditions equivalent. a) D is a Prüfer domain. b) Every D-torsion-free D-module is flat. c) Every ideal of D is flat as a D-module. d) Every finitely generated ideal of D is flat as a D-module. [83]. Let R be a ring. Prove the following. a) Every principal ideal of R is flat if and only if Rp is a domain for every prime ideal p of R. Such rings R are said to be PF rings. b) Every ideal of R is flat if and only if Rp is a valuation domain for every prime ideal p of R. Such rings R are said to be of weak global dimension at most one. c) Every R-torsion-free R-module is flat if and only if T (R) is von Neumann regular and R is of weak global dimension at most one. Such rings R are said to be semihereditary. d) R is von Neumann regular if and only if R is a semihereditary total quotient ring. Read [68] and verify that none of the following implications is reversible: PVMD ⇒ essential ⇒ v-domain ⇒ integrally closed.

Section 3.3 1. 2. 3. 4.

Verify the equivalence of statements (1)–(3) of Proposition 3.3.18. Prove the implications (4) ⇒ (6) ⇔ (7) ⇔ (8) ⇒ (9) of Proposition 3.3.2. Prove or give a counterexample: any divisorial ring is of finite character. Let R be a ring  and {Rλ : λ ∈ } is an indexed collection of overrings of R, and let S = {Rλ : λ ∈ }. Show that {Rλ : λ ∈ } is a locally finite indexed collection of R-algebras if and only if {Rλ : λ ∈ } is a locally finite indexed collection of S-algebras. 5. a) Prove that if S is a t-compatible extension of a ring R and T is a t-compatible extension of S, then T is a t-compatible extension of R.

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b) Prove that if S is a t-compatible extension of a ring R and T is a flat extension of R, then S ⊗ R T is a t-compatible extension of R. c) Are the analogous statements for t-linked extensions true? Prove or give a counterexample. Let R be a ring. a) Show that the intersection of any collection of semistar t-compatible overrings of R is a semistar t-compatible overring of R. b) Repeat part (a) for the following properties: t-compatible, semistar t-linked, and t-linked. Let R be a ring and M an R-module. Show that a prime ideal p lies in sK(M) if and only if every finitely generated ideal of Rp contained in pRp annihilates some nonzero element of Mp . Let R be a ring and M an R-module. Prove the following. a) The union of any chain of primes in sK(M) lies in sK(M). b) Any prime in sK(M) is contained in some maximal element of sK(M). Let R be a ring with total quotient ring K . Show that every prime ideal of R is a union of weak Bourbaki associated primes of K /R.  Let D be an integral domain with quotient field K . Prove that D= p∈wAss(K /D) Dp . Prove that any t-compatible overring of a Mori ring R that is also a fractional ideal of R is Mori. Prove that any flat overring of a Noetherian (resp., r-Noetherian) ring is Noetherian (resp., r-Noetherian). Prove that any flat overring of a Krull ring is Krull. Prove that any t-linked overring of a PVMR is a PVMR. Let R be a ring, M an R-module, and U a multiplicatively closed subset of R. Show that wAssU −1 R (U −1 M) = {pU −1 R ∈ wAss R (M) : p ∩ U = ∅} and sKU −1 R (U −1 M) = {pU −1 R ∈ sK R (M) : p ∩ U = ∅}. a) Let R be the ring of all algebraic integers. Show that Ass(T (R)/R) = v-Max(R) = ∅. (Hint: For any α ∈ T (R), show that (R : R α) is not a proper radical ideal and is therefore not prime.) Using the fact that R is a one dimensional Bézout domain, show that wAss(T (R)/R) = sK(T (R)/R) = Max(R) = Spec(R)reg . Conclude that R is not an H ring. 2 b) Let R be the ring F p [X, X 1/ p , X 1/ p , . . .], where p is a prime number. Show that R is a one dimensional Bézout domain with Ass(T (R)/R) = v-Max(R) = ∅ and wAss(T (R)/R) = sK(T (R)/R) = Max(R) = Spec (R)reg . (Hint: For any α ∈ T (R), show that (R : R α) is not a proper radical ideal and is therefore not prime.) Conclude that R is not an H ring. Show, moreover, that 2

Max(R) = {( f (X ), f (X 1/ p ), f (X 1/ p ), . . .) : f ∈ F p [X ] is irreducible} and therefore Spec(R) and Spec(F p [X ]) are order isomorphic. 17. Let p be a t-invertible t-prime of a ring R. Prove the following.

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a) There exists a finitely generated ideal I such that p = I v . b) p is t-maximal. (Hint: Let J be any ideal properly containing p, and let c ∈ J − p. Show that (I + c R)−1 = R, where I is as in part (a), and therefore J t = R.) c) Every regular t-invertible t-prime of R lies in Ass(T (R)/R) ∩ t-Max(R). d) If every regular t-maximal ideal of R is t-invertible, then R is an H ring. e) A PVMR R is an H ring if and only if every regular t-maximal ideal of R is t-invertible. Let D ⊆ D  be an extension of integral domains. Prove the following conditions equivalent. 1) D  is a t-linked extension of D. 2) For every t-prime q of D  , one has (q ∩ D)t = D. 3) For every q of D  , one has q ∩ D ⊆ p for some t-prime p of D.  t-prime   4) D = {Dp : p ∈ t-Max(D)}. Let  be a semistar operation on a ring R. In this exercise we show that any prime ideal minimal over a -closed ideal (or minimal over a regular principal ideal) of R is t -prime. Let p be a prime minimal over a -closed ideal I , and let J be any finitely generated subideal of p. Prove the following.  a) I Rp = pRp . b) c J n ⊆ I for some c ∈ R − p and some positive integer n. c) J  ⊆ p. d) p is t -prime. e) Any prime ideal minimal over a -closed ideal (or minimal over a regular principal ideal) of R is t -prime. f) Any ideal of R that is minimal among the regular primes of R is t-prime. g) Any height one prime of an integral domain is t-prime. h) The radical of any -closed ideal of R is t -closed. Let D be an integral domain. Consider the following conditions. 1) Every t-closed ideal of D is principal. 2) Every t-prime ideal of D is principal. 3) Every prime ideal of D minimal over a principal ideal is principal. 4) Every irreducible element of D is prime. Prove that (1) ⇒ (2) ⇒ (3) ⇒ (4) and that the four conditions are equivalent if D is atomic. You may use Exercise 19 to prove the implication (2) ⇒ (3). In this exercise we give a more direct proof of Theorem 2.9.19 using as few subsidiary results as possible. Let D be an integral domain. Prove the following seven conditions equivalent, via the implications (1) ⇒ (2) ⇒ (3) ⇒ (4) ⇒ (5) ⇒ (6) ⇒ (7) ⇒ (1), using only (i) the results of Exercises 19 and 20, (ii) the fact that Mori implies TV, and (iii) the fact that I v for any I ∈ F(D) is principal if and only if gcd(cI ) exists for all nonzero c ∈ D. 1) D is a UFD. 2) D is strong GCD and Mori. 3) D is strong GCD and TV. 4) Every t-closed ideal of D is principal. 5) D satisfies ACCP and every t-prime ideal of D is principal.

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6) D satisfies ACCP and every prime ideal of D minimal over a principal ideal is principal. 7) D satisfies ACCP and every irreducible element of D is prime. Krull’s principal ideal theorem (PIT) states that if R is a Noetherian ring and p is a prime ideal minimal over a principal ideal of R, then p has height at most one. Use PIT to prove that a Noetherian domain D is a UFD if and only if every height one prime ideal of D is principal. a) Prove that an integral domain D is a UFD if and only if every nonzero prime ideal of D contains a nonzero prime element of D. (Hint: For the “if” direction, consider the set U of products of all nonzero prime elements of D, which is a saturated multiplicative subset of D. If a ∈ D is not a product of nonzero prime elements of D, then show that there is an ideal p containing a that is maximal with respect to being disjoint from U , and p is in fact prime.) b) Use part (a) and Exercise 19 to give an alternative proof of the fact that if every t-closed ideal of D is principal then D is a UFD. Let R be a ring with total quotient ring K . Prove the following. a) (I v : K I v ) is a semistar t-linked overring of R containing (I : K I ) for every I ∈ K(R). b) The complete integral closure R  of R is a semistar t-linked overring of R. c) The complete integral closure of a PVMR is a PVMR. (∗∗) Must D[X ] be a TV domain if D is a TV domain? Prove or give a counterexample. Note that there exist Mori domains D such that D[X ] is not Mori [170].

Section 3.4  1. Prove that I v = {I Dp : p ∈ X 1 (D)} for any nonzero fractional ideal I of a Krull domain D. 2. Prove Proposition 3.4.4 and Corollary 3.4.8. 3. The t-dimension of a domain D is the supremum of the lengths n of all ascending chains (0)  p1  p2  · · ·  pn of t-prime ideals of D. Prove the following. a) A domain D is of t-dimension zero if and only if it is a field, if and only if t-Spec(D) is empty. b) If D is not a field, then the following conditions are equivalent. 1) D is of t-dimension one. 2) Every nonzero t-prime ideal of D is t-maximal. 3) Every t-maximal ideal of D is of height one. 4) Every height one prime of D is t-maximal. 5) X 1 (D) = t-Max(D). 6) X 1 (D) = t-Spec(D). c) The (Krull) dimension of a TD domain is equal to its t-dimension. d) Every one dimensional domain is of t-dimension one, but there exist domains of t-dimension one that are not one dimensional.

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4. Let D be a Krull domain. a) Let p be a height one prime ideal of D. Prove that (pn )v is equal to the nth symbolic power p(n) = pn Dp ∩ D of p. b) Let I and J be divisorial ideals of D. Show that no height one prime divides both I and J in Freg (D) if and only if (I + J )v = D, in which case (I J )v = I ∩ J. c)  Prove that any divisorial ideal of D can be expressed uniquely in the form (n p ) , where the n p are nonnegative integers, all but finitely many p∈X 1 (D) p of which are zero. 5. Show that an extension D ⊆ D  of Krull domains is t-linked if and only if q ∩ D has height at most one in D for all q ∈ X 1 (D  ). Such an extension of Krull domains is said to satisfy PDE. 6. Prove Theorem 3.4.9. 7. Let D be an integral domain with quotient field K , and let L be a field extension of K . Show that if {Dλ : λ ∈ }  is a locally finite collection of Krull domains contained in L such that D = {Dλ : λ ∈ }, then D is a Krull domain. Conclude that D is Krull if and only if D is of finite t-character and Dp is Krull for every t-maximal ideal p of D. 8. Let D be a Krull domain with quotient field K , and let K  be a subfield of K . Show that K  ∩ D is a Krull domain. 9. Let D be an integral domain with quotient field K . Prove that the following conditions are equivalent. a) D is a Krull domain. b) Every nonzero t-prime ideal of D is t-invertible. c) Every weak Bourbaki associated prime of K /D is t-invertible. 10. Let D be an integral domain. In this exercise we show that D[X ] is a Krull domain if and only if D is a Krull domain. Prove the following. a) If V is a DVR overring of D[X ], then V ∩ K is a DVR overring of D. b) If D[X ] is a Krull domain, then D is a Krull domain. c) Suppose that D is a Krull domain. i) Let p be a height one prime of D and v = vp : K −→ Z ∪ {∞} the associated discrete rank one valuation on K with K v = Dp . For any f = a0 + a1 X + · · · + an X n ∈ K [X ], set v( f ) = min{v(ai ) : i = 0, 1, . . . , n}. Show that v extends uniquely to a discrete rank one valuation on K (X ), and one has K (X )v = D[X ]pD[X ] and K [X ] ∩ K (X )v = Dp [X ]. In particular,   D[X ]pD[x] is a DVR. ii) D[X ] = K [X ] ∩ p∈X 1 (D) D[X ]pD[X ] . iii) D[X ] is a Krull domain. iv) The height one primes of D[X ] are precisely the primes of the form pD[X ], where p is a height one prime of D, and the primes of the form m ∩ D[X ], where m is a (necessarily principal) maximal ideal of K [X ]. The latter are characterized by the fact that their intersection with D is 0.

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11. Let D be a Krull domain with quotient field K . Prove the following. a) For each p ∈ X 1 (D) let vp : K −→ Z ∪ {∞} denote the normalized discrete rank one valuation on K with Dp = {x ∈ K : vp (x)  0}. Let p1 , p2 , . . . , pn be any n distinct height one primes of D, and let k1 , k2 , . . . , kn be integers. Then there exists an element x of K such that vpi (x) = ki for all i = 1, 2, . . . , n and vp (x)  0 for all p ∈ X 1 (D) − {p1 , p2 , . . . , pn }. This is known as the approximation theorem for Krull domains. b) If I is any divisorial fractional ideal of D, then for any nonzero a ∈ I there exists a b ∈ I such that (a, b)v = I . c) A nonzero fractional ideal of D is divisorial if and only if it is the intersection of two principal fractional ideals of D. d) Any Krull domain with only finitely many height one primes is a UFD. 12. [58]. Let D be an integral domain with quotient field K . Prove the following. a) For all I ∈ K(D) and all p ∈ t-Max(D), one has I Dp = (I w )Dp . b) If D is w-Noetherian, then Dp is Noetherian for all p ∈ t-Max(D). c) D is a Krull domain if and only if D is integrally closed and w-Noetherian. Section 3.5 1. 2. 3. 4.

Verify all the claims made in Definition 3.5.1. Prove statements (1)–(14) of Proposition 3.5.3. Prove statements (15)–(19) of Proposition 3.5.3. Let R be a subring of a ring S, let q be a prime ideal of S, and let p = q ∩ R. Show that [q]S[q] ∩ R[p] = [p]R[p] . 5. a) Show that, if R is an integrally closed ring, then R[U ] = {x ∈ T (R) : ux ∈ R for some u ∈ U } is an integrally closed overring of R for every multiplicative subset U of R. b) Prove Proposition 3.5.21. c)  Let R be a ring, and let  be a set of prime ideals of R such that R = p∈ R[p] . Show that R is integrally closed if and only if R[p] is integrally closed for all p ∈ . 6. Let m be a maximal ideal of a ring R. Show that [mn ]R[m] = ([m]R[m] )n for every nonnegative integer n. 7. Let p and q be prime ideals of a ring R with q ⊆ p, and let a be an ideal of R. Prove the following. a) If p is not regular, then [q]R[p] = qT (R) and R[p] /[q]R[p] = T (R)/qT (R). b) R[p] /[a]R[p] is isomorphic to a subring of Rp /aRp . c) R[p] /[(0)]R[p] is isomorphic to a subring of Rp . d) Unlike the field Rp /pRp , the ring R[p] /[p]R[p] may not be a field (and thus [p]R[p] may not be maximal), even if p is regular. e) (R[p] )[q]R[p] is isomorphic to Rq .

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8. √ Let p be a prime ideal of a ring R. An ideal a of R is said to be p-primary if a = p and ab ∈ a implies a ∈ a or b ∈ p for all a, b ∈ R. An ideal is primary if it is p-primary for some prime ideal p. Prove the following. a) A proper ideal a is primary if and only if ab ∈ a implies a ∈ a or bn ∈ a for some positive√ integer n, for all a, b ∈ R, and in case those conditions hold the ideal p = a is prime and a is p-primary. b) A proper ideal a of R is primary if and only if every zerodivisor of R/a is nilpotent. c) If the radical of an ideal a of R is a maximal ideal m, then a is m-primary. d) Every ideal a of R with radical p is contained in a smallest p-primary ideal, equal to aRp ∩ R = [a]R[p] ∩ R. e) The ideal p(n) = pn Rp ∩ p is the smallest p-primary ideal of R containing pn , for any positive integer n. The ideal p(n) is called the nth symbolic power of√p.  f) [ a]R[p] = [a]R[p] for any ideal a of R. g) Any primary ideal a of R contained in p satisfies condition (15) of Proposition 3.5.3. h) √ Any ideal a of R satisfying condition (15) of Proposition 3.5.3 satisfies a ⊆ p, and equality holds if and only if a is p-primary. i) An ideal a of R is q-primary if and only if aRp is qRp -primary, for any ideals a and q of R with q ⊆ p prime. j) There is an order-preserving bijection from the set of all primary ideals of R contained in p to the set of all primary ideals of Rp . k) An ideal a of R is q-primary if and only if [a]R[p] is [q]R[p] -primary, for any ideals a and q of R with q ⊆ p prime. l) There is an order-preserving bijection from the set of all primary ideals of R contained in p to the set of all primary ideals of R[p] contained in [p]R[p] . 9. Let S be an overring of a ring R. Show that the following conditions are equivalent. 1) S is flat  over R. 2) S = {R[q∩R] : q ∈ Spec(S)}. 3) S ⊆ {R[q∩R] : q ∈ Spec(S)}. Conclude that every flat overring of R is a subintersection of R.  10. Let R be a ring. Show that any subintersection of R has the form p∈ R[p] for some subset  of wAss(T (R)/R). 11. Let U be a multiplicative subset of a ring R. Let R[U ] = {x ∈ T (R) : ux ∈ R for some u ∈ U }, and for any ideal I of R let [I ]R[U ] = {x ∈ T (R) : ux ∈ I for some u ∈ U }. Generalize Proposition 3.5.3 to the ring R[U ] to the extent possible. 12. Let S be an R-torsion-free extension of a ring R.

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 a) Show that if S be a semistar t-linked extension of R, then S = {S[R−p] : p ∈ t-Max(R)}, where S[R−p] = {x ∈ T (S) : ux ∈ S for some u ∈ R − p} for any prime ideal p of R. b) Show that the converse of part (a) holds if R is an integral domain. c) (∗∗) Does the converse of part (a) hold for an arbitrary ring R? 13. Let D be a Krull domain. Prove the following.  a) Let  ⊆ X 1 (D), and let T = {Dp : p ∈ }. The kernel of the map Clt (D) −→ Clt (T ) is generated by the classes of the prime ideals in X 1 (D) − . b) Let U be a multiplicative subset of D. The kernel of the map Clt (D) −→ Clt (U −1 D) is generated by the classes of the prime ideals in X 1 (D) that meet U . 14. Let k be a field of characteristic not equal to 2, let D = k[X, Y, Z ](X Y − Z 2 ), and let x, y, z denote the image of X, Y, Z in D, respectively. By [69, Proposition 11.2], D is a Krull domain. Prove the following. a) D[x −1 ] is a UFD. b) Clt (D) is generated by the classes of the height one primes of D containing x. (Hint: Use Exercise 13(b).) c) The only height one prime of D containing x is (x, z). d) ((x, z)2 )t is principal. e) Clt (D) is isomorphic to Z/2Z. f) For every fractional ideal I of D, either I t is principal or there exists a nonzero c ∈ T (D) such that I t = c(x, z). 15. Let k be a field of characteristic not equal to 2, let D = k[X, Y, Z , W ](X Y − Z W ), and let x, y, z, w denote the image of X, Y, Z , W in D, respectively. By [69, Proposition 11.2], D is a Krull domain. Prove the following. a) D[z −1 ] is a UFD. b) Clt (D) is generated by the classes of the height one primes of D containing z. (Hint: Use Exercise 13(b).) c) The only height one primes of D containing z are (x, z) and (y, z). d) ((x, z)(y, z))t is principal. e) Clt (D) is cyclic. (In fact, Clt (D) ∼ = Z.) f) For every fractional ideal I of D there exists a nonzero c ∈ T (D) and an integer n such that I t = c((x, z)n )t . Section 3.6 1. 2. 3. 4. 5. 6. 7.

Fill in the missing details in the proof of Proposition 3.6.3. Fill in the missing details in the proof of Proposition 3.6.5. Prove Corollary 3.6.8. Prove Corollary 3.6.13. Prove Proposition 3.6.14. Prove Proposition 3.6.18. Prove the following.

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a) A TD valuation ring is equivalently a Prüfer valuation ring. b) A Mori valuation ring is equivalently a discrete rank at most one valuation ring. c) A Mori divisorial valuation ring is equivalently a Dedekind valuation ring. Let R be a valuation ring. Prove the following. a) For all regular x ∈ T (R) one has x ∈ R or x −1 ∈ R. b) The set of all regularly generated ideals of R is totally ordered. c) Every fractional ideal of R generated by finitely many regular elements of T (R) is principal. Let T be a ring. Prove the following. a) The set of all equivalence classes of trivial valuations on T is in one-to-one correspondence with SpecT . b) If T is a finite ring, then every valuation on T is trivial. Let R be a subring of a ring T , and let  be a T -semistar operation on R. Prove the following. a) Conditions (1) and (2) of Definition 3.6.20 hold if and only if {(x) : x ∈ T } − {(0) } is a totally ordered group, ordered by ⊇, under the operation of -multiplication. b) If conditions (1) and (2) of Definition 3.6.20 hold, then t is the largest valuative T -semistar operation that is less than or equal to , and condition (3) holds if and only if  is of finite type. Let R be a ring and (R, p) a valuation pair of T (R), and let q be a prime ideal of R contained in p. Show that (R[q] , [q]R[q] ) is also a valuation pair of T (R) = T (R[q] ). [130]. Let p be a prime ideal of a ring R. Show that (R[p] , [p]R[p] ) is a valuation pair of T (R) if and only if (Rp , pRp ) is a valuation pair of T (R)p . (Hint: Define a map v  on T (R)p by v  (x/u) = v(x) for all x ∈ T (R) and all u ∈ R − p, where v is the valuation of (R[p] , pR[p] ).) Let R be a ring. For any prime ideal p of R, let R(p) = {x ∈ T (R) : ux ∈ R for some regular u ∈ R − p}. Prove the following. a) Let U = (R − p)reg . Then R(p) = U −1 R is a flat overring of R contained in R[p] . b) pR(p) is a prime ideal of R(p) lying over p in R and under [p]R[p] in R[p] . c) R[p] = (R(p) )[pR(p) ] . d) R = {R(p) : p ∈ Max(R)reg }. e) [130]. The following conditions are equivalent. 1) (R[p] , [p]R[p] ) is a valuation pair of T (R). 2) There is a prime ideal q of R(p) containing every regular non-unit of R(p) such that (R(p) )[q] is a valuation ring. 3) For all ideals a and b of R, at least one of which is regular, either aRp ⊆ bRp or bRp ⊆ aRp .

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14. A subset X of a poset S is convex if for all s ∈ S with x  s  x  for some x, x  ∈ X one has s ∈ X . Let v : T −→  ∪ {∞} be a valuation on a ring T , and let R = Tv . In this exercise we establish a bijection between the set of all prime ideals of R that are closed under v and the convex subgroups of . Show the following. a) Let ϕ :  −→   be an (order-preserving) homomorphism of totally ordered abelian groups, and set ϕ(∞) = ∞. i) v  = ϕ ◦ v : T −→   ∪ {∞} is a valuation on T . ii) ker ϕ is a convex subgroup of . iii) pv is a prime ideal of R that is closed under v. b) Let  be a convex subgroup of . Then   = / has the structure of a totally ordered abelian group such that the canonical surjective group homomorphism ϕ :  −→   is order-preserving,  = ker ϕ, and v = ϕ ◦ v : T −→   ∪ {∞} is a valuation on T . c) There is a one-to-one order-reversing correspondence between the convex subgroups  of  and the prime ideals q of R that are closed under v. The correspondence is given by  −→ pv = {x ∈ R : v(x) > α for all α ∈ } and q −→ {α ∈  : max{α, −α} < v(x) for all x ∈ q}. d) A prime ideal q of R is closed under v if and only if v −1 (∞) ⊆ q ⊆ pv . e) The set of all ideals of R that are closed under v is totally ordered. f) The set of all ideals of R that are closed under v (resp., all prime ideals of R that are closed under v) is closed under arbitrary unions and intersections. 15. A totally ordered abelian group  is said to be Archimedean if for all γ, δ ∈  with γ > 0 there exists a positive integer n such that nγ  δ. A famous theorem of Hölder states that a totally ordered abelian group is Archimedean if and only if it has no nontrivial proper convex subgroups, if and only if it is order isomorphic to a subgroup of R. Let R be a valuation ring with valuation v : K −→  ∪ {∞}, and let p = pv . Using the previous exercise, prove that the following conditions are equivalent. 1)  is Archimedean. 2) R is completely integrally closed. 3) For any x ∈ K , if for some c ∈ R − (R : K K ) one has cx n ∈ R for all positive integers n, then x ∈ R. 4) t-Spec(R) = {v −1 (∞), p}. 5) |t-Spec(R)|  2. 6) R is of regular t-dimension at most one. 16. Let R be a valuation ring with nontrivial valuation v. Prove the following conditions equivalent. 1) R is Prüfer. 2) R is TD.

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24. 25.

3 Semistar Operations on Commutative Rings: Local Methods

3) Spec(R)reg = t-Spec(R)reg . 4) v −1 (∞) ⊆ q ⊆ pv for all q ∈ Spec(R)reg . 5) pv is the unique regular maximal ideal of R. 6) The set of all regular ideals of R is totally ordered. Let R be a ring and (R, p) a valuation pair of T (R) = R. Prove the following conditions equivalent. (Hint: Use Exercise 3.3.17(e).) 1) R is an H ring. 2) p is t-invertible. 3) p is divisorial. 4) p−1 = R. 5) p is t-finite. 6) p = (a)t for some a ∈ R. Also show that, if R is also a domain, then R is an H ring if and only if R is divisorial. Let R be a valuation ring with valuation v : T (R) −→  ∪ {∞}, and let p = pv . a) Let I ∈ K(R), and suppose that I v = I t . Show that there exists an x ∈ I v such that I v = (x)t and I t = (xp)t , or, equivalently, there exists a γ ∈  such − t that I v = aγ and I t = a− γ , where aγ = {x ∈ T (R) : v(x) > γ} = (aγ p) . b) Using part (a) and the previous exercise, prove the following conditions equivalent. a) R is an H ring. b) p is divisorial. c) p = (a)t for some a ∈ R. d) R is a TV ring. e) v = t on K(R). c) Show that a valuation ring is divisorial if and only if it is TV and Prüfer, if and only if it is H and Prüfer. Prove Proposition 3.6.21. Prove Theorem 3.6.22 and Corollaries 3.6.23, 3.6.24, 3.6.25, 3.6.26. Show that if R is a valuation ring, then Clt (R) and Clt (R) are trivial, where Clt (R) and Clt (R) are defined as in Exercise 2.5.7. Prove Propositions 3.6.27, 3.6.28, and 3.6.29. Let R be a valuation ring with valuation v : T (R) −→  ∪ {∞}. Show that R is t-Marot if and only if R has trivial t-class group, if and only if v maps the regular elements of R onto the nonnegative elements of . Deduce that R is Marot if and only if R is Prüfer and has trivial class group, if and only if R is Prüfer and v maps the regular elements of R onto the nonnegative elements of . Prove the result mentioned in Remark 3.6.6. For this exercise, you may use the results of Exercise 2.7.14. Let R be a ring and M an R-module, let U = R reg ∩ M reg , and let T = U −1 R. Prove the following. a) Let p be a prime ideal of R. Then (R(+)M, p(+)M) is a valuation pair of T (R(+)M) if and only if U −1 M = M and (R, p) is a valuation pair of T . b) R(+)M is a valuation ring if and only if U −1 M = M and (R, p) is a valuation pair of T for some prime ideal p of R.

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c) Suppose that U −1 M = M, and let I be a regular ideal of R. Then (I (+)M)v = I vT (+)M and (I (+)M)t = I tT (+)M, where vT and tT are the largest unital and largest finite type unital T -semistar operations on R, respectively. d) If  is a unital T -semistar operation on R, then we say that I ∈ Mod R (T ) is -invertible if (I J ) = R for some J ∈ Mod R (T ). When this holds, one has J  = I −1T and therefore (I I −1T ) = R. e) R(+)M is integrally closed if and only if U −1 M = M and R is integrally closed in T . f) R(+)M is Prüfer if and only if U −1 M = M and every finitely generated ideal of R that meets U is invertible. g) R(+)M is Dedekind if and only if U −1 M = M and every ideal of R that meets U is invertible. h) R(+)M is r-Bézout if and only if U −1 M = M and every finitely generated ideal of R that meets U is principal. i) R(+)M is an r-PIR if and only if U −1 M = M and every ideal of R that meets U is principal. j) R(+)M is a PVMR if and only if U −1 M = M and every finitely generated ideal of R that meets U is tT -invertible. k) R(+)M is Krull if and only if U −1 M = M and every ideal of R that meets U is tT -invertible. l) R(+)M is r-GCD if and only if U −1 M = M and the tT -closure of every finitely generated ideal of R that meets U is principal. m) R(+)M is an r-UFR if and only if U −1 M = M and the tT -closure of every ideal of R that meets U is principal. n) R(+)M is v-Prüfer if and only if U −1 M = M and every finitely generated ideal of R that meets U is vT -invertible. o) R(+)M is completely integrally closed if and only if U −1 M = M and every ideal of R that meets U is vT -invertible. p) Suppose that M is R-torsion-free and (R reg )−1 M = M . Then R(+)M has the following properties if and only if R does: integrally closed, completely integrally closed, v-Prüfer, PVMR, Krull, r-GCD, r-UFR, Prüfer, Dedekind, r-Bézout, and r-PIR. q) R(+)R ∼ = R[X ]/(X 2 ) = R[ε], where ε is the image of X in R[X ]/(X 2 ), and R(+)K ∼ = R + εK [ε]. r) R + εK [ε] has the following properties if and only if R does: integrally closed, completely integrally closed, v-Prüfer, PVMR, Krull, r-GCD, r-UFR, Prüfer, Dedekind, r-Bézout, and r-PIR. 26. Let R be a subring of a ring T . Let us say that a T -semistar operation  on R is  or (y) ⊆ (x) paravaluative if the following conditions hold: (1) (x) ⊆ (y)  for all x, y ∈ T ; (2)  is of finite type (or equivalently I = x∈I (x) for all I ∈ Mod R (T )); and (3) if (x) = (0) , then (x y) = (x z) implies (y) = (z) , for all x, y, z ∈ T . a) Let v : T −→  ∪ {∞} be a paravaluation on T with R = Tv . Then the map (−)v : I −→ Iv = {y ∈ T : v(y)  v(x) for some x ∈ I } is a unital paravaluative T -semistar operation on R.

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b) Let  be a paravaluative T -semistar operation on R. Let ∞ = (0) and  = {(x) : x ∈ T } − {∞}. Then  is a cancellative totally ordered commutative monoid under -multiplication and ordered by ⊇, and therefore  embeds in some totally ordered abelian group . Moreover, the map v : T −→  ∪ {∞} acting by x −→ (x) is a paravaluation on T such that R  = Tv , (−)v = , and  = v (T ) − {∞}. c) R is a paravaluation ring of T if and only if there exists a unital paravaluative T -semistar operation on R. d) Let K = T (R), and let v : K −→  ∪ {∞} be a paravaluation on T (R) with R = Tv . The following conditions are equivalent. 1) v is a valuation on K . 2) (x) is (−)v -invertible for all x ∈ K such that (x)v = (0)v . 3) (R : R K ) ⊆ (0)v and R is (−)v -Prüfer. Section 3.7 1. A ring R satisfies PIT if every prime ideal of R that is minimal over a principal ideal has height at most one. Krull’s principal ideal theorem states that every Noetherian ring satisfies PIT. Show that for any ring R that satisfies PIT one has 1 (R) = X 1 (R)reg . X reg 2. Let R be a ring. Prove the following. a) Every regular prime ideal of R contains a prime ideal in wAss(T (R)/R). b) Every regular prime ideal of R contains a regular t-prime ideal of R. c) The prime ideals of regular height one and the t-prime ideals of t-regular height one coincide. d) A prime ideal of R need not contain a t-prime ideal of R. 3. Prove statements (1)–(3) of Lemma 3.7.15 and statements (1)–(4), (6), and (7) of Proposition 3.7.17. 4. Show that an r-UFR is equivalently a Krull ring in which every regular t-maximal ideal is principal. 5. Show that a finite direct product R1 × R2 × · · · × Rn of rings R1 , R2 , . . . , Rn is a Krull ring if and only if Ri is a Krull ring for all i. 6. Let R be a ring with total quotient ring K . Let us say that R is nearly Krull if (R[p] , [p]R[p] ) is a discrete rank at most one valuation pair of K for every p ∈ t-Max(R). (An integral domain is nearly Krull if and only if it is an almost t-Dedekind domain in  the sense of Definition 3.4.3.) Let us also say that R is quasi-Krull if R = {Rλ : λ ∈ } for some family {(Rλ , pλ )}λ∈ of discrete rank at most one valuation pairs of K . a) Show that following conditions are equivalent. 1) R is a Krull ring. 2) R is a nearly Krull Mori ring. 3) R is a nearly Krull TV ring. 4) R is nearly Krull and of finite t-character. 5) R is nearly Krull and an H ring.

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b) Consider the following conditions. 1) R is Krull. 2) R is nearly Krull. 3) R is quasi-Krull.  4) R = {Rvλ : λ ∈ } for some family {vλ }λ∈ of discrete rank at most one valuations vλ : T (R) −→ Z ∪ {∞}.  5) R = {Rλ : λ ∈ } for some family {(Rλ , pλ )}λ∈ of discrete rank one valuation  pairs of K . 6) R = {Rλ : λ ∈ } for some family {Rλ }λ∈ of discrete rank one valuationoverrings of R. 7) R = {Rλ : λ ∈ } for some family {Rλ }λ∈ of Krull overrings of R. 8) R is completely integrally closed. Prove that (1) ⇒ (2) ⇒ (3) ⇔ (4) ⇔ (5) ⇔ (6) ⇔ (7) ⇒ (8). c) (∗∗) Is a quasi-Krull ring (or quasi-Krull domain) necessarily nearly Krull? Let p be a regular v-invertible v-prime ideal of a ring R. / p then y ∈ (pn )v , for a) Show by induction on n that if x y ∈ (pn )v and x ∈ all x, y ∈ R and all positive integers n. (In other words, the ideal (pn )v is p-primary.) Thus, if I J ⊆ (pn )v and I Ę p, then I J ⊆ (pn )v , for all ideals I and J of R. b) For all a ∈ R, let v(a) = sup{n ∈ Z0 : a ∈ (pn )v }. Show that v(x y) = v(x) + v(y) and v(x + y) v(y)} for all x, y ∈ R.   min{v(x), n v (p ) is not regular. Show that the map v : c) Suppose that the ideal ∞ n=1 R −→ Z extends uniquely to a discrete rank one valuation on T (R), and one has R[p] ⊆ T (R)v . d) Complete the proof of Proposition 3.7.25 by verifying that the map v : R −→ Z ∪ {∞} defined in the proof of the proposition extends uniquely to a paravaluation v : T (R) −→ Z ∪ {∞} on T (R). e) Show that if the t-operation in Proposition 3.7.25 is replaced everywhere with the v-operation then the resulting proposition is still true. (Hint: If I ∈ K(R) is t-invertible and t-closed, then I is v-invertible and  v-closed.) Let R be a ring and p a regular prime element of R such that n ( p n ) is not regular. Show that (R[( p)] , [( p)]R[( p)] ) is a discrete rank one valuation pair of T (R). Show that if  {Rλ : λ ∈ } is a locally finite collection of Krull overrings of a ring R, then {Rλ : λ ∈ } is Krull. Let R be a ring with total quotient ring K . Show that if there exists a family {(Rλ , pλ )}λ∈ of discrete rank one valuation pairs of K such that  for every regular x ∈ R one has x ∈ pλ for only finitely many λ ∈ , then {Rλ : λ ∈ M} is a Krull ring for every subset M of . Prove the following. a) If R is a t-Marot ring, then any t-compatible overring of R is also t-Marot. b) If R is a t-Marot ring, then R[p] is t-Marot for any prime ideal p of R. c) If R is a t-Marot PVMR, then R[p] is an r-GCD valuation ring for all p ∈ t-Max(R).

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d) If R is a t-Marot Krull ring, then R[p] is an r-UFR valuation ring for all 1 (R). p ∈ X reg 1 e) (∗∗) If R is a Krull ring and R[p] is an r-UFR for all p ∈ X reg (R), then is R necessarily t-Marot? 12. Let R be Krull ring with total quotient ring K . Prove the following. 1 (R) let vp : K −→ Z ∪ {∞} denote the discrete rank one a) For each p ∈ X reg valuation on K with R[p] = K vp and [p]R[p] = pvp . Let p1 , p2 , . . . , pn be 1 (R), and let k1 , k2 , . . . , kn be integers. Then any n distinct primes in X reg there exists an element x of K such that vpi (x) = ki for all i = 1, 2, . . . , n 1 (R) − {p1 , p2 , . . . , pn }. This is known as the and vp (x)  0 for all p ∈ X reg approximation theorem for Krull rings [162, Theorem 2.1]. b) If I is any regular divisorial fractional ideal of R, then for any regular a ∈ I there exists a b ∈ I such that (a, b)v = I . c) If I is any regular fractional ideal of a Dedekind ring R, then for any regular a ∈ I there exists a b ∈ I such that (a, b) = I . d) The quotient of any Dedekind ring by a regular ideal is a principal ideal ring. 13. a) In [115], B. G. Kang showed that a ring R is a Krull ring if and only if for every regular a ∈ R there exist regular primes p1 , p2 , . . . , pn of R such that (a) = (p1 p2 · · · pn )v . Read [115] and try to come up with a shorter and more streamlined version of the proof. b) Let R be a ring. We say that R is a t-UFR if R is a Krull ring and Clt (R) is trivial, where Clt (R) is defined as in Exercise 2.5.7. Let us say that I ∈ K(R) is t-principal if I = (x)t for some x ∈ T (R). Show that the following conditions are equivalent. 1) R is a t-UFR. 2) R is a Krull ring and every regular t-maximal ideal of R is t-principal. 3) The t-closure of every regular fractional ideal of R is t-principal and t-invertible. 4) For all regular a ∈ R there exist regular t-principal prime ideals p1 , p2 , . . . , pn of R (unique up to reordering the factors) such that (a) = (p1 p2 · · · pn )t . c) Show that an r-UFR is equivalently a t-UFR in which every regular t-principal ideal is principal. Conclude that a UFD is equivalently a domain that is a t-UFR. d) Suppose that R is not a total quotient ring. Show that the following conditions are equivalent. 1) R is a Krull valuation ring. 2) R is a discrete rank one valuation ring. 3) R is a valuation ring and a t-UFR. 4) R is a t-UFR with a unique (regular) t-maximal ideal. 5) There exists a regular t-principal prime ideal p of R such that for all regular a ∈ R one has (a) = (pn )t for some nonnegative integer n. 6) The group Invt (R) is cyclic and generated by a regular t-principal prime ideal of R.

Exercises

e) f) g) h) 14. Let 1) 2) 3) 4) 5)

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Find an example of a t-UFR that is not an r-UFR. Find an example of a factorial ring that is not a t-UFR. (∗∗) Must a t-UFR be factorial? (∗∗) Must a factorial t-UFR be an r-UFR? R be a ring. Prove the following conditions equivalent. R is a Krull ring. Every regular t-prime of R is t-invertible. Every prime ideal in wAss(T (R)/R) is t-invertible. Every prime ideal of R of regular height one is t-invertible. Every regular t-maximal ideal of R is t-invertible and of regular height one. (Hint: Suppose that the set of all regular t-ideals of R that are not t-invertible is nonempty. Use Zorn’s lemma to show that the set has a maximal element, and then show that any such maximal element is prime. Also use Exercise 3.3.17.) Let R be a ring. Using the previous exercise, prove the following. a) R is a Dedekind ring if and only if every regular prime of R is invertible. b) R is an r-UFR if and only if every regular t-prime of R is principal. c) R is an r-PIR if and only if every regular prime of R is principal. d) k[X, Y ]/(X Y ) is an r-PIR for any field k. e) Z[X ]/( p X ) is an r-PIR for every prime number p. f) Z[X ]/(n X ) is an r-PIR if and only if Z[X ]/(n X ) is integrally closed, if and only if n is squarefree, for all positive integers n. g) (∗∗) For which positive integers n is the integral closure of Z[X ]/(n X ) an r-PIR? Let R be a ring. In this exercise we show that R[X ] is a Krull ring if and only if R is a finite direct product of Krull domains. Prove the following. a) If R is a Krull domain, then R[X ] is a Krull domain. (See Exercise 3.4.10.) b) A finite direct product of Krull rings is a Krull ring. c) R is a finite direct product of Krull domains if and only if R is a reduced Krull ring with finitely many minimal primes. d) If R[X ] is a Krull ring, then R is reduced and has finitely many minimal primes. (Hint: Write (X ) = P1(n 1 ) ∩ P2(n 2 ) ∩ · · · ∩ Pk(n k ) , where Pi are the prime ideals of R[X ] that are minimal over X .) e) R[X ] is a Krull ring if and only if R is a reduced Krull ring with finitely many minimal primes, if and only if R is a finite direct product of Krull domains. Show that if D is a Krull domain, then D[X] for any nonempty set X of indeterminates is a Krull domain. Conclude from the previous exercise that R[X] for any ring R and any nonempty set X of indeterminates is a Krull ring if and only if R is a finite direct product of Krull domains. Read [32] and digest the proof of [32, Theorem 13], which is stated without proof as Theorem 2.6.28. Try to come up with a shorter and more streamlined version of the proof. (∗∗) Must any t-linked overring of a Krull ring be a Krull ring?

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Section 3.8 1. Show that any overring of a Prüfer ring is Prüfer and any overring of a Dedekind ring is Dedekind. 2. Prove the following. a) A ring R is a t-PIR if and only if it is a w-PIR. b) An integral domain D is a UFD if and only if every w-closed ideal of D is principal. 3. Show that a quasi-Marot ring R is a PVMR if and only if R is integrally closed and (I ∩ J )t = I t ∩ J t for all regular ideals I and J of R. 4. Show that an integral domain D is Dedekind if and only if D is integrally closed and every fractional ideal of D is a finite intersection of principal fractional ideals of D. 5. Show that a Prüfer ring is equivalently a PVMR in which every regular maximal ideal is t-closed. 6. Let R be a ring, and let S be an R-torsion-free extension of R. a) Show that the following conditions are equivalent. 1) S is a t-linked extension of R. 2) S is a w-compatible extension of R: one has I w S ⊆ (I S)w , or equivalently (I S)w = (I w S)w , for all I ∈ Freg (R). 3) S is a w-linked extension of R: if I w = R then (I S)w = S, for all I ∈ Freg (R). b) Let R be a PVMR. Show that S is a t-linked extension of R if and only if S is a t-compatible extension of R. 7. Prove that the following implications hold for any ring R. PVMR ⇒ P-ring ⇒ essential ⇒ integrally closed. 8. Use Proposition 3.8.3, Theorem 3.3.23, and Proposition 3.5.17 to provide an alternative proof that (1) ⇒ (3) in Theorem 3.7.9. 9. Let us say that a ring R is nearly Krull if (R[p] , [p]R[p] ) is a discrete rank at most one valuation pair of T (R) for every p ∈ t-Max(R). Show that a ring R is nearly Krull if and only if R is a PVMR and R[p] is Krull for every t-maximal ideal p of R, if and only if R is a PVMR and R[p] is Mori for every t-maximal ideal p of R. 10. An ideal I of a ring R is said to be cancellative if I H = I J implies H = J for all ideals H, J of R. For example, any invertible ideal of a ring R is cancellative. Prove that the following conditions on any ring R are equivalent. 1) R is Prüfer. 2) Every finitely generated regular ideal of R is cancellative. 3) R is integrally closed and (I + J )n = I n + J n for all finitely generated ideals I, J of R such that I + J is regular and all positive integers n. 4) R is integrally closed and (I + J )2 = I 2 + J 2 for all finitely generated ideals I, J of R such that I + J is regular.

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11. 12.

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5) R is integrally closed and (a, b)2 = (a 2 , b2 ) for all a, b ∈ R such that (a, b) is regular. Show that an overring R of a Prüfer domain D is a valuation domain if and only if R = Dp for some prime ideal p of D. Show that if R is a ring and p is a regular prime ideal of R such that I R[p] is v-closed in R[p] for every v-finite v-closed regular ideal I of R, then [p]R[p] is t-prime. Use this fact and Lemma 3.5.9 to give an alternative proof of Proposition 3.5.10. Verify the claims made in Remark 3.8.12. Let R be an integrally closed ring, and let a, b ∈ R with a regular. Prove by induction that, if a n−1 b ∈ (a n , bn )w for some integer n > 1, then (a, b) is w-invertible. (Hint: Use ideas from the proof of Theorem 3.8.4.) [25]. A ring R is said to be locally Prüfer if Rp is Prüfer for every prime ideal p of R. Prove the following. a) Every locally Prüfer ring is Prüfer. b) Let k be a field. The ring R = k[W, X, Y, Z ]/(W 2 , W X, W Y, W Z ) is Prüfer (in fact, a total quotient ring) but not locally Prüfer. (Hint: Let p = (w, x, y), where w, x, y are the images of W, X, Y , respectively, in R. Then p is a prime ideal of R such that Rp ∼ = k(Z )[X, Y ](X,Y ) is a two dimensional local Noetherian domain.) A ring R is said to be arithmetical if H ∩ (I + J ) = (H ∩ I ) + (H ∩ J ) for every ideal H, I, J of R. Prove the following. a) A ring R is Prüfer if and only if H ∩ (I + J ) = (H ∩ I ) + (H ∩ J ) for every ideal H, I, J of R, at least one of which is regular. b) Every arithmetical ring is Prüfer. c) A ring R is arithmetical if and only if H + (I ∩ J ) = (H + I ) ∩ (H + J ) for every ideal H, I, J of R. d) Let k be a field. The ring k[X ]/(X )2 is arithmetical but the ring k[X, Y ]/ (X, Y )2 is not. e) Any localization of an arithmetical ring at a multiplicative subset is arithmetical. f) An integral domain is arithmetical if and only if it is Prüfer. g) A ring R is arithmetical if and only if Rp is arithmetical for every prime ideal p of R. h) A ring R is said to be of weak global dimension at most one if every ideal of R is flat. Using Exercise 3.2.11, show that every ring of weak global dimension at most one is arithmetical. i) A ring is of weak global dimension at most one if and only if it is a reduced arithmetical ring. Let R be a ring. a) Show that, if R[X ] is Prüfer, then R is von Neumann regular. b) Show that R[X ] is a Prüfer domain if and only if R is a field. c) Show that R[X, Y ] is a Prüfer ring if and only if R is trivial. d) (∗∗) Show that R[X ] is Prüfer if and only if R is von Neumann regular. (This was proved by P.J. McCarthy in 1973.)

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18. A ring R is said to be arithmetical if H ∩ (I + J ) = (H ∩ I ) + (H ∩ J ) for every ideal H, I, J of R. A ring R is said to be a chained ring if the set of all ideals of R is totally ordered by inclusion. Prove the following. a) A ring R is a chained ring if and only if R is a local arithmetical ring. b) Let R be a ring. The following conditions are equivalent. 1) R is arithmetical. 2) Rp is a chained ring for every maximal (or prime) ideal p of R. 3) Every finitely generated ideal of R is locally principal. 4) For all ideals I, J of R with I ⊇ J and I finitely generated, there exists an ideal H of R such that J = H I . c) If R is chained, then every homomorphic image of R, and every localization of R at a multiplicative subset, is chained. d) Every homomorphic image of a valuation domain is chained. e) If R is chained, then R has few zerodivisors and R is a Marot Prüfer valuation ring. 19. Prove all of the claims made in Remark 3.8.13. 20. a) Use statements (10)–(12) of Corollary 3.6.23 and statement (5) of Proposition 3.6.28 to show that a valuation ring is Mori if and only if it is discrete rank at most one. b) Use part (a), Theorem 3.3.23, and Propositions 3.5.17 and 3.8.3 to provide an alternative proof of the fact that, if R is a Krull ring, then R is of finite t-character and (R[p] , [p]R[p] ) is a discrete rank at most one valuation pair of T (R) for every t-prime ideal of R. 21. Let X be any infinite set of indeterminates. Show that R[X] is a PVMR if and only if R[X 1 , X 2 , . . . , X n ] is a PVMR for every positive integer n. 22. Let R be a ring. a) (∗∗) In [135, Theorem 7.8], it is shown that R[X ] is a PVMR if and only if R is a PVMR and T (R) is von Neumann regular. Try to come up with a shorter and more streamlined version of the proof. b) Use any of the results in [135] to show that R[X 1 , X 2 , . . . , X n ] is a PVMR if and only if R[X ] is a PVMR, for any positive integer n. c) Using parts (a) and (b) and the previous exercise, deduce that R[X] for any nonempty set X of indeterminates is a PVMR if and only if R is a PVMR and T (R) is von Neumann regular. 23. (∗∗) In [41, Theorem 2.10] it is shown that a domain D is a PVMD if and only if every t-linked overring of D is integrally closed. Does this generalize to rings with zerodivisors? In particular, is it true that a ring R is a PVMR if and only if every t-linked overring of R is integrally closed? If so, does this generalize to a characterization of the t -Prüfer rings for any semistar operation ? 24. (∗∗) In [150], it is shown that a domain D is a P-domain if and only if every localization of D at a prime ideal of D is essential. If possible, generalize this to a characterization of the P-rings.

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Section 3.9 1. Let R be a ring and T a (commutative) R-algebra, let f, g ∈ T [X ], and let a ∈ R. Verify the following. a) cT (a f ) = acT ( f ). b) cT ( f ) ⊇ cT ( f (a X )), and equality holds if a is a unit of R. c) cT ( f ) + cT (g) ⊇ cT ( f + g). d) cT ( f )cT (g) ⊇ cT ( f g). √ 2. Let R be a ring. Show that c( f g) ⊆ c( f )c(g) ⊆ c( f g) for all f, g ∈ R[X ]. Conclude that if c( f ) = R and c(g) = R, then c( f g) = R. 3. a) Let R be a ring. Let a, b, c, d ∈ R, and let f = a + bX and g = c + d X . Show that c R ( f )c R (g) = c R ( f g) if and only if ad ∈ (ac, ad + bc, bd). b) Suppose that D is a UFD that is not a PID. Show that there exist nonassociate primes p and q of D with ( p, q) = D, and then f = p + q X and g = p − q X are linear polynomials in D[X ] such that c D ( f )c D (g) = c D ( f g). 4. Let R be a ring. For all f ∈ R[X ], let C R ( f ) denote the gcd of the coefficients of f , if the gcd exists. Prove the following. a) R is a GCD ring if and only if C R ( f ) exists for all f ∈ R[X ] such that c R ( f ) is regular. b) If R is an r-GCD ring, then C R ( f g)R = C R ( f )C R (g)R for all f, g ∈ R[X ] with c R ( f ) and c R (g) regular. c) (∗∗) If R is a GCD ring, then does it follow that C R ( f g)R = C R ( f )C R (g)R for all f, g ∈ R[X ] with c R ( f ) and c R (g) regular? 5. Use Exercise 4(b) to prove the following. a) Let D be a GCD domain with quotient field K . A nonconstant polynomial f ∈ D[X ] is irreducible in D[X ] if and only f is irreducible in K [X ] and the gcd of the coefficients of f is 1. b) Let D be an integral domain. Then D[X ] is a UFD if and only if D is a UFD. (Hint: Use the fact that K [X ] is a UFD if K is a field.) 6. Let R be a ring. a) Show that ann R c( f ) = (0 : R f R[X ]) for all f ∈ R[X ]. b) Use the Dedekind–Mertens lemma to show that a polynomial f ∈ R[X ] is regular if and only if ann R c( f ) = (0), if and only if there is no nonzero c ∈ R such that c f = 0. This was first proved by McCoy in 1942 and is known as McCoy’s theorem [146]. c) An ideal I of R is said to be dense if ann R I = (0). An ideal I of R is said to be semiregular if I contains a finitely generated dense ideal of R. Show that if I is regular, then I is semiregular, and if I is semiregular, then I is dense. d) Show that, by McCoy’s theorem, a polynomial f ∈ R[X ] is regular if and only if its content c R ( f ) is semiregular. e) A ring R is said to be McCoy, or satisfy Property A, if every semiregular ideal of R is regular. Show that every overring of a McCoy ring is McCoy. f) Use McCoy’s theorem to show that the following conditions are equivalent.

286

7.

8.

9.

10.

11.

12.

3 Semistar Operations on Commutative Rings: Local Methods

1) R is McCoy. 2) T (R) is McCoy. 3) A polynomial f ∈ R[X ] is regular if and only if its content c R ( f ) is regular. g) Show that any dense ideal of a Noetherian ring is regular. h) Show that every overring of a Noetherian ring is McCoy. Let k be a field and D = k[[X, Y ]]. Let M = p∈X 1 (D) D/p. Use Exercise 2.7.14 to show the following. a) D(+)M is a local total quotient ring with maximal ideal (X, Y )(+)M, which is equal to the set of all zerodivisors of D(+)M. b) The maximal ideal of D(+)M is semiregular but not regular, and therefore the ring D(+)M is not McCoy. (See the previous exercise for the definition of a semiregular ideal and of a McCoy ring.) c) The ring D(+)M has few zerodivisors, and therefore, by Exercise 2.7.19, is Marot. Let R be a ring. Prove the following. a) A polynomial f ∈ R[X ] is Gaussian if and only if the image of f in Rm [X ] is Gaussian for every maximal ideal m of R. b) More generally, μ R ( f ) = max{μ Rm ( f ) : m ∈ Max(R)} for all f ∈ R[X ], where μ R ( f ) denotes the Dedekind–Mertens number of f . c) A ring R is Gaussian if and only if Rm is Gaussian for every maximal ideal m of R, if and only if Rp is Gaussian for every prime ideal p of R. [25]. Show that, for any field k, the ring k[X, Y ]/(X, Y )2 is Gaussian but the rings k[X, Y ]/(X 2 , Y 2 ) and k[X, Y ]/(X, Y )3 are not. Also show that all three of these rings are local total quotient rings and therefore are Prüfer. [86]. Let R be a local Gaussian ring, and let a, b ∈ R. a) Show that (a, b)2 = (a 2 , ab) or (a, b)2 = (b2 , ab). (Hint: Consider the polynomials a X + b and bX + a. Show that one can write a 2 = rab + s(a 2 + b2 ) with r, s ∈ R. Consider the two cases: s is a unit, or 1 − s is a unit.) b) Without loss of generality, assume that (a, b)2 = (a 2 , ab), and write b2 = pa 2 + qab with p, q ∈ R. Show that (a, b)2 = (a 2 ). (Hint: Consider the polynomials a X + b and a X + (qa − b).) c) Conclude that either (a, b)2 = (a 2 ) or (a, b)2 = (b2 ). d) Deduce that every local Gaussian ring is Prüfer. e) Deduce that every Gaussian ring is locally Prüfer. f) Show that a ring R is a reduced Gaussian ring if and only if Rp is a valuation domain for every prime ideal p of R. Thus, by Exercise 3.2.11, a ring of weak global dimension at most one is equivalently a reduced Gaussian ring. A ring R is said to be arithmetical if H ∩ (I + J ) = (H ∩ I ) + (H ∩ J ) for every ideal H, I, J of R. A result of Tsang states that, for any ring R, any polynomial f ∈ R[X ] such that c( f ) is locally principal is Gaussian [180]. Using this result and Exercise 3.8.16, prove that every arithmetical ring is Gaussian. In [99], it is shown that, for any ring R and any f ∈ R[X ], if c R ( f ) can be generated by k elements of R, then μ R ( f )  k.

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a) Explain how this result from [99] implies the Dedekind–Mertens lemma, and provide an example showing that it is a stronger result. b) Assuming the result of [99], show that, if c R ( f ) is locally principal, then f is Gaussian (even if c R ( f ) is not regular). c) More generally, show that, if c Rm ( f ) can be generated by k elements of Rm for every maximal ideal m of R, then μ R ( f )  k. d) Distill a proof of the result from [99] mentioned above and present it as simply as possible. Section 3.10 1. a) b) 2. Let a)

3. 4. 5.

6. 7. 8. 9. 10. 11.

12.

Show that any overring of a Marot valuation ring is a Marot valuation ring. Show that any overring of a Prüfer valuation ring is a Prüfer valuation ring. R be a Marot ring. Show that the following conditions are equivalent. 1) R is Krull. 2) R is of finite t-character and R[p] for all p ∈ t-Max(R) is a Dedekind valuation ring. 3) R[p] for every p ∈ t-Max(R) is a Dedekind valuation ring, and every regular element of R lies in [p]R[p] for only finitely many p ∈ t-Max(R). 4) There exists  a family {Rλ }λ∈ of Dedekind valuation overrings of R such that R = {Rλ : λ ∈ } and for every regular x ∈ R one has x ∈ pλ for only finitely many λ ∈ , where pλ is the unique regular maximal ideal of Rλ for all λ. b) If R is not assumed Marot, then how do the four conditions relate to one another? Prove Proposition 3.10.1. Prove Corollaries 3.10.14 and 3.10.15. Let R be a ring. Show that the set of all regular principal ideals of R is totally ordered if and only if the set of all regularly generated ideals of R is totally ordered. Verify that conditions (1)–(4) in Proposition 3.10.7 are equivalent. Fill in the missing details of the proof of Proposition 3.10.18. Prove Proposition 3.7.32. Prove Proposition 3.10.24. Prove Proposition 3.10.25. a) State and prove as many equivalences as you can for a ring to be an r-GCD valuation ring. b) State and prove as many equivalences as you can for a ring to be an r-Bézout valuation ring. Let R be a ring and p a prime ideal of R. Prove the following. a) If (R, p) is a valuation pair of T (R), then the following conditions are equivalent. 1) R is Prüfer. 2) a = [a]R[p] for every regular ideal a of R.

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3 Semistar Operations on Commutative Rings: Local Methods

3) If ab ∈ a, then a ∈ p or b ∈ a, for all a, b ∈ R and all regular ideals a of R. 4) Every regular ideal of R is of the form [a]R[p] for some ideal a of R. 5) Every finitely generated regular ideal of R is of the form [(a)]R[p] for some a ∈ R. b) If R is a Prüfer ring, then I R[p] = [I ]R[p] for every regular ideal I of R. 13. [108, Section 6]. Let R be a ring with total quotient ring K , and let I be an ideal of R. One defines the I -saturation of R in K , or (Nagata) transform of I in K , to be (R : K I ∞ ) =

14. 15. 16.

17.



{(R : K I n ) : n ∈ Z0 } = {x ∈ K : x I n ⊆ R for some n ∈ Z0 }.

Prove the following. a) (R : K I ∞ ) is an overring of R. b) If I is invertible, then I (R : K I ∞ ) = (R : K I ∞ ). c) For any prime ideal p of R, if I Ę p, then (R : K I ∞ ) ⊆ R[p] , and the converse holds if I is invertible. d) Let R be a Prüfer ring. For all regular primes p and q of R, one has R[p] ⊆ R[q] if and only if p ⊇ q. (Hint: Suppose that p Ğ q. Let b ∈ q − p, let a be a regular element of q, let I = (a, b), and consider the ring (R : K I ∞ ).) Use the examples provided in Sections 3.10 and 2.10 to show that none of the five conditions in Proposition 3.10.18 is equivalent. Verify all the claims made in Example 3.10.10. [107, Section 27, Example 7] provides an example of a ring R and a valuation pair (R, p) of T (R) such that p is regular but not maximal and thus R is not Prüfer. Read and digest the example and then provide your own. Try to make your example as simple as possible. (∗∗) If possible, prove any part of Conjecture 3.10.11 or solve any part of Open Problem 3.10.26 using Exercise 3.6.25.

Section 3.11 1. Prove Lemma 3.11.13. 2. Prove Proposition 3.11.17. 3. Show that any almost Dedekind ring is completely integrally closed and of regular dimension at most one. Also show that, if R is an almost Dedekind ring, then [pn ]R[p] = pn R[p] for every regular prime ideal p of R and every positive integer n. 4. Show that any prime ideal contained in an essential prime is essential, and any prime ideal contained in a strongly essential prime is strongly essential. 5. Show that a ring R is a near PVMR if and only if R[p] is a PVMR for every regular t-maximal ideal p of R. 6. Let R be a ring. Prove the following conditions equivalent. 1) R is a PVMR. 2) R is a v-coherent near PVMR.

Exercises

7.

8.

9.

10.

289

3) R is a near PVMR and sK(T (R)/R) = t-Spec(R)reg . 4) R is a v-coherent K-ring. 5) R is a K-ring and sK(T (R)/R) = t-Spec(R)reg . Let  be a semistar operation on a ring R. Let us say that  is r-spectral if    for some spectral semistar operation  on R. Prove the following. a)  is r-spectral if and only if  is r-stable and R has enough regular -prime ideals. for any set  ⊆ Spec R a set of prime b) If R is Marot, then  is r-spectral  ideals of R such that R = {R[p] : p ∈ }. c) R is essential if and only if R is -Prüfer for some r-spectral semistar operation  on R. Let R be a ring. Prove the following conditions equivalent. 1) R is DW, that is, w  d. 2) The only r-finite type r-stable semistar operation on R is the trivial semistar operation. 3) Every regular maximal ideal of R is w-closed. 4) Every regular maximal ideal of R is t-closed. 5) Max(R)reg = t-Max(R)reg . 6) If I ∈ Ireg (R) and I t = R, then I = R. Let R be a ring with total quotient ring K . Using Exercise 8, prove the following. 1 (R), a) R has regular dimension at most one if and only if Max(R)reg = X reg if and only if R is DW and of regular t-dimension at most one, if and only 1 (R) = wAss(K /R) = sK(K /R) = t-Spec(R)reg = if Max(R)reg = X reg reg t-Max(R) . b) Must a domain of dimension at most one be an H domain? c) Suppose that R is a domain, or more generally that R is additively regular. Then R has only finitely many regular t-maximal ideals if and only if R is DW and R has only finitely many regular maximal ideals. (Hint: Consider the union of the regular t-maximal ideals, and use Exercise 2.7.18) d) Suppose that R is Marot. Then R has only one regular t-maximal ideal if and only if R is DW and R has only one regular maximal ideal. e) There exists a ring with zerodivisors that has only one t-maximal ideal but is not DW. Let D be an integral domain with quotient field K = D, and let T = D + X K [X ]. Prove the following. a) The nonzero prime ideals (resp., maximal ideals) of T are the ideals p + X K [X ], where p is a prime ideal (resp., maximal ideal) of D, and the principal prime ideals f T = f K [X ] ∩ T , where f ∈ K [X ] is irreducible and f (0) = 1. b) The height one primes of T are X K [X ] and the principal prime ideals f T of T . c) The t-primes of T are the ideals p + X K [X ], where p is a t-prime ideal of D, and the principal prime ideals f T of T . d) The t-maximal ideals of T are the ideals p + X K [X ], where p is a t-maximal ideal of D, and the principal prime ideals f T of T .

290

e) f) 11. Let a)

3 Semistar Operations on Commutative Rings: Local Methods

Prove statements (1)–(8) of Theorem 1.2.6. Prove statement (9) of Theorem 1.2.6. R be a valuation ring with regular t-maximal ideal  p. Prove the following. If  is  a set of regular t-primes of R, then R = {R[p] : p ∈ } if and only if p = .  b) R is strongly essential if and only if p = SEss(R). c) If R is of finite regular t-dimension, then R is strongly essential if and only if R is Prüfer. d) (∗∗) If R is strongly essential, must R be Prüfer? 12. Fill in the missing details in the proof of Theorem 3.11.24. 13. Show that any one dimensional local essential domain is a valuation domain, but there exists a local essential domain that is not a valuation domain. 14. (∗∗) Must every regular essential prime of an essential ring R be contained in a prime that is maximal among the essential primes of R?

Chapter 4

Semiprime, Star, and Semistar Operations on Commutative Rings

This chapter develops the theories of semiprime, star, and nonunital semistar operations on commutative rings. We show that nonunital semistar operations generalize star operations, semistar operations, regular semiprime operations, and standard finite type semiprime operations. We also discuss the problem of determining which semiprime operations are induced by some unital (resp., nonunital) semistar operation. Particular attention is given to the integral closure and tight closure semiprime operations. Convention 4.0.1. In Chapters 4–6, semistar operations are not assumed to be unital, while in Chapters 2 and 3, semistar operations are assumed to be unital.

4.1 Star Operations and Fractional and Nonunital Star Operations In Chapter 1, we surveyed the theory of star operations on integral domains, which are unital sub-multiplicative closure operations on fractional ideals. Chapters 2 and 3 were concerned with unital semistar operations on commutative rings, which are unital sub-multiplicative closure operations on Kaplansky fractional ideals. Any such closure operation on K(R), where R is a ring, restricts to closure operations on (1) the ordered submonoid I(R) of all ideals of R, (2) the ordered submonoid Freg (R) of all regular fractional ideals of R, and (3) the ordered submonoid F(R) of all fractional ideals of R. Such closure operations are also considered in the literature, thus necessitating the following definitions. Definition 4.1.1. Let R be a ring. (1) A semiprime operation on R is a nucleus on the ordered monoid I(R) of all ideals of R, that is, a semiprime operation on R is a closure operation s on the poset I(R) such that I s J s ⊆ (I J )s for all I, J ∈ I(R) (or equivalently such that a I s ⊆ (a I )s for all I ∈ I(R) and all a ∈ R). © Springer Nature Switzerland AG 2019, corrected publication 2020 J. Elliott, Rings, Modules, and Closure Operations, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-24401-9_4

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4 Semiprime, Star, and Semistar Operations …

(2) A star operation on R is a unital nucleus on the ordered monoid Freg (R) of all regular fractional ideals of R, that is, a star operation on R is a closure operation ∗ on the poset Freg (R) such that R ∗ = R and I ∗ J ∗ ⊆ (I J )∗ for all I, J ∈ Freg (R). (3) A fractional star operation on R is a unital nucleus on the ordered monoid F(R) of all fractional ideals of R, that is, a fractional star operation on R is a closure operation ∗ on the poset F(R) such that R ∗ = R and I ∗ J ∗ ⊆ (I J )∗ for all I, J ∈ F(R) (or equivalently such that R ∗ = R and a I ∗ ⊆ (a I )∗ for all I ∈ F(R) and all a ∈ T (R)). A moral of this chapter is that the theory of semiprime operations provides motivation for the study of possibly nonunital semistar operations, defined as follows. Definition 4.1.2. Let R be a ring. (1) A (nonunital) semistar operation on R is a nucleus on the ordered monoid K(R) of all Kaplansky fractional ideals of R, that is, a semistar operation on R is a closure operation  on the poset K(R) such that I  J  ⊆ (I J ) for all I, J ∈ K(R) (or equivalently such that a I  ⊆ (a I ) for all I ∈ K(R) and all a ∈ T (R)). (2) A semistar operation  on R is unital if R  = R. (3) A semistar operation  on R is reduced if (0) = (0). (4) Semistar(R) (resp, USemistar(R), RSemistar(R)) denotes the poset of all semistar operations (resp., unital semistar operations, reduced semistar operations) on R. Convention 4.1.3. (1) To avoid confusion, we denote operations on K(R) using symbols in bold, operations on Freg (R) using symbols in italics, and operations on I(R) using symbols in ordinary roman typeface. Thus, for example, the semistar operations v, t, w, d as studied in Chapters 2 and 3 hereafter are denoted v , t , w , d , respectively, while v, t, w, d denote star operations and v, t, w, d denote semiprime operations. (2) , ∗, and s typically denote a generic semistar operation, a generic star operation, and a generic semiprime operation, respectively. Proposition 4.1.4. Let  be a semistar operation on a ring R. (1) One has R  R  = R  , and therefore R  is an overring of R. Moreover, one has K(R  ) = {I ∈ K(R) : I R  = I } ⊆ K(R). (2)  restricts to a semistar operation, a fractional star operation, a star operation, and a semiprime operation on R  . (3) If  is a unital semistar operation, then  restricts to a fractional star operation, a star operation, and a semiprime operation on R. (4) sprim : I −→ I  ∩ R is a semiprime operation on R. (5) v : I −→ I  ∩ I v is a unital semistar operation on R. (6) star : I −→ I  ∩ I v is a star operation on R. (7) Any fractional star operation on R restricts to a star operation on R.

4.1 Star Operations and Fractional and Nonunital Star Operations

293

Definition 4.1.5. Let  be a semistar operation on a ring R. (1) The semiprime operation sprim : I −→ I  ∩ R on R is called the semiprime operation induced by . (2) The unital semistar operation v : I −→ I  ∩ I v on R is called the unital semistar operation induced by . (3) The star operation star : I −→ I  ∩ I v on R is called the star operation induced by . In general, given a star operation (resp., fractional star operation, semiprime operation), a natural question arises as to whether or not it is induced by some semistar operation. We prove in this section that all star operations and all fractional star operations are indeed induced by appropriate semistar operations. Thus, the theories of star operations and fractional star operations can be subsumed under the theory of semistar operations. Section 4.3 is devoted to the corresponding problem for semiprime operations. Note in particular that not every semiprime operation is induced by some semistar operation. It is well-known and easy to check that, if ∗ is any star operation on an integral domain R with quotient field K , then, setting (0) = (0) and I  = I ∗ for all I ∈ Freg (R) and I  = K for all I ∈ K(R) − F(R), the operation  is the largest semistar operation on R that restricts to ∗. It is perhaps surprising that such a semistar operation exists even if R is a ring with zerodivisors. To prove this, we need first to provide analogues for star operations of our previous results on semistar operations. Lemma 4.1.6. Let ∗ be a star operation on a ring R, let I, J ∈ Freg (R), and let {Iλ : λ ∈ } be a nonempty subset of Freg (R).   (1) ( λ∈ Iλ )∗ = Iλ∗ )∗ if {Iλ : λ ∈ } is bounded above in Freg (R). ( λ∈ ∗ ∗ ∗ reg (2) λ∈ Iλ = ( λ∈ Iλ ) if {Iλ : λ ∈ } is bounded below in F (R). reg (3) The intersection of a collection of ∗-closed elements of F (R) that is bounded below is ∗-closed. (4) (I J )∗ = (I ∗ J )∗ = (I J ∗ )∗ = (I ∗ J ∗ )∗ . (5) (I : K J )∗ ⊆ (I ∗ : K J ) = (I ∗ : K J ∗ ) = (I ∗ : K J )∗ , and therefore (I ∗ : K J ) is ∗-closed. (6) (I −1 )∗ = (I ∗ )−1 = I −1 , and therefore I −1 is ∗-closed. (7) If I is invertible, then I ∗ = I and (I J )∗ = I J ∗ . (8) (a I )∗ = a I ∗ and (a R)∗ = a R for all regular elements a of T (R). Proposition 4.1.7. Let R be a ring. The poset of all star operations on R is complete. The infimum inf  of a set  of star operations acts by I −→ I inf  =

 {I ∗ : ∗ ∈ }

and the supremum sup  acts by I −→ I sup  =

 {J ∈ Freg (R) : J ⊇ I and J ∗ = J for all ∗ ∈ }

294

4 Semiprime, Star, and Semistar Operations …

for all I ∈ Freg (R). Moreover, for all I ∈ Freg (R), one has I sup  = I if and only if I ∗ = I for all ∗ ∈ . Definition 4.1.8. A star operation ∗ on a ring R is of finite type if I∗ =

 {J ∗ : J ∈ Freg (R) is f.g. and J ⊆ I }

for all I ∈ Freg (R). Definition 4.1.9. A star operation ∗ on a ring R with total quotient ring K is stable if (I ∩ J )∗ = I ∗ ∩ J ∗ for all I, J ∈ Freg (R) and (I : K J )∗ = (I ∗ : K J ) for all I, J ∈ Freg (R) with J finitely generated. Definition 4.1.10. (1) For any star operation ∗ on a ring R, we let  {J ∗ : J ∈ Freg (R) is f.g. and J ⊆ I },  {(I : K J ) : J ∈ I(R)reg , J ∗ = R}, I∗ =  = {(I : K J ) : J ∈ I(R)reg is f.g. and J ∗ = R},

I ∗t =

I ∗w

for all I ∈ Freg (R). The operations ∗t , ∗, and ∗w are star operations on R. (2) The v-operation is the star operation v : I −→ (I −1 )−1 . (3) The t-operation is the star operation t = vt . (4) The w-operation is the star operation w = vw . Proposition 4.1.11. Let ∗ be a star operation on a ring R. (1) ∗t is the largest finite type star operation less than or equal to ∗. (2) ∗ is largest stable star operation less than or equal to ∗. (3) ∗w is the largest stable finite type star operation less than or equal to ∗, and one has ∗w = ∗t  (∗)t . In particular, one has ∗w  ∗t  ∗ and ∗w  ∗  ∗. Corollary 4.1.12. Let  be a semistar operation on R. (1) (star )t = start . (2) star = star . (3) (star )w = starw . Corollary 4.1.13. Let R be a ring. Then v = starv is the largest star operation, t = start is the largest finite type star operation, v = starv is the largest stable star operation, and w = starw is the largest stable finite type star operation, on R.

4.1 Star Operations and Fractional and Nonunital Star Operations

295

Proposition 4.1.14. Let R be a ring and J ∈ Freg (R). starv (J ) is the largest star operation ∗ on R such that J ∗ = J . start (J ) is the largest finite type star operation ∗ on R such that J ∗ = J . starv (J ) is the largest stable star operation ∗ on R such that J ∗ = J . starw (J ) is the largest stable finite type star operation ∗ on R such that J ∗ = J .

(1) (2) (3) (4)

Proposition 4.1.15. Let R be a ring and S ⊆ Freg (R). starv (S) is the largest star operation ∗ on R such that S ⊆ Freg (R)∗ . start (S) is the largest finite type star operation ∗ on R such that S ⊆ Freg (R)∗ . starv (S) is the largest stable star operation ∗ on R such that S ⊆ Freg (R)∗ . starw (S) is the largest stable finite type star operation ∗ on R such that S ⊆ Freg (R)∗ . (5) starv (S) = inf{starv (J ) : J ∈ S} and starv (S) = inf{starv (J ) : J ∈ S}. (6) start (S) = inf{start (J ) : J ∈ S} and starw (S) = inf{starw (J ) : J ∈ S} if S is finite. (1) (2) (3) (4)

Moreover, for any star operation ∗ on R, one has the following. (7) (8) (9) (10)

∗ = starv (Freg (R)∗ ) = inf{starv (J ) : J ∈ Freg (R)∗ }. ∗t = start (Freg (R)∗t ) = inf{start (J ) : J ∈ Freg (R)∗t }. ∗ = starv (Freg (R)∗ ) = inf{starv (J ) : J ∈ Freg (R)∗ }. ∗w = starw (F(R)∗w ) = inf{starw (J ) : J ∈ F(R)∗w }.

Now we have what we need to show that every star operation is induced by some semistar operation. Definition 4.1.16. For any star operation ∗ on a ring R, let e ∗ = v (Freg (R)∗ ) = inf{vv (I ) : I ∈ Freg (R)∗ } and d ∗ = inf{ ∈ USemistar(R) : star = ∗}. Proposition 4.1.17. Let ∗ be a star operation on a ring R. (1) (2) (3)

e ∗ is the largest unital semistar operation on R that restricts to ∗. d ∗ is the smallest unital semistar operation on R that restricts to ∗. A unital semistar operation  on R restricts to ∗ if and only if d ∗    e ∗ .

Proof. We prove (1); the proofs of statements (2) and (3) are left as an exercise. Clearly, e ∗ is a unital semistar operation on R. By Proposition 4.1.15(7),one has ∗ = inf{starv (J ) : J ∈ Freg (R)∗ }. Thus, for any I ∈ Freg (R), one has I ∗ = {I v (J ) : J ∈ Freg (R)∗ } = I e ∗ . Therefore e ∗ restricts to ∗. Now, suppose that  is any unital semistar operation on R that restricts to ∗. Let I ∈ Freg (R)∗ . Then I  = I ∗ = I , so I is -closed and therefore   v (I ). Therefore   inf{vv (I ) : I ∈ Freg (R)∗ } = e ∗ . This proves (1).  For example, one has e v = v and, more generally, the following.

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4 Semiprime, Star, and Semistar Operations …

Corollary 4.1.18. Let R be a ring, and let S ⊆ K(R). (1) (2) (3) (4)

e starv (S) = v (S). (ee starv (S) )t = t (S). e starv(S) = e starv (S) = v (S). (ee starv (S) )w = w (S).

Since every star operation ∗ can be considered in a canonical way to be a unital semistar operation (viz., e ∗ ), or alternatively as an interval of unital semistar operations (viz., d ∗    e ∗ ), the theory of semistar operations is a proper generalization of the theory of star operations. One may try to repeat all of the analysis above for fractional star operations. For example, since F(R), like Freg (R), is bounded complete, the poset of all fractional star operations is complete, with infima and suprema computed in the obvious way. One also has the following. Definition 4.1.19. For any fractional star operation ∗ on a ring R, let e ∗ = v(F(R)∗ ) = inf{v(I ) : I ∈ F(R)∗ }, and let Id∗ =



{J ∈ K(R) : J ⊇ I and ∀H ∈ F(R) (H ⊆ J ⇒ H ∗ ⊆ J )}

for all I ∈ K(R). Proposition 4.1.20. Let ∗ be a fractional star operation on a ring R. (1) e ∗ is the largest unital semistar operation on R that restricts to ∗. (2) d ∗ is the smallest unital semistar operation on R that restricts to ∗. (3) A unital semistar operation  on R restricts to ∗ if and only if d ∗    e ∗ . Proof. The proof of (1) is similar to the proof of statement (1) of Proposition 4.1.17. Statement (2) follows from Proposition 6.2.9 of Chapter 6, since F(R) is a supspanning subset of K(R).  By Propositions 4.1.17 and 4.1.20, the theory of semistar operations generalizes the theory of fractional star operations, which in turn generalizes the theory of star operations. Unfortunately, however, complications arise with fractional star operations that do not arise with star or semistar operations. All of these complications stem from the fact that I −1 and (I : K J ) may not be fractional if I and J are fractional, and, moreover, the ordered monoid F(R) need not be residuated. For example, one naturally says that I ∈ F(R) is ∗-invertible if there exists a J ∈ F(R) such that (I J )∗ = R, in which case it follows that I −1 = J ∗ is also fractional. It follows that I ∈ F(R) is ∗-invertible if and only if (I I −1 )∗ = R and I −1 is fractional, but unfortunately the additional assumption that I −1 is fractional cannot be removed. The most serious of these complications is that the construction of the stable fractional star operation ∗ associated to a fractional star operation ∗ cannot be carried out

4.1 Star Operations and Fractional and Nonunital Star Operations

297

without the use of semistar operations. Naturally one says that a fractional star operation ∗ is stable if I ∗ ∩ J ∗ = (I ∩ J )∗ for all I, J ∈ F(R) and (I : K J )∗ = (I ∗ : K J ) for all I, J ∈ F(R) such that J is finitely generated and (I : K J ) is fractional. One may attempt to define ∗ in the obvious way: I∗ =

 {(I : K J ) : J ∈ I(R), J ∗ = R}.

It is easy to show that ∗  ∗ is a stable star operation. However, it is not possible to show conversely that if ∗ is stable then ∗ = ∗. This is because the proof that I ∗ ⊆ I ∗ requires one to show that (I : R x)∗ = R for all x ∈ I ∗ , but unfortunately (I : R x)∗ is undefined if (I : R x) is not fractional. The way around this is to use the stable semistar operation e ∗ and instead define ∗ = e ∗ |F(R) (a definition that is not equivalent to the previous definition of ∗). This yields I∗ =



{(I : K J ) : J ∈ I(R), J e ∗ = R}

for all I ∈ F(R), which provides the correct definition of the largest stable fractional star operation less than or equal to ∗. This solution is somewhat unsatisfactory because it requires the use of the theory of semistar operations, implying that the theory of fractional star operations is not self-contained. At any rate, the fact that the theory of semistar operations generalizes the theories of star and fractional star operations eliminates the need for the latter and provides a moral justification for our treatment of semistar operations in Chapters 2 and 3, despite the fact that many of the results in those chapters (in particular those that don’t require defining the -closure of a non-regular or non-fractional Kaplansky ideal) can easily be recast in terms of star operations simply by replacing every semistar operation  with the star operation star . The remainder of this chapter focuses on results that cannot be recast in those terms. Recall that Krull’s original 1935 definition of a star operation (which he called a  -operation) on an integral domain D left out the requirement D  = D, although Krull later added it as a hypothesis in a paper he published a year later. If we leave out the requirement R ∗ = R for a star operation ∗ on a ring R, then we obtain the following notion. Definition 4.1.21. A nonunital star operation on a ring R is a nucleus on the ordered monoid Freg (R) of regular fractional ideals of R. In other words, a nonunital star operation on R is a self-map ∗ : I −→ I ∗ of the set Freg (R) such that ∗ is (1) (2) (3) (4)

order-preserving: I ⊆ J implies I ∗ ⊆ J ∗ , expansive: I ⊆ I ∗ , idempotent: (I ∗ )∗ = I ∗ , and sub-multiplicative: I ∗ J ∗ ⊆ (I J )∗ ,

where each of these axioms holds for all I, J ∈ Freg (R). As nonunital star operations are nuclei, all of the results on star operations stated earlier in this section generalize to nonunital star operations. (For example,

298

4 Semiprime, Star, and Semistar Operations …

the nonunital star operation v(J ) defined by I v(J ) = (J : K (J : K I )) is the largest nonunital star operation ∗ on R such that J ∗ = J .) Since for any nonunital star operation ∗ on R the star operation inf{∗, v} is the largest star operation less than or equal to ∗, and ∗ is a star operation if and only if ∗  v, the results on star operations can be recovered easily from their respective generalizations to nonunital star operations. For example, since Freg (R) is bounded complete, Proposition 6.2.7 of Chapter 6 implies that the poset NUStar(R) of all nuclei on Freg (R), that is, the poset of all nonunital star operations on R, is bounded complete. From this we recover the fact that the poset Star(R) = NUStar(R)v of all star operations on R is complete. Moreover, it follows that NUStar(R) is itself complete if and only if there is a largest nonunital star operation on R. We show below that NUStar(R) is complete if and only if the complete integral closure R  of R is a fractional ideal of R, in which case R  is completely integrally closed and is the largest overring of R that is a fractional ideal of R. As we have seen, in general, the complete integral closure of a ring need not be completely integrally closed. However, since the intersection of completely integrally closed overrings of a ring is obviously completely integrally closed, it follows that every ring R is contained in a unique smallest completely integrally closed overring R ⊇ R  , which we call the complete complete integral closure of R. Clearly, R  is completely integrally closed if and only if R  = R . The ring R is in some sense a “better” overring of R than is the complete integral closure R  , although there are no known general methods to compute it. Proposition 4.1.23 below shows that if R  is a fractional ideal of R, then R  is completely integrally closed. Lemma 4.1.22. Let R be a ring with total quotient ring K . If S is an overring of R that is a fractional ideal of R, then R and S have the same complete integral closure. Proof. Obviously, one has R  ⊆ S  . Let x ∈ S  . Then S[x] is a fractional ideal of S, so that (S : K S[x]) is regular. Let b be any regular element of (S : K S[x]), so bS[x] ⊆ S. By hypothesis there also exists a regular element a of (R : K S), so that aS ⊆ R. Then one has ab R[x] ⊆ abS[x] ⊆ aS ⊆ R, and therefore ab is a regular element of (R : K R[x]). Therefore R[x] is a fractional ideal of R, whence x ∈ R  .  It follows, then, that R  = S  . Proposition 4.1.23. Let R be a ring. If the complete integral closure R  of R is a fractional ideal of R, then R  is completely integrally closed. Both conditions hold, in particular, if R  is finitely generated as an R-algebra, and hence also if R is Noetherian and R is module-finite over R. Proof. If R  is a fractional ideal of R, then by the lemma R and R  have the same  complete integral closure, namely, R  . The rest of the proposition is clear. Proposition 4.1.24. Let R be a ring with total quotient ring K . If ∗ is a nonunital star operation on R, then R ∗ ⊆ R  . Moreover, the following conditions are equivalent.

4.1 Star Operations and Fractional and Nonunital Star Operations

(1) (2) (3) (4)

299

NUStar(R) is a complete lattice. There is a largest nonunital star operation e on R. There is a largest overring R  of R that is a fractional ideal of R. R  is a fractional ideal of R, that is, (R : K R  ) is regular.

If the conditions above hold, then one has R  = R e = R  = R and e = v(R e ). Moreover, the conditions above hold if R  is finitely generated as an R-algebra, and hence also if R is completely integrally closed or if R is Noetherian and R is modulefinite over R. Finally, one has Star(R) = NUStar(R) if and only if NUStar(R) is complete and e = v, if and only if R is completely integrally closed. Proof. If ∗ is a nonunital star operation on R, then R ∗ is a fractional ideal of R, so (R : K R ∗ ) is regular, which implies that (R : K R[x]) is regular for all x ∈ R ∗ , and therefore R ∗ ⊆ R  . Clearly, (1) and (2) are equivalent since NUStar(R) is bounded complete. If e is a largest nonunital star operation on R and E is any overring of R such that E is a fractional ideal of R, then the operation ∗ E : I −→ I E is a nonunital star operation on R, and therefore ∗ E  e and so E = R ∗ E ⊆ R e . Thus R e is the largest overring of R that is a fractional ideal of R. Therefore (2) implies (3). Suppose that (3) holds. Then for any nonunital star operation ∗, one must have R ∗ = R  and therefore ∗  v(R  ), so that e = v(R  ) is the largest nonunital star operation on R. Furthermore, if x ∈ R  , then the overring R[x] of R is a fractional ideal of R, so that R[x] ⊆ R  and therefore x ∈ R  . Conversely, one has R  = R ∗ R ⊆ R  , so that R  = R  . Therefore (3) implies both (2) and (4). Since also (4) implies (3), all four conditions are equivalent. Suppose, now, that the four conditions hold. We have seen already that R e =  R = R  and e = v(R e ). Finally, since R  is a fractional ideal of R, the rest of the proposition follows from Proposition 4.1.23.  Corollary 4.1.25. A ring R is Dedekind if and only if every nonunital star operation on R is trivial. Thus far in this section, we have discussed star and semistar operations (both unital and nonunital), fractional star operations, and semiprime operations. The most general of these are the semistar and semiprime operations, neither of which fully generalizes the other. In Section 4.3 we discuss the problem of determining when a semiprime operation is induced by some semistar operation, and in Chapter 5 we discuss a natural generalization of both semiprime and semistar operations to modules and algebras over noncommutative rings.

4.2 Nonunital Semistar Operations In this section, we develop nonunital semistar analogues of notions defined previously for unital semistar operations, including finite type, stable, and spectral semistar operations and semistar invertibility, semistar class groups, and -Prüfer rings.

300

4 Semiprime, Star, and Semistar Operations …

There are many subtle differences between nonunital semistar operations and unital semistar operations. In most respects, nonunital semistar operations are more complicated than unital semistar operations. For example, if  is a nonunital semistar operation on a ring R and I ∈ K(R), then I −1 may not be -closed (although (R  : K I ) is -closed), and I may not be -closed if I is invertible. However, in a few respects, nonunital semistar operations are less complicated than unital semistar operations. For example, the divisorial semistar operations v (J ) are given more sim(J ) = I v ∩ (J : K (J : K I )); and, ply by I v (J ) = (J : K (J : K I )), rather than by I v similarly, the nonunital semistar operation I −→ {I S : S ∈ O}  induced by a set O of overrings of R does not require the assumption that R = {S : S ∈ O}. Various other differences are revealed in this section and in the extensive literature of semistar operations on integral domains; see [44, 50, 62, 64, 67, 161, 164, 165], for example. As the ordered monoid K(R) is a multiplicative lattice, Proposition 6.2.7 of Chapter 6 yields the following. Proposition 4.2.1. Let R be a ring with total quotient ring K . The poset of all semistar operations on R is complete. The infimum inf  of a set  of semistar operations on R acts by I −→ I inf  =

 {I  :  ∈ }

and the supremum sup  of  acts by I −→ I sup  =

 {J ⊇ I : J ∈ K(R), J  = J for all s ∈ }

for all I ∈ K(R). Moreover, for all I ∈ K(R), one has I sup  = I if and only if I  = I for all  ∈ . Recall that v : I −→ (I −1 )−1 is the largest unital semistar operation on R for any ring R. By contrast, one has the following. Definition 4.2.2. Let R be a ring with total quotient ring K . The operation e : I −→ K is the largest semistar operation on R. It is useful to observe that any semistar operation  on a ring R induces a unital semistar operation ˙ = |K(R  ) on the overring R  of R. This sometimes allows one to reduce a given problem concerning semistar operations to the unital case. Definition 4.2.3. For any semistar operation  on a ring R, we let ˙ = |K(R  ) denote the induced unital semistar operation on the ring R  . Definition 4.2.4. Let  and  be semistar operations on a ring R. We write     if I  = I  for all regular (fractional) ideals I of R. We write   if I  ⊆ I  for all regular (fractional) ideals I of R.

4.2 Nonunital Semistar Operations

301

Definition 4.2.5. A semistar operation  on a ring R is of finite type if I =



{J  : J ∈ K(R) is f.g. and J ⊆ I }

for all I ∈ K(R). Definition 4.2.6. A semistar operation  on a ring R with total quotient ring K is stable if (I ∩ J ) = I  ∩ J  for all I, J ∈ K(R) and (I : K J ) = (I  : K J ) for all I, J ∈ K(R) with J finitely generated. Definition 4.2.7. For any semistar operation  on a ring R, we let  {J  : J ∈ K(R) is f.g. and J ⊆ I },  {(I : K J ) : J ∈ I(R), J  = R  }, I  = I s =  {(I : K J ) : J ∈ I(R) is f.g. and J  = R  }, I w = I t =

for all I ∈ K(R). The operations t ,  = s , and w are semistar operations on R. Proposition 4.2.8. Let  be a semistar operation on a ring R. (1) t is the largest finite type semistar operation less than or equal to . (2)  = s is largest stable semistar operation less than or equal to . (3) w is the largest stable finite type semistar operation less than or equal to , and one has w = t  ()t . In particular, one has w  t   and w    . Proof. See Theorems 6.5.7, 6.6.7, and 6.6.9.



Proposition 4.2.9. Let R be a ring with total quotient ring K , and let  be a semistar operation on R. The following conditions are equivalent. (1)  is stable. (2) (I : K J ) = (I  : K J ) and (I ∩ R) = I  ∩ R  for all I, J ∈ K(R) with J finitely generated. (3) (I : R J ) = (I  : R  J ) for all I, J ∈ K(R) with J finitely generated. (4) (I : R x) = (I  : R  x) for all I ∈ K(R) and all x ∈ K . (5)  = . Proof. See Theorem 6.6.7.



The proof of the following proposition is left as an exercise. Proposition 4.2.10. Let R be a ring. (1) For any collection  of semistar operations on R, one has inf  = inf{ :  ∈ }. Consequently, the infimum of a collection of stable semistar operations on R is stable.

302

4 Semiprime, Star, and Semistar Operations …

(2) For any finite collection  of semistar operations on R, one has (inf )t = inf{t :  ∈ }. Consequently, the infimum of a finite collection of finite type semistar operations on R is of finite type. (3) Let S be an overring of a ring R. The operation I −→ I S on K(R) is a finite type semiprime operation on R, and it is stable if and only if S is flat over R. (4) Let O be a collection ofoverrings of R. If S is flat over R for all S ∈ O, then the operation O : I −→ {I S : S ∈ O} on K(R) is a stable semiprime operation on R. If O is finite, then O is of finite type. The divisorial semistar operations v (J ) and v (J ) are defined, and constructed, as follows. Proposition 4.2.11. Let R be a ring with total quotient ring K (1) For any J ∈ K(R), the operation v (J ) defined by I v (J ) = (J : K (J : K I )) for all I ∈ K(R) is the largest semistar operation  on R such that J  = J . Moreover, I ∈ K(R) is v (J )-closed if and only if I = (J : K H ) for some H ∈ K(R). (2) Let J ⊆ K(R). The operation v (J ) = inf{vv (J ) : J ∈ J } is the largest semistar operation  on R such that J  = J for all J ∈ J . For any semistar operation  on R, one has   v (J ) if and only if J  = J for all J ∈ J . (3) One has  = v (K(R) ) = inf{vv (J ) : J ∈ K(R) } for any semistar operation  on R. Thus, every semistar operation on R is of the form v (J ) for some J ⊆ K(R). Proof. See Proposition 6.3.6.



Corollary 4.2.12. Let R be a ring with total quotient ring K , and let  be a semistar operation on R. (1) r = v ((0)) : I −→ ann K (ann K (I )) is the largest reduced semistar operation on R. (2) red = inf{, r } is the largest reduced semistar operation on R less than or equal to . Note that if D is an integral domain with quotient field K , then e : I −→ K is the only non-reduced semistar operation on D. Next, we discuss -primality and -maximality. Complications in studying these notions arise from the fact that, if  is nonunital, then I  is not necessarily an ideal of R for every ideal I of R; indeed, R is an ideal of R, but R   R is not. This motivates the definition of a -ideal, defined below. Definition 4.2.13. Let s be a semiprime operation on a ring R. (1) An ideal of R is s-prime if it is s-closed and prime. (2) An ideal of R is s-maximal if it is maximal among the proper s-closed ideals of R.

4.2 Nonunital Semistar Operations

303

Definition 4.2.14. Let  be a semistar operation on a ring R. (1) A -ideal of R is an ideal of R of the form b ∩ R for some -closed ideal b of R  , or equivalently an ideal of the form b ∩ R for some ideal b of R, or, equivalently still, an ideal a of R such that a = a ∩ R, that is, such that a is sprim -closed. (2) An ideal p of R is -prime if p is a prime -ideal of R, or equivalently if p is sprim -prime. (3) An ideal p of R is -maximal if p is maximal among the proper -ideals of R, or equivalently if p is sprim -maximal. (4) -Spec(R) denotes the set of all -prime ideals of R. (5) -Max(R) denotes the set of all -maximal ideals of R. (6) An element I of K(R) is -stable if (I ∩ R) = I  ∩ R  . Remark 4.2.15. Let  be a semistar operation on a ring R. In the literature, because -ideals may not be -closed, -ideals are called quasi--ideals, and -maximal ideals are called quasi--maximal ideals. We feel, however, that the prefix “quasi-” is unnecessary and, in the interest of simplicity, we have chosen to avoid the prefix in this particular context. We leave the proofs of the following two lemmas as an exercise. Lemma 4.2.16. Let R be a ring. If s is a semiprime operation on R, then every s-maximal ideal of R is s-prime. Consequently, if  is a semistar operation on R, then every -maximal ideal of R is -prime. Lemma 4.2.17. Let  be a semistar operation on a ring R. (1) For any ideal a of R, the ideal a ∩ R is the smallest -ideal of R containing a. (2) An ideal a of R is contained in some proper -ideal of R if and only if a  R  . (3) If  is of finite type, then any proper -ideal of R is contained in some -maximal ideal of R. (4) If  is of r-finite type, then any proper regular -ideal of R is contained in some -maximal ideal of R. (5) Any ideal of R is -stable. (6) An ideal b of R  is -stable if and only if b = (b ∩ R) . (7) For any ideal b of R  , the ideal (b ∩ R) is the largest -stable ideal of R  contained in b . Definition 4.2.18. Let  be a semistar operation on a ring R. We say that R has enough -maximal ideals if any proper -ideal of R is contained in some -maximal ideal of R. Corollary 4.2.19. Let  be a semistar operation on a ring R. If  is of finite type, then R has enough -maximal ideals. Lemma 4.2.20. Let  be a semistar operation on a ring R. (1) The set of all -closed -stable ideals of R  is equal to {a : a is a  -ideal of R}.

304

4 Semiprime, Star, and Semistar Operations …

(2) The map  restricts to a bijection from the set of all -ideals of R to the set of all -closed -stable ideals of R  , with inverse b −→ b ∩ R. (3) Suppose that  is stable or R  is flat over R. Then every ideal of R  is -stable, and the map  restricts to a bijection from the set of all -ideals of R to the set of all -closed ideals of R  . Proposition 4.2.21. Let  be a semistar operation on a ring R, and let p be an ideal of R. The following conditions are equivalent. (1) p is -maximal. (2) p = a ∩ R for some ideal a of R  that is maximal among the proper -closed -stable ideals of R  (in which case a = p is uniquely determined by p). (3) p is a -ideal of R such that p is maximal among the proper -closed -stable ideals of R  . (4) p is maximal in the set {a ∈ I(R) : a  R  }. Proof. Suppose that (1) holds, that is, let a = p . Then a is a proper -closed and -stable ideal of R  with p = a ∩ R. Let I be any proper -closed -stable ideal of R  containing a. Let J = I ∩ R, so that J  = I since I is -closed and -stable. Since J is a proper -ideal of R containing p, one has J = p and therefore I = J  = p = a. Thus (1) implies (2). Conversely, suppose that (2) holds. Then p = a since a is -closed and -stable. Let I be a proper -ideal of R containing p. Then a = p ⊆ I  and I  is a proper, -closed, and -stable ideal of R  , so a = I  by maximality of a. Therefore p = a ∩ R = I  ∩ R = I , since I is a -ideal. Therefore p is -maximal. Thus (2) implies (1). Moreover, (2) and (3) are clearly equivalent. Next, to prove that (1) is equivalent to (4), observe that p is -maximal if and only if p is maximal among the -ideals in the set S = {a ∈ I(R) : a  R  }. But by Lemma 4.2.17(1) every ideal in the set S is contained in some -ideal in the set S. It follows that p is -maximal if and only if p is maximal in the set S. Thus (1) and (4) are equivalent.  Corollary 4.2.22. Let  be a semistar operation on a ring R, and suppose that every ideal of R  is -stable, which holds, for example, if  is stable or R  is flat over R. Then, the following conditions are equivalent for any ideal p of R. (1) p is -maximal. (2) p = q ∩ R for some ˙ -maximal ideal q of R  , that is, for some ideal q of R  that is maximal among the proper -closed ideals of R  . (3) p is a -ideal and p is ˙ -maximal, that is, p = p ∩ R and p is maximal among the proper -closed ideals of R  . By the following proposition, if  is of finite type, then condition (1) of the corollary implies condition (2). Proposition 4.2.23. Let  be a semistar operation on a ring R. Suppose that  is of finite type, or more generally that the ring R  has enough ˙ -maximal ideals. If p is a -maximal ideal of R, then p = q ∩ R for some ˙ -maximal ideal q of R  , that is, for some ideal q of R  that is maximal among the proper -closed ideals of R  .

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305

Proof. If  is of finite type, then ˙ is also of finite type and therefore R  has enough ˙ -maximal ideals. Suppose that R  has enough ˙ -maximal ideals and that p is maximal, so that p is maximal in the set {a ∈ I(R) : a  R  }. Since p  R  , one has p ⊆ q for some ˙ -maximal ideal q of R  . Then p = p ∩ R ⊆ q ∩ R, and q ∩ R is a proper -ideal of R, whence p = q ∩ R.  The following example shows that the converse of the proposition is false, even if R = k[X, Y ] for some field k. Example 4.2.24. Let R be a ring and S an overring of R. The map  = {S} : I −→ I S is a finite type semistar operation on R with R  = S. An ideal of R is -closed if and only if it is also an ideal of S, every ideal of S is -closed, and an ideal of R is a -ideal if and only if it lies under some ideal of S. Therefore, an ideal of S is maximal among the proper -closed ideals of S if and only if it is a maximal ideal of S. Let m be a maximal ideal of S, and let p = m ∩ R. Then p is -maximal if and only if p is maximal among the proper ideals of R that lie under some ideal of S. Therefore, if there also exists a proper ideal n of S such that q = n ∩ R properly contains p, then p is not -maximal. If this holds, then by Corollary 4.2.22 the ring S is not flat over R and  is not stable. For an explicit example, let R = k[X, X Y ] ∼ = k[X, Z ], where k is a field, and let S0 = k[X, Y ] ∼ = k[X, Z / X ]. Let m0 = (Y ) and n0 = (X, Y − 1) in S0 . Then p = m0 ∩ R = (X Y ) and q = n0 ∩ R = (X, X Y ), so that q properly contains p. However, m0 is not maximal. But let S = (S0 )m0 ∩ (S0 )n0 ⊆ k(X, Y ). One can check that S is a semilocal ring with exactly two maximal ideals m = m0 S and n = n0 S, which lie, respectively, over m0 and n0 in S0 and over p and q in R. Therefore m is maximal among the proper -closed ideals of S = R  , but p = m ∩ R is not -maximal. Moreover, every proper ideal I of R lying under some ideal of S is contained in some maximal ideal of S and is therefore contained in m or n, and therefore I is contained in p or q, and hence I is contained in q. Therefore q is the unique -maximal ideal of R. Next, we generalize Corollaries 3.11.6 and 3.11.16 to nonunital semistar operations as follows. Theorem 4.2.25. Let  be a semistar operation on a ring R. One has the following. (1) w -Spec(R) = {q ∈ Spec(R) : q ⊆ p for some p ∈ t -Max(R)} = {q ∈ Spec(R) : qt = R t }. (2) w -Max(R)  = t -Max(R). (3) I w = {[I ]R[p] : p ∈ t -Max(R)}for all I ∈ K(R).  (4) If R is Marot, then I w = {I R[p] : p ∈ t -Max(R)} = {I R[p] : p ∈ t -Max(R)reg } for all regular I ∈ K(R). Proof. We may suppose without loss of generality that  = t is of finite type.  First, we claim that I  = {[I  ]R[p] : p ∈ -Max(R)} for all I ∈ K(R).  To prove the claim, it suffices to assume that I is -closed and show that {[I ]R[p] : p ∈ -Max(R)} ⊆ I . Let x ∈ K = T (R) lie in the given intersection. For each

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p ∈ -Max(R), we can write bp x = ap with ap ∈ I and bp ∈ R − p. Then, since bp ∈ (I : R x) − p, we have (I : R x) Ę p, for all p ∈ -Max(R). It follows that (I : R x) = R  . But since (I : K x) is -closed, we have R  = (I : R x) = ((I : K x) ∩ R) ⊆ (I : K x) ∩ R  = (I : K x) ∩ R  = (I : R  x), and so R  = (I : R  x) and therefore x ∈ I .  Now, I −→ I c = {[I ]R[p] : p ∈ -Max(R)} defines a stable semistar operation c on R. By our claim proved above, we have (I  )c = I  for all I ∈ K(R) and therefore c  . But then since c is stable we have also c   = w . Now, if J  = R  , or equivalently if J w = R w , where J ∈ K(R), then we claim that J c = R w . For suppose that J c = R w . Then there exists x ∈ R w − [J ]R[p] for some -maximal ideal p of R. Then (J : R x) ⊆ p, so, since R w = (R w : R w x) = (J w : R w x) = (J : R x)w , one has R = (J : R x)w ∩ R ⊆ pw ∩ R ⊆ p ∩ R = p, which is a contradiction. Now, let I, J ∈ K(R) with J finitely generated and J  = R  . Then since J c = w R , we have (I : K J ) ⊆ (I : K J )c ⊆ (I c : K J ) = (I c : K J c ) = (I c : K R w ) = I c . Taking the union over all such J , we see that I w ⊆ I c . Therefore w  c, so equality holds. This proves (3). Next, statements (1) and 3.11.3.  (2) follow from (3) as in the proof of Theorem  Now, let c : I −→ {I R[p] : p ∈ t -Max(R)} or c : I −→ {I R[p] : p ∈ t -Max(R)reg }. In either case c is a stable semistar operation on R. Moreover, one  has J w = R w if and only if J c = R w for all regularly generated J ∈ I(R), and hence for all regular J ∈ I(R) if R is Marot. Suppose, then, that R is Marot, and let I ∈ K(R) be regular. Since w and c are stable one has 

  {(I : K J ) : J ∈ Ireg (R), J c = R w }  = {(I : K J ) : J ∈ Ireg (R), J w = R w }

Ic =

= I w . 

This proves (4). Corollary 4.2.26. Let  be a semistar operation on a ring R. One has R w =

 {R[p] : p ∈ t -Max(R)}.

Moreover, an ideal p of R is t -maximal if and only if p = q ∩ R for some ideal q of R w that is maximal among the proper w -closed ideals of R w , if and only if p = pw ∩ R and pw is maximal among the proper w -closed ideals of R w . For any such ideals q of R w and p = q ∩ R of R, one has (R w )[q] = R[p] and [q](R w )[q] = [p]R[p] . Example 4.2.27. Consider the domains R ⊆ S and the finite type semistar operation  : I −→ I S of Example 4.2.24. The maximal ideal q of R is the unique -maximal ideal of R. Therefore one has R w = Rq  S = R  and I w = I Rq for all I ∈ K(R). Theorems 3.11.3 and 3.11.5 also generalize to nonunital semistar operations, as in Theorems 4.2.30 and 4.2.31 below. The proofs are left as exercises.

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Definition 4.2.28. Let R be a ring and  ⊆ Spec R a set of prime ideals of R. We let  {[I ]R[p] : p ∈ } I [] = for all I ∈ K(R). The operation [] : I −→ I [] is a semistar operation on R. Definition 4.2.29. A semistar operation  on a ring R is spectral if  = [] for some set  ⊆ Spec R. Theorem 4.2.30. Let R be a ring and  ⊆ Spec R a set of prime ideals of R. One has the following.  (1) [] : I −→ {[I ]R[p] : p ∈ } is a stable semistar operation on R. (2) Every proper [] -ideal of R is contained in some prime in . (3) [] -Spec(R) = {q ∈ Spec(R) : q ⊆ p for some p ∈ }. (4) [] -Max(R) is the set of all maximal elements of . Theorem 4.2.31 (cf. [161, Theorem 22]). Let  be a semistar operation on a ring R. The following conditions are equivalent. (1)  is spectral. (2)  is stable and every proper -ideal of R is contained in some -prime of R.  (3)  = [-Spec(R)] , that is, I  = {[I ]R[p] : p ∈ -Spec(R)} for all I ∈ K(R). Next, recall that the set Spec R of all prime ideals of any ring R is endowed with a natural topology called the Zariski topology, where a subset  of Spec R is closed if and only if it is of the form V (I ) = {p ∈ Spec R : p ⊇ I } for some ideal I of R. [176, Proposition 3.13] states  that, if D is an integral domain and  is a set of prime ideals of D such that D = {Dp : p ∈ }, then the spectral semistar operation [] on D associated to  is of finite type if and only if  is a compact subset of the topological space Spec D. (Here, compact subsets are not assumed to be Hausdorff.) The proposition (and its proof in [176]) easily generalizes to rings with zerodivisors as follows. Proposition 4.2.32. Let R be a ring and  a set of prime ideals of R. The following conditions are equivalent. (1) (2) (3) (4)

The spectral semistar operation [] on R associated to  is of finite type.  is a compact subset of Spec R. [] -Spec(R) is a compact subset of Spec R. [] -Max(R) is a compact subset of Spec R.

Corollary 4.2.33. Let  be a semistar operation on a ring R. The sets t -Spec(R) and t -Max(R) are compact subsets of Spec R. Proposition 4.2.34. Let Rbe a ring and  a nonempty subset of w -Spec(R) (or of t -Spec(R)) such that R = {R[p] : p ∈ }. The following conditions are equivalent.

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 is a compact subset of Spec R. [] = w . [] is of finite type. []  t . t -Max(R) ⊆ .

Proof. If  is compact, then [] is a finite type stable unital semistar operation on R with w  []  v , so that [] = w . Therefore (1) ⇒ (2) ⇒ (3) ⇒ (4). Suppose that t -Max(R) Ę . Let p be a t -maximal ideal of R that does not lie in . If []  t , then p[] ⊆ pt = p, and yet p is not contained in any element of  and thus p[] = R, which is a contradiction. Thus (4) ⇒ (5). As is well-known, a subset  of Spec R is compact if and only if its generization  = {q ∈ Spec R : q ⊆ p for some p ∈ } is compact. Suppose that t -Max(R) ⊆ . Then the generization   of  is equal to w -Spec(R), which is compact since w is of finite type. It follows, then, that  is also compact. Therefore (5) ⇒ (1).  Corollary 4.2.35. A PVMR is equivalently an essential ring R such that Ess(R) is a compact subset of Spec R. Corollary 4.2.36. Let R be a ring with total quotient ring K . Then sK(K /R) (resp., wAss(K /R)) is a compact subset of Spec R if and only if t -Max(R) ⊆ sK(K /R) (resp., t -Max(R) ⊆ wAss(K /R)). For example, if R is a v -coherent ring such that every t -maximal ideal is regular, then sK(T (R)/R) is compact. Similarly, if R is an H ring such that every t -maximal ideal is regular, then wAss(T (R)/R) = sK(T (R)/R) is compact. Remark 4.2.37. A domain D with quotient field K such that t -Spec(D) ⊆ sK(K /D) (resp., t -Max(D) ⊆ sK(K /D)) is said to be well behaved (resp., conditionally well behaved). For example, any v-coherent domain is well behaved, and any H domain is conditionally well behaved. A domain need not be conditionally well behaved, and a conditionally well behaved domain need not be well behaved [189]. By the corollary above, a domain D with quotient field K is conditionally well behaved if and only if sK(K /D) is a compact subset of Spec D. By contrast, both t -Spec(R) and t -Max(R) are compact subsets of Spec R for any ring R. A major complication that arises with nonunital semistar operations is that there are several competing notions of “semistar invertibility.” Definition 4.2.38. Let  be a semistar operation on a ring R, and let I ∈ K(R). (1) I is quasi -invertible [67] if I satisfies the following equivalent conditions. (a) (I J ) = R  for some J ∈ K(R). (b) (I (R  : K I )) = R  . (c) I  (or I R  ) is ˙ -invertible.

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(2) I is -invertible in the sense of Fontana, Jara, and Santos [64], or FJS -invertible, if I satisfies the following equivalent conditions. (a) (I I −1 ) = R  . (b) I  = I  for some I  , J  ∈ K(R) such that I J  ⊆ R and (I  J  ) = R  . (c) I  = I  for some I  ⊆ I in K(R) such that I J  ⊆ R and (I  J  ) = R  for some J  ∈ K(R). (3) I is -invertible, or FJS -invertible up to -closure, if I satisfies the following equivalent conditions. (a) I  = I  for some FJS -invertible I  ∈ K(R). (b) I  = I  for some I  ∈ K(R) such that (I  I −1 ) = R  . (c) I  = I  for for some I  , J  ∈ K(R) such that I  J  ⊆ R and (I  J  ) = R  . (4) I is strictly -invertible if I satisfies the following equivalent conditions. (a) I  = I  for some FJS -invertible I  ⊆ I in K(R). (b) I  = I  for some I  ⊆ I in K(R) such that (I  I −1 ) = R  . (c) I  = I  for some I  ⊆ I in K(R) such that I  J  ⊆ R and (I  J  ) = R  for some J  ∈ K(R). Convention 4.2.39. The reader should be warned that, in the literature, “-invertible” has meant “FJS -invertible” as we have defined it above [6, 64, 67]. All four of the classes of “semistar invertible” Kaplansky fractional ideals above are closed under multiplication. Moreover, one has the following implications: FJS  -invertible ⇒ strictly  -invertible ⇒ -invertible ⇒ quasi  -invertible. At first sight, the notion of quasi -invertibility seems to be the most natural. However, one reason that it is not the most interesting notion of semistar invertibility is because it mostly reflects properties of the ring R  rather than the ring R, since I is quasi -invertible in R if and only if I R  is (quasi) ˙ -invertible in R  . Although the notion of FJS -invertibility is also very natural, it suffers from a defect that is not shared by the other three notions of semistar invertibility: the closure of an FJS -invertible Kaplansky fractional ideal need not be FJS -invertible, and in fact R  need not be FJS -invertible. Consequently, the -closed FJS invertible Kaplansky fractional ideals of R do not necessarily form a group under -multiplication. Proposition 4.2.40. Let  be a semistar operation on a ring R. The following conditions are equivalent. (1) The -closure of an FJS -invertible Kaplansky fractional ideal of R is FJS -invertible. (2) Every -invertible I ∈ K(R) is FJS -invertible. (3) Every quasi -invertible I ∈ K(R) is FJS -invertible. (4) All four semistar invertibility conditions in Definition 4.2.38 are equivalent. (5) R  is FJS -invertible.

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(6) (R : R R  ) = R  (or equivalently (R :T (R) R  ) = R  ). (7) The -closed FJS -invertible Kaplansky fractional ideals form a group under -multiplication. Moreover, if  is stable and R  is finitely generated as an R-module, then all of the conditions above hold. The proposition above corrects an error in [6, p. 282–283] and [67, Remark 2.22(a)], which incorrectly state that the condition (D : K D  ) = 0 for any domain D is necessary and sufficient for the -closed FJS -invertible Kaplansky fractional ideals of D to form a group. The following is an example of a semistar operation on an integral domain D with quotient field K such that (D : K D  ) = 0 and yet the -closed FJS -invertible Kaplansky fractional ideals do not form a group under -multiplication. Example 4.2.41 (cf. [67, Example 2.20 and Remark 2.22(a)]). As in [67, Example 2.20], let D be a pseudo-valuation domain with quotient field K and maximal ideal m such that V = m−1 is a DVR. (For instance, let V = l[[X ]], let m = Xl[[X ]], and let D = k + m, where l/k is a proper extension of fields.) Consider the finite type semistar operation  : I −→ I V on D. One has D  = V and (m(D  : K m)) = m(V : K m)V = V since V is a DVR. In particular, m is quasi -invertible. On the other hand, one has m = mV = m, so mm−1 = m and therefore (mm−1 ) = m = m, so m is not FJS -invertible. Therefore one must have (D : K D  ) = D  , that is D  cannot be FJS -invertible. Indeed, one has (D : K D  ) = (D : K V ) = m = (0), and yet also (D : K D  ) = (D : K V )V = mV = m  D. Thus, contrary to claims made in [6, p. 282–283] and [67, Remark 2.22(a)], D  is not FJS -invertible, and thus the -closed FJS -invertible Kaplansky fractional ideals do not form a group under -multiplication, even though (D : K D  ) = 0. We note also that m is strictly -invertible. Indeed, since m = πV for some π ∈ V , letting I = π D, we see that m = I V = I  , where I ⊆ m is nonzero and principal, hence invertible, hence FJS -invertible. Thus, m is strictly -invertible but not FJS -invertible. Example 4.2.42. Let D be a UFD and S any Prüfer overring of D that is not Bézout, and let  be the semistar operation I −→ I S on D. Let J be any finitely generated ideal of S that is not principal. (For example, we may let D = Z[X ], S = Int(Z), and J = (2, X ) Int(Z).) Since J is invertible in S, one has J (S : K J ) = S, where K = T (D) = T (S), and therefore J ∈ K(R) is -closed and quasi -invertible. We claim that J is not -invertible. Suppose otherwise. Then there exists I ∈ K(R) such that J = I S and I I −1 S = S, where I −1 = (R : K I ). Clearly I and I −1 are nonzero, so I is regular and fractional, whence I v = I t = (x) is principal since D is a UFD. Therefore I −1 = (1/x) is also principal, whence S = I I −1 S = (1/x)I S, so that J = I S = x S, contradicting the fact that J is not principal. The main deficiencies in the notion of FJS -invertibility considered above leads one to consider more generally the Kaplansky fractional ideals of R that are FJS -invertible up to -closure, which are precisely what we have called the -invertible

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Kaplansky fractional ideals. Those that are -closed then indeed form a group under -multiplication. Since a -closed Kaplansky fractional ideal is -invertible if and only if it is strictly -invertible, these two notions lead to the same groups, thus motivating the following definition. Definition 4.2.43. Let  be a semistar operation on a ring R. Then Inv (R) of R denotes the group of -closed (strictly) -invertible Kaplansky fractional ideals of R under -multiplication. Although the -closed quasi -invertible Kaplansky fractional ideals of R always form a group, denoted QInv (R), this group coincides with the group Inv˙ (R  ), and it contains the group Inv (R) as a subgroup. The containment Inv (R) ⊆ Inv˙ (R  ) = QInv (R) may be proper, since in Example 4.2.42 above the -closed regular fractional ideal J of the domain R = D is quasi -invertible but not -invertible. By contrast, as shown in Proposition 4.2.40, an analogue of Inv (R) for FJS -invertibility exists if and only if R  is FJS -invertible, but in that case that group coincides with Inv (R) since then all four invertibility conditions are equivalent. Lemma 4.2.44. Let  be a semistar operation on a ring R, and let I ∈ K(R). Each of the following conditions implies the next. (1) (2) (3) (4)

I I I I

is FJS -invertible. is strictly -invertible. is -invertible. is quasi -invertible.

Moreover, if  is stable and I is finitely generated, then all four conditions are equivalent. Proof. The given implications are clear. Suppose that  is stable and I is finitely generated, and suppose that I is quasi -invertible, so that (I (R  : K I )) = R  . Then one has (I I −1 ) = (I (R : K I ) ) = (I (R  : K I )) = R  , whence I is FJS -invertible.  Lemma 4.2.45. Let  be a semistar operation on a ring R, and let I ∈ K(R). (1) If I ∈ I(R), then I  = R  if and only if I  = R  . (2) I is FJS -invertible if and only if I is FJS -invertible. By the previous two lemmas, one has the following. Proposition 4.2.46. Let  be a semistar operation on a ring R, and let I ∈ K(R) be finitely generated. The following conditions are equivalent. (1) (2) (3) (4) (5)

I I I I I

is FJS -invertible. is FJS -invertible. is strictly -invertible. is -invertible. is quasi -invertible.

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Definition 4.2.47. Let  be a semistar operation on a ring R. (1) R is FJS -Prüfer if every finitely generated regular I ∈ K(R) is FJS -invertible. (2) R is strictly -Prüfer if every finitely generated regular I ∈ K(R) is strictly -invertible. (3) R is -Prüfer if every finitely generated regular I ∈ K(R) is -invertible. (4) R is quasi -Prüfer if every finitely generated regular I ∈ K(R) of R is quasi -invertible. Clearly one has (1) ⇒ (2) ⇒ (3) ⇒ (4), with equivalences if (R : K R  ) = R  or if  is stable. The proof of the following lemma is left as an exercise. Lemma 4.2.48. Let  be a semistar operation on a ring R. Then R is quasi -Prüfer if and only if R  is ˙ -Prüfer. We note the following consequence of Proposition 4.2.46 and Lemma 4.2.48. Proposition 4.2.49. Let  be a semistar operation on a ring R. The following conditions are equivalent. (1) (2) (3) (4) (5) (6)

R is FJS -Prüfer. R is FJS -Prüfer. R is strictly -Prüfer. R is -Prüfer. R is quasi -Prüfer. ˙ R  is -Prüfer.

Corollary 4.2.50. Let  be a semistar operation on a ring R. The following conditions are equivalent. (1) (2) (3) (4) (5) (6)

R is FJS t -Prüfer. R is FJS w -Prüfer. R is strictly w -Prüfer. R is w -Prüfer. R is quasi w -Prüfer. R w is (˙w )-Prüfer.

Lemma 4.2.51. Let  be a semistar operation on a ring R, and let I ∈ K(R). Consider the following conditions. (1) I is FJS -invertible. (2) I Rp is regular and principal (or invertible) in Rp for every -prime ideal p of R. (3) For every -prime p of R there exists an a ∈ T (R) such that [I ]R[p] = [(a)]R[p] . One has (1) ⇒ (2) ⇒ (3); if I is regular, then one has (2) ⇔ (3); and if I is finitely generated and regular and every proper regular -ideal of R is contained in some -prime, then all three conditions are equivalent.

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Proof. Since [I ]R[p] = [J ]R[p] if and only if I Rp = J Rp , for all I, J ∈ K(R) and all prime ideals p, one has (2) ⇒ (3), and (2) ⇔ (3) if I is regular. Suppose that I is FJS -invertible, and let p be a -prime ideal of R. Since (I I −1 ) = R  , one has I I −1 Ę p. It follows that Rp = I I −1 Rp = (I Rp )(I −1 Rp ) and therefore I Rp is invertible in Rp (since also T (R)p ⊆ T (Rp )). Suppose, conversely, that I Rp is invertible in Rp for every -prime ideal p of R, where I is finitely generated and regular and every proper regular -ideal of R is contained in some -prime. Then I I −1 Rp = (I Rp )(I −1 Rp ) = (I Rp )(I Rp )−1 = Rp , whence I I −1 Ę p, for all -prime ideals p  of R, so that (I I −1 ) = R  , whence I is FJS -invertible. Theorem 4.2.52. Let  be a semistar operation on a ring R. The following conditions are equivalent. (1) (2) (3) (4) (5)

R is FJS t -Prüfer. R is w -Prüfer. R w is (˙w )-Prüfer. (R[p] , [p]R[p] ) is a valuation pair of T (R) for every t -maximal ideal p of R. I Rp is principal (or invertible) in Rp for every finitely generated regular fractional ideal I and every t -maximal (or t -prime) ideal p of R.

Proof. Statements (1)–(3) are equivalent by Proposition 4.2.49. Let K = T (R), and let R  = R w and  = (˙w ). Since  is unital, stable, and of finite type, by Theorem   , [q]R[q] ) is a valuation pair of T (R) 3.8.4 the ring R  is  -Prüfer if and only if (R[q]   for every  -maximal ideal q of R , that is, for every ideal q of R w that is maximal among the proper w -closed ideals of R w . For any such ideal q, the ideal p = q ∩ R is   , [q]R[q] ) = (R[p] , [p]R[p] ) by Corollary 4.2.26, and conversely t -maximal and (R[q] any t -maximal ideal p of R is of the form q ∩ R for a unique  -maximal ideal q = pw of R  . It follows, then, that statements (3) and (4) are equivalent. Finally, statements (1) and (5) are equivalent by Lemma 4.2.51.  A domain D is said to be a PMD if R is FJS t -Prüfer, that is, if every nonzero finitely generated fractional ideal of D is FJS t -invertible [64]. Corollary 4.2.53 (cf. [165, Propositions 2.1 and 2.2]). Let  be a semistar operation on an integral domain D. The following conditions are equivalent. (1) (2) (3) (4)

D is a PMD. D is w -Prüfer. D w is (˙w )-Prüfer. Dp is a valuation domain for every t -maximal ideal p of D.

A domain D is said to be a QPMD if D is quasi t -Prüfer, that is, if every nonzero finitely generated fractional ideal of R is quasi t -invertible [165]. An example of a QPMD that is not a PMD is provided in [165, Example 2.3]. Thus, a quasi t -Prüfer domain need not be t -Prüfer. Also, by [165, Lemma 2.4] or Proposition 4.2.48, a domain D is a QPMD if and only if D  is a P˙MD.

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Lemma 4.2.54. Let  be a semistar operation on a ring R, and let c : ˙ -Spec(R  ) −→ -Spec(R) denote the contraction map P −→ P ∩ R. (1) If  is stable or R  is flat over R, or more generally if every ideal of R  is -stable, then the contraction map c is a poset embedding. (2) If  is spectral, then the contraction map c is a poset isomorphism with inverse acting by p −→ p , and ˙ = [c−1 ()] is also spectral. Proof. The first statement is clear since under the given hypotheses one has (P ∩ R) = P for all P ∈ ˙ -Spec(R  ). To prove the second statement, suppose that  = [] is spectral, where  ⊆ Spec R. We must show that c is surjective. Let q ∈ -Spec(R), so that by Theorem 4.2.30 there exists a p ∈  such that q ⊆ p. Then [q]R[p] ⊆ R[p] is prime and R  ⊆ R[p] , whence Q = [q]R[p] ∩ R  is prime with Q ∩ R = q. Moreover, one has Q  = ([q]R[p] ∩ R  ) = ([q]R[p] ) ∩ R  , while ([q]R[p] ) ⊆ [q]R[p] since p ∈ , whence Q  = Q and thus Q ∈ ˙ -Spec(R  ). Therefore c is surjective. Finally, for all I ∈ K(R  ) ⊆ K(R), one easily checks that [I ]R[p] = [I ](R  )[p ] , where p = c−1 (p) = [p]R[p] ∩ R  , for all p ∈ , whence ˙ = [c−1 ()] is spectral. Theorem 4.2.55. Let  = [] be a spectral semistar operation on a ring R, where  ⊆ Spec(R). The following conditions are equivalent. R is FJS -Prüfer. R is strictly -Prüfer. R is -Prüfer. R is quasi  -Prüfer. R  = {R[p] : p ∈ } is ˙ -Prüfer.  ⊆ Ess(R), that is, (R[p] , [p]R[p] ) is a valuation pair of T (R) for all p ∈ . I Rp is principal (or invertible) in Rp for every finitely generated regular fractional ideal I of R and every prime p ∈  (or every -prime p of R). (8) For every finitely generated regular ideal I of R and every prime p ∈  (or every -prime p of R) there exists an a ∈ R such that [I ]R[p] = [(a)]R[p] . (1) (2) (3) (4) (5) (6) (7)

Moreover, the contraction map c : ˙ -Spec(R  ) −→ -Spec(R) is a poset isomorphism with inverse acting by p −→ p , and the unital semistar operation ˙ = [c−1 ()] on R  is also spectral. In particular, if the conditions above hold, then the ring R  is essential. Proof. Since  is stable, the equivalence of statements (1)–(5) follows from Proposition 4.2.49. Moreover, since (R  )[p ] = R[p] and [p ](R  )[p ] = [p]R[p] for all p ∈ , one has (5) ⇔ (6) by Theorem 3.11.3 (5). One also has (1) ⇔ (7) ⇔ (8) by Lemma 4.2.51. The rest of the theorem then follows from Lemma 4.2.54. Applying the theorem to the case where  is a singleton, we obtain the following. Corollary 4.2.56. Let R be a ring and p a prime ideal of R, and let [{p}] be the semistar operation I −→ [I ]R[p] on R. The following conditions are equivalent.

4.2 Nonunital Semistar Operations

315

(R[p] , [p]R[p] ) is a valuation pair of T (R). R is [{p}] -Prüfer. ˙ )-Prüfer, where ([{p}] ˙ ) : I −→ [I ]R[p] for all I ∈ K(R[p] ). R[p] is ([{p}] I Rp is principal (or invertible) in Rp for every finitely generated regular fractional ideal I of R. (5) For every finitely generated regular ideal I of R there exists an a ∈ R such that [I ]R[p] = [(a)]R[p] . (6) p is not regular, or [p]R[p] is the unique w -maximal ideal of R[p] and for every finitely generated regular ideal I of R[p] there exists an a ∈ R[p] such that I w = (a)w .

(1) (2) (3) (4)

Moreover, (R, p) is a valuation pair of T (R) if and only if the equivalent conditions above hold and R = R[p] . ˙ ) in statement (3) of the corollary is the Note that the semistar operation ([{p}] w -operation on the valuation ring R[p] . Corollary 4.2.57. Let R be a ring and p a regular prime ideal of R. Then (R, p) is a valuation pair of T (R) if and only if p is the unique w -maximal ideal of R and for every finitely generated regular ideal I of R there exists an a ∈ R such that I w = (a)w . We may also apply the theorem to the set  = Ess(R) of all essential primes of semistar operation [Ess(R)] : I −→  R. For any ring R, let ε denote the spectral {[I ]R[p] : p ∈ Ess(R)}. Since Ess(R)reg ⊆ t -Spec(R)reg , one has w ε . Corollary 4.2.58. For any ring R, one has the following. (1) ε is the smallest spectral semistar operation  on R such that R is FJS -Prüfer (resp., strictly -Prüfer, -Prüfer, quasi -Prüfer). (2) ε -Spec(R) = Ess(R), and the contraction map ε˙ -Spec(Rε ) −→ ε -Spec(R) is a poset isomorphism. ε (3) The unital  semistar operation ε˙ on R is spectral. ε (4) R = {R[p] : p ∈ Ess(R)} is ε˙ -Prüfer and essential. (5) R is a PVMR if and only if ε w , if and only if ε t . (6) R is Prüfer if and only if ε d , if and only if ε = d . (7) R is essential if and only if ε is unital, if and only if ε  v . Definition 4.2.59. Let R be a ring. (1) Let O a collection of overrings  of R. We let O denote the semistar operation on R acting by I −→ I O = {I S : S ∈ O} for all I ∈ K(R). (2) Let  ⊆ Spec R a set of prime ideals of R. We let  denote the semistar operation on R acting by I −→ I  = {I R[p] : p ∈ } for all I ∈ K(R). The two theorems below generalize Theorems 3.2.3 and 3.11.10, respectively, to nonunital semistar operations. Theorem 4.2.60. Let R be a ring and O a collection of overrings of R.

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(1) If every S ∈ O is flat over R, then the semistar operation O is stable. (2) If I ∈ K(R) is FJS O -invertible, then I S is invertible in S for all S ∈ O; moreover, the converse holds if I is finitely generated and every S ∈ O isflat over R. (3) If R is FJS O -Prüfer, then S is Prüfer for all S ∈ O; moreover, the converse holds if every S ∈ O is flat over R. Theorem 4.2.61. Let R be a ring and  a subset of Spec R. (1) If R[p] is flat over R for all p ∈ , then the semistar operation  on R is stable. (2) If I ∈ K(R) is FJS  -invertible, then I R[p] is invertible in R[p] for all p ∈ ; moroever, the converse holds if I is finitely generated and R[p] is flat over R for all p ∈ . (3) The following conditions are equivalent. (a) R is FJS  -Prüfer. (b) R[p] is Prüfer for all p ∈ . (c)  ⊆ SEss(R). (d) R is strictly  -Prüfer (or  -Prüfer, or quasi  -Prüfer) and R[p] is flat over R  for all p ∈ . (e) R  = {R[p] : p ∈ } is (˙ )-Prüfer and R[p] is flat over R for all p ∈ . (f) I R[p] is invertible in R[p] for every finitely generated regular fractional ideal I of R and every prime p ∈ . (g) For every finitely generated regular ideal I of R and every prime p ∈  there exists an a ∈ R such that I R[p] = [(a)]R[p] . Corollary 4.2.62. Let R be a ring and p a prime ideal of R, and let {p} be the semistar operation I −→ I R[p] on R. The following conditions are equivalent. R is FJS {p} -Prüfer. R[p] is Prüfer. (R[p] , [p]R[p] ) is a Prüfer valuation pair of T (R). R is strictly {p} -Prüfer (or {p} -Prüfer, or quasi {p} -Prüfer) and R[p] is flat over R. ˙ )-Prüfer and R[p] is flat over R. (5) R[p] is ({p} (6) I R[p] is invertible in R[p] for every finitely generated regular fractional ideal I of R. (7) For every finitely generated regular ideal I of R there exists an a ∈ R such that I R[p] = [(a)]R[p] .

(1) (2) (3) (4)

Moreover, (R, p) is a Prüfer valuation pair of T (R) if and only if the equivalent conditions above hold and R = R[p] . Corollary 4.2.63. Let R be a ring and p a regular prime ideal of R. Then (R, p) is a Prüfer valuation pair of T (R) if and only if p is the unique w -maximal ideal of R and for every finitely generated regular ideal I of R there exists an a ∈ R such that I = (a)w . Let  be a semistar operation on a ring R. Consider also the following conditions.

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317

(1) R is FJS -Dedekind if every regular fractional I ∈ K(R) is FJS -invertible. (2) R is strictly -Dedekind if every regular fractional I ∈ K(R) is strictly -invertible. (3) R is -Dedekind if every regular fractional I ∈ K(R) is -invertible. (4) R is quasi -Dedekind if every regular fractional I ∈ K(R) of R is quasi -invertible. Clearly one has (1) ⇒ (2) ⇒ (3) ⇒ (4), with equivalences if (R : K R  ) = R  . Example 4.2.64. The following examples show, respectively, that a strictly -Dedekind (or strictly -Prüfer) domain need not be FJS -Prüfer, and a quasi -Prüfer domain need not be -Prüfer. (1) Let D and V be domains as in Example 4.2.41, and let  = {V } : I −→ I V . Then D is quasi -Dedekind since D  = V is a DVR. In fact, D is strictly -Dedekind. Indeed, if I is any nonzero ideal of D, then I V = aV for some nonzero a ∈ I (more specifically, for any a ∈ I of least valuation), so that I  = I V = J V = J  , where J = a D ⊆ I is invertible and therefore FJS -invertible, whence I is strictly -invertible. However, D is not FJS -Dedekind since D  = V is regular and fractional but not FJS -invertible. Moreover, if V is finitely generated as a D-module (e.g., if l/k is a finite extension of fields), then D is not FJS -Prüfer. (2) Let D and S be domains as in Example 4.2.42, and let  = {S} : I −→ I S. Then D is quasi -Prüfer since D  = S is Prüfer, but D is not -Prüfer since D has a finitely generated regular fractional ideal J that is not -invertible. Note also that S is not flat over D (and  is not stable), and therefore the flatness assumption in statement (3) of Theorem 4.2.60 can not be removed. Open problem 4.2.65. Generalize our results on -Dedekind rings in Sections 2.6 and 2.11 to FJS -Dedekind, strictly -Dedekind, -Dedekind, and/or quasi -Dedekind rings. Open problem 4.2.66. Generalize further results in Chapters 2 and 3, as well as in the extensive literature on semistar operations, to nonunital semistar operations on rings with zerodivisors.

4.3 Semiprime Operations Recall that a semiprime operation on a ring R is a closure operation s on the poset I(R) of all ideals of R such that I s J s ⊆ (I J )s for all I, J ∈ I(R) (or equivalently such that a I s ⊆ (a I )s for all I ∈ I(R) and all a ∈ R). Semiprime operations abound in commutative algebra. Example 4.3.1. Let R be a ring. The following operations are semiprime operations on R.

318

4 Semiprime, Star, and Semistar Operations …

(1) The operation sprim : I −→ I  ∩ R, for any semistar operation  on R. (2) The restriction to I(R) of any unital semistar operation or fractional star operation on R. (3) The operation I −→ I + J , for any ideal J of R. (4) The operation I −→ (J : R (J : R I )), for any ideal J of R [119, Examples 1(4)]. (5) The operation I −→ ann R (M/I M) √ = (I M : R M),n for any R-module M. I = {a ∈ R : a ∈ I for some n ∈ Z>0 } = (6) The radical operation I  − →  {p ∈ Spec(R) : p ⊇I }. (7) The operation I −→ {p ∈  : p ⊇ I }, for any set  of prime ideals of R. In fact, for any such semiprime operation s, one has (I J )s = (I ∩ J )s = I s ∩ J s for all ideals I and J of R, whereas for an arbitrary semiprime operation s one has (I J )s ⊆ (I ∩ J )s ⊆ I s ∩ J s and the containments may be proper. (8) The operation I −→ ϕ−1 (ϕ(I )S), for any ring homomorphism ϕ : R −→ S. (9) The operation I −→ {I + Jλ : λ ∈ }, for any collection {Jλ : λ ∈ } of ideals of R. (Apply the previous example to the ring homomorphism R −→  λ∈ R/Jλ .) (10) The operation I −→ (I : R J ∞ ), for any finitely generated ideal J of R, where ∞ n n (I : R J ) = ∞ n=0 (I : R J ) = {a ∈ R : a J ⊆ I for some n ∈ Z0 } for any ideal I is the J -saturation of I [55, Example 2.1.2(4)]. Saturation is important in local cohomology: one has H J0 (R/I ) = (I : R J ∞ )/I for any ideal I of R [182, Section 4.6].  (11) The operation I −→ (I : R J ∞ ) = {(I : R H ∞ ) : H ⊆ J is f.g.}, for any ideal J of R. (12) The integral closure operation I −→ I [178], which is studied in Section 4.4. (13) The tight closure operation I −→ I ∗ , defined if R is Noetherian and contains a field [102, 109], and which is studied for Noetherian rings of prime characteristic in Section 4.6. (14) The plus closure operation I −→ I + , defined if R is a domain [109]. Proposition 4.3.2. Let R be a ring. The poset of all semiprime operations on R is complete. The infimum inf  of a set  of semiprime operations on R acts by I −→ I inf  =

 {I s : s ∈ }

and the supremum sup  of  acts by I −→ I sup  =

 {J ⊇ I : J ∈ I(R), J s = J for all s ∈ }

for all I ∈ I(R). Moreover, for all I ∈ I(R), one has I sup  = I if and only if I s = I for all s ∈ . Definition 4.3.3. Let R be a ring. The smallest semiprime operation d : I −→ I on R is called the trivial semiprime operation on R. The largest semiprime operation on R is denoted e and acts by e : I −→ R.

4.3 Semiprime Operations

319

Proposition 4.3.4. Let R be a ring. (1) For any J ∈ I(R), the operation v(J ) defined by I v(J ) = (J : R (J : R I )) for all I ∈ I(R) is the largest semiprime operation s on R such that J s = J . Moreover, I ∈ I(R) is v(J )-closed if and only if I = (J : R H ) for some H ∈ I(R). (2) Let J ⊆ I(R). The operation v(J ) = inf{v(J ) : J ∈ J } is the largest semiprime operation s on R such that J s = J for all J ∈ J . For any semiprime operation s on R, one has s  v(J ) if and only if J s = J for all J ∈ J . (3) One has s = v(I(R)s ) = inf{v(J ) : J ∈ I(R)s } for any semiprime operation s on R. Thus, every semiprime operation on R is of the form v(J ) for some J ⊆ I(R). Definition 4.3.5. A semiprime operation s on a ring R is reduced if (0)s = (0). Corollary 4.3.6. Let R be a ring, and let s be a semiprime operation on R. (1) r = v((0)) : I −→ ann R (ann R (I )) is the largest reduced semiprime operation on R. (2) sred = inf{s, r} is the largest reduced semistar operation on R less than or equal to s. Definition 4.3.7. A semiprime operation s on a ring R is of finite type if Is =



{J s : J ∈ I(R) is f.g. and J ⊆ I }

for all I ∈ I(R). Definition 4.3.8. A semiprime operation s on a ring R is stable if (I : R J )s = (I s : R J ) for all I, J ∈ I(R) with J finitely generated. Proposition 4.3.9. A semiprime operation s on a ring R is stable if and only if (I : R x)s = (I s : R x) for all I ∈ I(R) and all x ∈ R. Moreover, if s is a stable semiprime operation on R, then (I ∩ J )s = I s ∩ J s for all I, J ∈ I(R). √ Example 4.3.10. The radical operation s : I −→ I on any ring R satisfies (I J )s = (I ∩ J )s = I s ∩ J s for all I, J ∈ I(R) but is stable if and only if every ideal of R is a radical ideal, if and only if R is von Neumann regular. Consequently, the condition (I ∩ J )s = I s ∩ J s is necessary but not sufficient for a semiprime operation s to be stable. Definition 4.3.11. For any semiprime operation s on a ring R, we let I st = Is =

 {J s : J ∈ I(R) is f.g. and J ⊆ I },

 {(I : R J ) : J ∈ I(R), J s = R} = {x ∈ R : (I : R x)s = R}, I sw =

 {(I : R J ) : J ∈ I(R) is f.g. and J s = R},

320

4 Semiprime, Star, and Semistar Operations …

for all I ∈ I(R). The operations st , s, and sw are semiprime operations on R. Proposition 4.3.12. Let s be a semiprime operation on a ring R. (1) st is the largest finite type semiprime operation less than or equal to s. (2) s is largest stable semiprime operation less than or equal to s. (3) sw is the largest stable finite type semiprime operation less than or equal to s, and one has sw = st  (s)t . In particular, one has sw  st  s and sw  s  s. Recall that if  is a semistar operation on a ring R, then the semiprime operation on R induced by  is the semiprime operation sprim : I −→ I  ∩ R. We note the following. Proposition 4.3.13. Let R be a ring. (1) For any collection  of semiprime operations on R, one has inf  = inf{s : s ∈ }. Consequently, the infimum of a collection of stable semiprime operations on R is stable. (2) For any finite collection  of semiprime operations on R, one has (inf )t = inf{st : s ∈ }. Consequently, the infimum of a finite collection of finite type semiprime operations on R is of finite type. (3) For any semistar operation  on R, one has (sprim )t = sprimt and sprim = sprim . Consequently, if  is of finite type (resp., stable), then sprim is of finite type (resp., stable). (4) Let S be an overring of a ring R. The operation I −→ I S ∩ R on I(R) is a finite type semiprime operation on R, and it is stable if S is flat over R. (5) Let O be a collection of overrings  of R. If S is flat over R for all S ∈ O, then the operation sO : I −→ R ∩ {I S : S ∈ O} on I(R) is a stable semiprime operation on R. If O is finite, then sO is of finite type. The following proposition shows that there is a natural correspondence between the stable semiprime operations and the stable semistar operations on a ring. Proposition 4.3.14. Let R be a ring with total quotient ring K . For any semiprime operation s on R, let sstar s be the operation on K(R) acting by sstar s : I −→

 {(I : K J ) : J ∈ I(R), J s = R}

for all I ∈ K(R), which is a stable semistar operation on R. (1) One has sprimsstar s = s for every semiprime operation s on R. (2) One has sstar sprim =  for every semistar operation  on R. (3) The association  −→ sprim defines an isomorphism from the poset of all stable semistar operations on R to the poset of all stable semiprime operations on R, with inverse s −→ sstar s .

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321

The proof of the proposition is left as an exercise. It follows from the proposition that every stable semiprime operation on a ring R is induced by a unique stable semistar operation on R. We now address the more difficult problem of determining which semiprime operations are induced by some semistar operation, or alternatively by some unital semistar operation. For this purpose, we introduce the following definition. Definition 4.3.15. A semiprime operation on a ring R is rigid if it is induced by some semistar operation on R, that is, if it is of the form sprim for some semistar operation  on R. Thus, we wish to characterize the rigid semiprime operations. Definition 4.3.16. A prenucleus on an ordered magma M is a preclosure + on M such that x + y  (x y)+ and x y +  (x y)+ for all x, y ∈ M. Clearly, a nucleus is equivalently an idempotent prenucleus. It sometimes happens that one can define a natural preclosure that is not a closure operation. However, our next two propositions can often be used to repair this deficiency. For their proofs, see Proposition 6.2.4 in Chapter 6. Proposition 4.3.17. Let S be a bounded complete poset, and let + be a preclosure on S that is bounded above by some closure operation on S. (1) There exists a smallest closure operation +∞ on S that is larger than +, and ∞ ∞ one has x + = inf{y ∈ S : y  x and y + = y} for all x ∈ S and S + = {x ∈ S : x + = x}. 0 α+1 α (2) Define x + = x, and define x + = (x + )+ for all successor ordinals α + 1 α β and x + = sup{x + : β < α} for all limit ordinals α, for all x ∈ S. One has +∞ = +α for all sufficiently large ordinals α. (3) If + is a prenucleus on a bounded complete and residuated ordered magma M = S, then +∞ is the smallest nucleus on M that is larger than +. Definition 4.3.18. For any preclosure + on a bounded complete partially ordered set S, the idempotent hull +∞ of + is the smallest closure operation on S that is larger than +, which exists by Proposition 4.3.17. Proposition 4.3.19. Let R be a ring, and let p be a preclosure on K(R). (1) There exists a smallest closure operation p ∞ on K(R) that  is larger than p , p∞ p p∞ and one has K(R) = {I ∈ K(R) : I = I } and I = {J ∈ K(R) : J ⊇ I and J p = J } for all I ∈ K(R). 0 α+1 α (2) Define  I p = I , and define I p = (I p ) p for all successor ordinals α + 1 and α β I p = {I p : β < α} for all limit ordinals α, for all I ∈ K(R). One has p ∞ = p α for all sufficiently large ordinals α. (3) If p is a prenucleus on K(R), then p ∞ is the smallest semistar operation R larger than p .

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4 Semiprime, Star, and Semistar Operations …

We also note an analogue for semiprime operations. Proposition 4.3.20. Let R be a ring, and let p be a preclosure on I(R). (1) There exists a smallest closure operation p∞ on I(R)  that is larger than p, and one ∞ ∞ has I(R)p = {I ∈ I(R) : I p = I } and I p = {J ∈ I(R) : J ⊇ I and J p = J } for all I ∈ I(R). 0 α+1 α (2) Define  I p = I , and define I p = (I p )p for all successor ordinals α + 1 and α β I p = {I p : β < α} for all limit ordinals α, for all I ∈ I(R). One has p∞ = pα for all sufficiently large ordinals α. (3) If p is a prenucleus on I(R), then p∞ is the smallest semiprime operation on R larger than p. Definition 4.3.21. Let R be a ring, and let p be a self-map of I(R). We denote by d p the smallest semistar operation  on R such that I p ⊆ I  ∩ R for all ideals I of R. The following proposition is immediate. Proposition 4.3.22. Let R be a ring, and let p be a self-map of I(R). (1) (2) (3) (4)

d sprim   for any semistar operation  on R. d sprimd p = d p . sprimd p is the smallest rigid semiprime operation greater than or equal to p. p is a rigid semiprime operation if and only if p = sprimd p .

We would like to find a workable characterization of the rigid semiprime operations, along with some way of computing d p . It turns out that the following condition is sufficient (but not necessary). Definition 4.3.23. A semiprime operation s on a ring R is regular if (a I )s = a I s for all regular a ∈ R and and all I ∈ I(R). Remark 4.3.24. A semiprime operation s on a ring R is said to be a prime operation on R if (a I )s = a I s for all a ∈ R and and all I ∈ I(R). This condition, however, is much more restrictive than regularity. Lemma 4.3.25. Let ∗ : F(R)−→ F(R) be a fractional star operation on a ring R. For all I ∈ K(R), set I π = {J ∗ : J ∈ F(R), J ⊆ I }. Then π : K(R) −→ K(R) is a prenucleus on K(R), and π ∞ is the smallest unital semistar operation on R that restricts to ∗. Moreover, if ∗ is of finite type, then π = π ∞ is the unique finite type unital semistar operation on R that restricts to ∗. If a semiprime operation s is the restriction of some unital semistar operation, then clearly s must be regular. The converse is also true. Theorem 4.3.26. Let s be a regular semiprime operation on a ring R. (1) There exists a largest unital semistar operation e s on R that restricts to s. (2) d s is the smallest unital semistar operation on R that restricts to s.

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(3) A unital semistar operation  on R restricts to s if and only if d s    e s . (4) For each fractional I ∈ K(R), choose a regular  c ∈ R such that cI ⊆ R, and set I π = c−1 (cI )s . For all I ∈ K(R), set I π = {J π : J ∈ K(R) is fractional and ∞ J ⊆ I }. Then d s is equal to the operation π ∞ on K(R) given by I −→ I π =  π ds es π {J ∈ K(R) : J ⊇ I and J = J }. Moreover, one has I = I = I for all fractional I ∈ K(R). type semis(5) If s is of finite type, then d s = (ee s )t , and d s is the unique unital finite tar operation on R that restricts to s. Moreover, one has I d s = {J π : J ∈ K(R) is f.g. and J ⊆ I } for all I ∈ K(R). Proof. For each I ∈ F(R), choose a regular c ∈ R such that cI ⊆ R, and set I ∗ = c−1 (cI )s . One easily checks that the map ∗ : F(R) −→ F(R) is a well-defined fractional star operation on R that restricts to s. Consequently, the proposition follows from Lemma 4.3.25 and Proposition 4.1.20.  Corollary 4.3.27. A semiprime operation on a ring R is regular if and only if it is the restriction of some unital semistar operation on R. In particular, every regular semiprime operation on R is rigid. Corollary 4.3.28. Let R be a ring. The association s −→ d s , with inverse  −→ |I(R) , defines an isomorphism between the poset of all regular finite type semiprime operations s on R and the poset set of all finite type unital semistar operations  on R. We wish to generalize Theorem 4.3.26 to possibly nonunital semistar operations. Definition 4.3.29. Let R be a ring. For any preclosure p on K(R), we let pcl p denote the preclosure pcl p : I −→ I p ∩ R on I(R). Definition 4.3.30 ([54, Definition 2.2]). Let R be a ring. A preclosure p on I(R) is standard if ((cI )p : R c) = I p for all ideals I of R and all regular c ∈ R. Every rigid semiprime operation is standard. Indeed, if s = sprim for some semistar operation  on R, then I s ⊆ ((cI )s : R c) = ((cI ) ∩ R : R c) = (cI  ∩ R : R c) ⊆ (cI  : K c) ∩ R = I  ∩ R = I s

for all ideals I of R and all regular c ∈ R. More generally, if p is a prenucleus on K(R), then pcl p is a standard prenucleus on I(R). Definition 4.3.31 (cf. [54, Definition 3.4]). Let R be a ring and p a prenucleus on I(R). Let  I πp = c−1 (cI )p ∈ K(R) c∈R reg ∩I −1

for all I ∈ F(R), and let I πp = for all I ∈ K(R).

 {J π p : J ∈ F(R) and J ⊆ I }

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The significance of π p for any standard prenucleus on I(R) is that it is the smallest prenucleus p on K(R) such that p = pcl p . Moreover, if s is a regular semiprime operation on R, then π s coincides with the operation π defined in Theorem 4.3.26. Theorem 4.3.32. Let R be a ring and p a standard prenucleus on I(R). (1) The operation π p : K(R) −→ K(R) is well-defined. For all fractional I ∈ K(R), if x = a/u with a, u ∈ R and u regular, and if c is any regular element of R with cI ⊆ R, then x ∈ I π p if and only if ac ∈ (ucI )p . (2) π p is the smallest prenucleus p on K(R) such that p = pcl p (that is, such that I p = I p ∩ R for all ideals I of R). (3) If p is standard finite type prenucleus on I(R), then π p is the unique finite type prenucleus p on K(R) such that p = pcl p . π p )∞ is the smallest semistar operation  on R such that p  sprim . (4) d p = (π (5) sprimd p is the smallest rigid semiprime operation greater than or equal to p. (6) If s = p is a standard finite type semiprime operation on R, then d s = π s , and d s is the unique finite type semistar operation  on R such that s = sprim . Proof. The proof of statement (1) is left as an exercise. It follows easily that π p is a preclosure on K(R), and one has I πp ∩ R =

 c∈R reg ∩I −1

(c−1 (cI )p ∩ R) =



((cI )p : R c) = I p

c∈R reg

for all I ∈ I(R) since p is standard, whence pclπ p = p. We show that π p is a prenucleus. Let I, J ∈ K(R). Suppose first that I and J are fractional. Then cI ⊆ R and d J ⊆ R for some regular c, d ∈ R, so that cd I J ⊆ R. Let x = a/u and y ∈ b/v with a, b, u, v ∈ R and u, v regular. Suppose that x ∈ I π p and y ∈ J π p . Then ac ∈ (ucI )p and bd ∈ (vd J )p , and therefore (ab)(cd) = (ac)(bd) ∈ (ucI )p (vd J )p ⊆ (ucI vc J )p = (uvcd I J )p . Therefore x y = (ab)/(uv) ∈ (I J )π p . Thus I J π p ⊆ (I J )π p if I, J are fractional. From this one can show easily that I J π p ⊆ (I J )π p if just I is fractional, and then finally for all I, J . Thus, π p is a prenucleus. Now, let p be any prenucleus on K(R) with p = pcl p . Let J ∈ K(R) be fractional, and choose a regular c ∈ R such that c J ⊆ R. Let x = a/u, where a, u ∈ R with u regular. If x ∈ J π p , then ac ∈ (uc J )p = (uc J ) p ∩ R = uc J p ∩ R, so ac ∈ uc J p and therefore x = a/u ∈ J p . Therefore J π p ⊆ J p for all fractional J ∈ K(R). It follows, then, that I π p ⊆ I p for all I ∈ K(R), that is, π p  p . This proves (2). π s )t is a prenucleus Next, we prove (3). Suppose that p is of finite type. Clearly, (π π s )t . Therefore π s = (π π s )t is of finite on K(R) with p = pt = pcl(ππ s )t and thus π s  (π type. Let p be any finite type prenucleus on K(R) such that p = pcl p . Then π s  p . Let I ∈ K(R) be finitely generated (hence fractional). Choose a regular c ∈ R such that cI ⊆ R. Let x = a/u, where a, u ∈ R with u regular. If x ∈ I p , then a ∈ u I p , so ac ∈ ucI p ∩ R = (ucI ) p ∩ R = (ucI )p and therefore x ∈ I π p . Therefore I p ⊆ I π p for all finitely generated I ∈ K(R). Thus, since p is of finite type, it follows that p  π p and therefore p = π p . This proves (3).

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Recall that for any prenucleus p on K(R) the operation  = p ∞ is the smallest π p )∞ , so  is the smallest semistar operation semistar operation   p . Let  = (π larger than π p . Clearly sprim = pcl  pclπ p = p. Let  be any semistar operation on R with sprim  p. Since sprim is a standard semiprime operation, one has   π p )∞ = . This proves (4), and then (5) follows π sprim  π p and therefore   (π easily. Finally, to prove (6) we must show that  = π s is idempotent (hence a semistar operation) if s = p is idempotent (hence a semiprime operation) and of finite type. For this we follow the proof of [54, Proposition 3.5]. Let x = a/u ∈ (I  ) . Then x ∈ J  for some finitely generated J = (x1 , . . . , xn ) ∈ K(R) with J ⊆ I  . Since each xi ∈ I  , one has xi ∈ Hi for some finitely generated Hi ∈ K(R) with Hi ⊆ I . Let H = H1 + · · · + Hn . Since H and J are finitely generated, there exists a regular c ∈ R such that cH ⊆ R and c J ⊆ R. Let ai = cxi ∈ R for all i. Since xi = ai /c ∈ Hi = Hiπ s , one has ai c ∈ (ccHi )s and therefore ai ∈ (cHi )s for all i. Hence n

s n   s (cHi ) ⊆ cHi = (cH )s . c J = (a1 , . . . , an ) ⊆ i=1

i=1

But then since x = a/u ∈ J  , we have ac ∈ (uc J )s ⊆ (u(cH )s )s = (ucH )s . Therefore x = a/u ∈ H π s , where H ∈ K(R) is finitely generated with H ⊆ I . Thus we  have x ∈ I (ππ s )t = I  . Corollary 4.3.33 (cf. [54, Theorem 3.1]). Let R be a ring. The association s −→ d s = π s , with inverse  −→ sprim , defines an isomorphism between the poset of all standard finite type semiprime operations s on R and the poset set of all finite type semistar operations  on R. Corollary 4.3.34. A finite type semiprime operation on a ring is rigid if and only if it is standard. Corollary 4.3.35. Let R be a ring. A prenucleus on I(R) is standard if and only if it is of the form pcl p for some prenucleus p on K(R). Remark 4.3.36. Statement (5) of Theorem 4.3.32, and Corollary 4.3.33, can be deduced from [54, Theorems 3.1 and 3.6 and Proposition 3.5], which are proved through slightly different means. There, the author N. Epstein does not require the nucleus condition for closures on I(R) or K(R). One needs only to check that his results descend appropriately for nuclei. Theorem 4.3.32 solves the problem of determining which finite type semiprime operations are induced by semistar operations. In Sections 4.4–4.6, we apply the theorem, respectively, to the integral closure, complete integral closure, and tight closure operations on ideals. A complete characterization of the rigid semiprime operations is provided in Theorem 5.2.48 of Section 5.2. Next, we discuss possibly nonstandard preclosures on I(R). The key tool is a result of Epstein.

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Definition 4.3.37. Let R be a ring and p a prenucleus on I(R). For any ideal I of R, let  ((cI )p : R c). I pst = c∈R reg

Proposition 4.3.38 ([54, Proposition 2.5(2)]). Let R be a ring and p a prenucleus on I(R). Then the map pst : I −→ I pst on I(R) is the smallest standard prenucleus on R larger than p. Proof. For the proof that pst is the smallest standard preclosure on R larger than p, we refer the reader to [54, Proposition 2.5(2)]. We show that pst is a prenucleus. Let I, J ∈ I(R), and let x ∈ I pst and y ∈ J pst . Then there exist regular c, d ∈ R such that cx ∈ (cI )p and dy ∈ (d J )p . Then cd x y ∈ (cI )p (d J )p ⊆ (cd I J )p , so that x y ∈ ((cd I J )p : R cd) ⊆ (I J )pst . Therefore I pst J pst ⊆ (I J )pst , so pst is a prenucleus.  Combining statements (4) and (5) of Theorem 4.3.32 with the proposition above, and noting that sprimπ p = pst for any prenucleus p on K(R), we obtain the following. Proposition 4.3.39. Let R be a ring and p a prenucleus on I(R). Then π p is the smallπ p )∞ is the smallest est prenucleus  on K(R) such that pst = sprim , and d p = (π p  semistar operation  on R such that I ⊆ I ∩ R for all I ∈ I(R) (that is, such that p  sprim ). Moreover, sprimd p is the smallest rigid semiprime operation greater than or equal to p. Finally, we note the following. Proposition 4.3.40. Let R be a ring, and let p be a prenucleus on K(R). (1) I π pcl p = I p for all fractional I ∈ K(R). (2) Suppose that R p = R. Then sprim p ∞ = (pcl p )∞ , that is, sprim p ∞ is the smallest semiprime operation s  pcl p . ∞ ∞ (3) Suppose that R p is a fractional ideal of R. Then R p = R p , and I p = I d pcl p for all fractional I ∈ K(R). In particular, sprim p ∞ = sprimd pcl p is the smallest rigid semiprime operation greater than or equal to pcl p . Proof. Let I ∈ K(R) with I fractional, say, cI ⊆ R, where c ∈ R is regular. By definition of π pcl p , one has I π pcl p = {a/u : a, u ∈ R with u regular and ac ∈ (ucI )pcl p }, where J pcl p = J p ∩ R for all J ∈ I(R). Now, a/u ∈ I π pcl p ⇔ ac ∈ (ucI )pcl p ⇔ ac ∈ ucI p ∩ R ⇔ a ∈ u I p ∩ R ⇔ a/u ∈ I p , and therefore I π pcl p = I p .

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Now, suppose that R p = R. Let I be any ideal of R. Then I pcl p = I p ∩ R = I p ∞ ∞ ∞ is also an ideal of R, so I (pcl p ) = I p = I p ∩ R = I sprim p ∞ . Therefore (pcl p )∞ = sprim p ∞ . Finally, suppose that R p is a fractional ideal of R. Then both I π pcl p and I p are fractional, and thus equal, whenever I is fractional. Therefore, transfinite iteration of the operations π pcl p and p applied to any fractional ideal of R are the same, so I d pcl p = ∞ I p for all fractional I ∈ K(R). Moreover, since J π pcl p = J p for all fractional J ∈ K(R), one has (Rπ pcl p )π pcl p =

 {J p : J ∈ K(R) is fractional and J ⊆ R p } = R p = Rπ pcl p , ∞

and therefore R p = Rd pcl p = R (ππ pcl p ) = Rπ pcl p = R p . ∞



4.4 Integral Closure of Ideals and Submodules The operation I −→ I of integral closure on the ideals of a ring is a well-studied semiprime operation that was introduced by Prüfer in 1932 [167]. In this section, we study the unique finite type semistar operation b on any ring R such that the integral closure I of any ideal I is given by I b ∩ R. For integral domains R, the semistar operation b is well-known, having been studied, for example, by Krull in [124, 125] and by Zariski and Samuel in [192, Appendix 4]. Definition 4.4.1. Let R be a ring and T an R-algebra, let I be an R-submodule of T , and let x ∈ T . (1) Mod R (T ) denotes the ordered monoid of all R-submodules of T . (2) x is integral over I (in T ) if there exists a positive integer n and elements ai ∈ I i such that x n + a1 x n−1 + · · · + an−1 x + an = 0. Such an equation is called an equation of integral dependence of x over I , and the minimal degree of such an equation is called the degree deg I (x) of x over I . We set deg I (x) = ∞ if x is not integral over I . T (3) The integral closure of I in T , denoted I b R , is the set of all elements of T that are integral over I . T (4) I is integrally closed in T if I b R = I . T (R) (5) The integral closure I b R of I in T (R) is denoted I b R , or just I b when R is understood. R (6) For any ideal I of R, the integral closure I = I b R = I b ∩ R of I in R is called the integral closure of I . (7) I is integrally closed if I = I (that is, if I is integrally closed in R). We warn the reader that the notation “R” can refer to either the integral closure of R as a ring or the integral closure of R as an ideal (which is just R), but which is intended is clear in any given context.

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Lemma 4.4.2. Let R be a ring and T an R-algebra, let I be an R-submodule of T T T , and let x ∈ T . Then x ∈ I b R if and only if x ∈ J b R for some finitely generated R-submodule J of I . T

Proof. Let x ∈ I b R , so that there is an equation x n + a1 x n−1 + · · · + an = 0 of integral dependence of x over I . Then it is straightforward to show that there exists a finitely generated R-submodule J of I such that ai ∈ J i for i = 1, 2, . . . , n. Thus T the equation is one of integral dependence over J , so x ∈ J b R .  Assuming for now that the map b : I −→ I on I(R) is indeed a semiprime operation, we have the following. Theorem 4.4.3. Let R be a ring. The semiprime operation b : I −→ I on R is standard and of finite type. Therefore there is a unique finite type semistar operation  = π b on R such that I = I  ∩ R for all ideals I of R, and it is also the smallest semistar operation  = d b on R such that I ⊆ I  ∩ R for all ideals I of R. Moreover, I  = I b for all I ∈ K(R), so b =  is a finite type semistar operation. Proof. Let I be an ideal of R and x, c ∈ R with c regular. If cx ∈ cI , then there exists a positive integer n and elements ai ∈ (cI )i = ci I i such that (cx)n + a1 (cx)n−1 + · · · + an−1 cx + an = 0. We may write ai = ci bi with bi ∈ I i for all i. Then cn (x n + b1 x n−1 + · · · + bn−1 x + bn ) = (cx)n + a1 (cx)n−1 + · · · + an−1 cx + an = 0, and therefore x n + b1 x n−1 + · · · + bn−1 x + bn = 0. This is an equation of integral dependence of x over I , and therefore x ∈ I . Therefore (cI : R c) = I . Thus the semiprime operation b is standard of finite type. Therefore, by Corollary 4.3.33, there is a unique finite type semistar operation  = π b on R such that I = I  ∩ R for all ideals I of R, and one has  = d b . Now, let I ∈ K(R) be fractional, with cI ⊆ R for some regular c ∈ R. By Theorem 4.3.32(1), one has I π b = {a/u ∈ K : a, u ∈ R with u regular and ac ∈ ucI }. Let x = a/u ∈ I π b , so that ac ∈ ucI . Then there exists a positive integer n and elements ai ∈ I i and bi = u i ci ai ∈ (ucI )i such that (ac)n + b1 (ac)n−1 + · · · + bn−1 ac + bn = 0 and thus (dividing by (uc)n ) x n + a1 x n−1 + · · · + an−1 x + an = 0, and therefore x ∈ I b . Reversing this argument shows that, conversely, if x = I π b = I b . Finally, by the lemma, a/u ∈ I , then ac ∈ ucI . Therefore  πb  for general πb I ∈ K(R) one has I = {J ∈ K(R) : J is f.g. and J ⊆ I } = {J b ∈ K(R) :  J is f.g. and J ⊆ I } = I b . The above proof uses the fact that b : I −→ I is a semiprime operation, which requires proof. The most difficult parts of the proof are showing that I is closed under addition and that I = I for all ideals I . One method of proving this is via the method of ideal reductions [178], initiated in the influential 1954 paper [159] of Northcott and Rees. We now generalize the method of reductions to R-submodules of T . This has interest in its own right and allows for a self-contained proof that b is a semistar operation that does not rely on the fact that b is a semiprime operation. Everything that follows is known in the situation where T = R [178].

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Definition 4.4.4. Let R be a ring and T an R-algebra, and let I and J be Rsubmodules of T . We say that J is a reduction of I if J ⊆ I and J I n = I n+1 for some nonnegative integer n, and the reduction number red J (I ) of I with respect to J is the least such integer n, where we set red J (I ) = ∞ if J is not a reduction of I . We write J I if J is a reduction of I . The following result provides a useful criterion, in terms of reductions, for an element of T to be integral over an R-submodule I of T . Proposition 4.4.5. Let R be a ring and T an R-algebra, let I be an R-submodule of T , and let x ∈ T . Then deg I (x) = 1 + red I (I + x R). In particular, x is integral over I if and only if I is a reduction of I + x R. Proof. Note that (I + x R)n = I n + x I n−1 + x 2 I n−2 + · · · + x n−1 I + x n R, so I (I + x R)n = I n+1 + x I n + x 2 I n−1 + · · · + x n−1 I 2 + x n I and (I + x R)n+1 = I n+1 + x I n + x 2 I n−1 + · · · + x n I + x n+1 R. Therefore the equality I (I + x R)n = (I + x R)n+1 holds if and only if x n+1 R ⊆ I n+1 + x I n + x 2 I n−1 + · · · + x n I . The proposition follows.  The following result provides another useful criterion for an element of T to be integral over I . Proposition 4.4.6. Let R be a ring and T an R-algebra, let I be an R-submodule of T , and let x ∈ T . Then x is integral over I if and only if there exists a finitely generated R-submodule J of T such that x J ⊆ I J and such that, whenever a J = 0 for some a ∈ T , one has ax ∈ nilrad(T ) (or, alternatively, whenever a J = 0 for some a ∈ T one has ax n = 0 for some positive integer n). Proof. Let x ∈ T be integral over I , so that by Lemma 4.4.2 there is a finitely generated R-submodule H of I such that x is integral over H . Then, by Proposition 4.4.5, one has H (H + x R)n = (H + x R)n+1 for some n. Let J = (H + x R)n , so J is finitely generated and x J = x(H + x R)n ⊆ (H + x R)n+1 = H (H + x R)n = H J ⊆ I J . Also, if a J = 0 for some a ∈ T , then ax n = 0 since x n ∈ J , and therefore ax ∈ nilrad(T ) as well. R-submodule of T such Conversely, let J = (y1 , . . . , ym ) be a finitely generated  that x J ⊆ I J . For each i = 1, . . . , m, write x yi = mj=1 ai j y j for some ai j ∈ I . Let A be the matrix (δi j x − ai j ), where δi j is the Kronecker delta function. Let y be the transpose of the vector (y1 , . . . , ym ). By construction Ay = 0, so that det(A)y = adj(A)Ay = 0. Hence, for all i one has det(A)yi = 0, so that det(A)J = 0. Supposing also that a J = 0 for a ∈ T implies that ax ∈ nilrad(T ), we have (det(A)x)k = 0 for some integer k, and then an expansion of the equation (det(A)x)k = 0 yields an equation of integral dependence of x over I .  The following proposition provides some useful properties of reductions and reduction numbers. Proposition 4.4.7. Let R be a ring and T an R-algebra, and let H, J, I be Rsubmodules of T .

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(1) red J (I ) is the least nonnegative integer n such that J ⊆ I and I n+m = I n J m for all positive integers m (and equals ∞ if the condition fails for all n). (2) red H (I )  red H (J ) + red J (I ). In particular, if H J and J I , then H I . (3) is a partial ordering on the set Mod R (T ) of all R-modules of T that is a refinement of the subset relation. (4) red J (I )  red H (I ) if H ⊆ J ⊆ I . In particular, if H I and H ⊆ J ⊆ I , then J I. (5) red H (I + J )  red H (I ) + red H (J ). In particular, if H I and H J , then H I + J. (6) If I and J are finitely generated, H + J ⊆ I , and H I , then H H + J . Proof. Let n = red J (I ). We may assume without loss of generality that n < ∞. Then I n+m = I n+m−1 J = I n+m−2 J 2 = · · · = I n+1 J m−1 = I n J m for all positive integers m. Conversely, if the given equality holds for all positive integers m, then in particular I n+1 = I n J . This proves (1). Let m = red H (J ) and n = red J (I ). We may assume without loss of generality that m, n < ∞. It follows that H ⊆ J ⊆ I , and also I n+m+1 = I n J m+1 = I n J m H ⊆ I n+m H ⊆ I n+m+1 so equality holds throughout. Therefore H ⊆ I and I n+m+1 = I n+m H , so red H (I )  m + n. This proves (2), and then statement (3) is obvious. Next, suppose that H is a reduction of I and H ⊆ J ⊆ I , and let k = red H (I ) < ∞. Then I k+1 = H I k ⊆ J I k ⊆ I k+1 , so I k+1 = J I k and therefore red J (I )  k. This proves (4). Next, let n = red H (I ) and m = red H (J ). We may suppose without loss of generality that n, m < ∞. Then H ⊆ I and H ⊆ J , so H ⊆ I + J . Moreover, we have I n+1 = H I n and J m+1 = H J m . Therefore (I + J )n+m+1 = I n+m+1 + I n+m J + I n+m−1 J 2 +· · ·+ I n+1 J m + I n J m+1 +· · · + I J n+m + J n+m+1 = H I n+m + H I n+m−1 J + · · · + H I n J m + I n H J m + · · · + I H J n+m−1 + H J n+m = H (I n+m + I n+m−1 J + · · · + I n J m + · · · + I J n+m−1 + J n+m ) = H (I + J )n+m . Therefore red H (I + J )  red H (I ) + red H (J ). This proves (5). Finally, we prove (6). Let J = (x1 , . . . , xk ), and let n = red H (I ), so that H I n = n+1 I . By (4), we have H + (x1 , . . . , xi−1 ) I for all i. As xi ∈ I , we have xi I n ⊆ I n+1 = H I n ⊆ (H + (x1 , . . . , xi−1 ))I n , so that xi I n ⊆ (H + (x1 , . . . , xi−1 ))I n . Also, if a I n = 0 for some a ∈ T , then, as xi ∈ I , also axin = 0, so axi ∈ nilrad(T ). Since also I n is finitely generated, xi is integral over H + (x1 , . . . , xi−1 ) by Proposition 4.4.6, so H + (x1 , . . . , xi−1 ) H + (x1 , . . . , xi ). Therefore, by (3) and induc tion on k, one has H H + (x1 , . . . , xk ) = H + J . Definition 4.4.8. Let R be a ring and T an R-algebra. A T -semistar operation on R is a nucleus  on the ordered monoid Mod R (T ), that is, it is a closure operation  on the poset Mod R (T ) such that I  J  ⊆ (I J ) for all I, J ∈ Mod R (T ). Example 4.4.9. A semistar operation on a ring R is equivalently a T (R)-semistar operation on R. A semiprime operation on a ring R is equivalently an R-semistar operation on R. The following theorem implies that b TR is a finite type T -semistar operation on R, thus providing an alternative proof of Theorem 4.4.3.

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Theorem 4.4.10. Let R be a ring and T an R-algebra. Then the association I −→ T I b R defines a finite type T -semistar operation b TR on R. In other words, one has the following. (1) (2) (3) (4)

T

I b R ∈ Mod R (T ) for all I ∈ Mod R (T ). b TR is a closure operation on the poset Mod R (T ). T T T I b R J b R ⊆ (I J )b R for all I, J ∈ Mod R (T ). T For all x ∈ I b R , there is a finitely generated R-submodule J of I such that T x ∈ JbR . T

Proof. We first prove (1). Let x, y ∈ I b R . By Lemma 4.4.2, there exists a finitely b TR

generated H1 ⊆ I such that x ∈ H1 . Likewise, there exists a finitely generated bT

T

H2 ⊆ I such that y ∈ H2 R . Thus x, y ∈ H b R , where H = H1 + H2 ⊆ I is finitely generated. Since x and y are integral over H , one has H H + x R and H H + y R, and therefore H H + x R + y R by Proposition 4.4.7(5). Let J = (x + y)R. As H + J ⊆ H + x R + y R and J and H + x R + y R are finitely generated, one has H H + J = H + (x + y)R by Proposition 4.4.7(6). Therefore x + y is integral T over H and hence over I , so that x + y ∈ I b R . T Next, let x ∈ I b R and c ∈ R. Let x n + a1 x n−1 + · · · + an = 0 be an equation of integral dependence of x over I . Then (cx)n + ca1 (cx)n−1 + · · · + cn an = 0 is an T equation of integral dependence of cx over I , so cx ∈ I b R . T This proves (1). It is then trivial to check that the operation I −→ I b R is a preclosure on Mod R (T ). To prove that the operation is idempotent and therefore a closure T T operation on Mod R (T ), let I ∈ Mod R (T ) and x ∈ (I b R )b R . Then there exists a finitely T T generated J ⊆ I b R with x ∈ J b R . Write J = (a1 , . . . , ak ). Then there exists a finitely T generated H ⊆ I such that each ai ∈ H b R for all i. Then H H + J , and also, since x is integral over J , hence over H + J , one has H + J H + J + x R. Thus we have H H + J + x R. Therefore, by Proposition 4.4.7(6), since H + J + x R and x R are finitely generated, we have H H + x R. Thus x is integral over H and T hence over I , so x ∈ I b R . This proves (2). T Next, let I, J ∈ Mod R (T ), and let x ∈ I and y ∈ J b R . Let y n + a1 y n−1 + a2 y n−2 · · · + an = 0 be an equation of integral dependence of y over J . Then (x y)n + xa1 (x y)n−1 + x 2 a2 (x y)n−1 + · · · + x n an = 0 is an equation of integral dependence T of x y over I J , since x i ai ∈ I i J i = (I J )i for all i. Therefore x y ∈ (I J )b R . Therefore T T  I J b R ⊆ (I J )b R . This proves (3), and, finally, (4) is Lemma 4.4.2. Corollary 4.4.11. Let R be a ring. Then the map I −→ I b defines a finite type semistar operation b on R. Moreover, for any ideal I of R, the ideal I sprimb = I b ∩ R is the integral closure I of the ideal I . In particular, the semiprime operation b : I −→ I on R is the standard finite type semistar operation sprimb on R associated to the semistar operation b , and conversely b is the unique finite type semistar operation  = d b = π b on R with b = sprim . We also note the following.

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Proposition 4.4.12. Let R be a ring and T an R-algebra, and let I and J be Rsubmodules of T with I finitely generated and J ⊆ I . Then J is a reduction of I if T T and only if J b R = I b R . Proof. If J is a reduction of I , then by Proposition 4.4.7(6) J is a reduction of J + x R T T T for every x ∈ I . Thus I ⊆ J b R by Proposition 4.4.5, so that J b R = I b R . To prove the T T T converse, suppose that J b R = I b R , so that I = (x1 , . . . , xn ) ⊆ J b R , where each x j ∈ T . Then each x j is integral over J and hence over J + (x1 , . . . , x j−1 ). Then by Proposition 4.4.5 one has J J + (x1 ) J + (x1 , x2 ) · · · J + (x1 , . . . , xn ) = I , so that J I is a reduction by Proposition 4.4.7(3).  Many important properties of integral closure generalize to semistar integral closure. An example is the following. Lemma 4.4.13. Let R be a ring. (1) If S is an integral extension of R, then I = I S ∩ R for all ideals I of R. (2) R is integrally closed if and only if the semistar operation b is unital, if and only if the semiprime operation of integral closure is regular (that is, a I = a I for all regular a ∈ R and all ideals I of R). Proof. Statement (1) is [178, Proposition 1.6.1], and statement (2) follows from Corollary 4.3.27 or is easy to check directly. (Also see Exercise 4.4.14.)  Proposition 4.4.14. Let R be a ring with total quotient ring K and S an integral extension of R with total quotient ring L. Then S is R-torsion-free and therefore K ⊆ L. Moreover, one has I b = (I S)b ∩ K for all I ∈ K(R). Proof. It is elementary to check that every regular element of R is a regular element of S and therefore S is R-torsion-free and K ⊆ L. The operation  : I −→ I  = (I S)b ∩ K is a finite type semistar operation on R. Moreover, one has I  ∩ R = ((I S)b ∩ K ) ∩ R = ((I S)b ∩ S) ∩ R = I S ∩ R = I for all ideals I of R by Lemma 4.4.13(1). Therefore  is the semistar operation b on R by Corollary 4.4.11.  We also note the following alternative characterization of the semistar operation b , whose proof is left as an exercise. Corollary 4.4.15. On any ring R, the semistar operation b is the unique finite type semistar operation  on R such that the integral closure in R of any ideal I of R is equal to I  . It is well-known that, if D is an integral domain, then Ib =

 {I V : V is a valuation overring of D}

for all I ∈ K(D) [192, Appendix 4, Theorem 1]. We wish to generalize this to rings with zerodivisors.

4.4 Integral Closure of Ideals and Submodules

333

Lemma 4.4.16. Let R be a ring and T an R-algebra, let I be an R-submodule of T , and let x be a regular element of T . Then x is integral over I if and only if x −1 I extends to the unit ideal of R[x −1 I ], where R[x −1 I ] is the R-subalgebra of T [x −1 ] generated by x −1 I . Proof. Let R  = R[x −1 I ]. Suppose that x is integral over I with equation of integral dependence x n + a1 x n−1 + · · · + an−1 x + an = 0. Then −1 = ax1 + ax 22 + · · · + xann is in the extension of the ideal x −1 I to R  . Conversely, suppose that the ideal x −1 I R  is the unit ideal of R  . Any element of R  can be written in the form a0 + ax1 + ax 22 + · · · + ax nn for some ai ∈ I i . Thus every element of x −1 I R  can be written in the form a1 + ax 22 + · · · + ax nn for some ai ∈ I i . Since −1 ∈ x −1 I R  , one has −1 = ax1 + ax 22 + x · · · + ax nn for some ai ∈ I i , and therefore x n + a1 x n−1 + · · · + an−1 x + an = 0 is an equation of integral dependence of x over I .  Proposition 4.4.17. Let R be a ring and T an R-algebra, let I be an R-submodule of T , and let x be an element of T . Let J = (0 :T (x T )∞ ) = {a ∈ T : ax n = 0 for some n  1}, which is an ideal of T . Let T = T /J (not to be confused with integral closure), and let x denote the image of x in T . Then x is regular in T . Let R = (R + J )/J and I = (I + J )/J denote, respectively, the images of R and J in T . The following conditions are equivalent. (1) x is integral over I in T . (2) x is integral over I in T . (3) x −1 I extends to the unit ideal of R[x −1 I ] ⊆ T [x −1 ]. Proof. Conditions (2) and (3) are equivalent by the lemma. Moreover, any equation of integral dependence of x over I can be reduced to an equation of integral dependence of x over I ; and conversely any equation of integral dependence of x over I in T can be lifted to an equation of the form x n + a1 x n−1 + · · · + an−1 x + an = b in T , where ai ∈ I i for all i and b ∈ J , and multiplying by x m for some m with bx m = 0 results in an equation of integral dependence of x over I . Therefore (1) and (2) are equivalent.  Theorem 4.4.18. Let R be a ring and T an R-algebra, and let I be an R-submodule  of T . The integral closure of I in T contains  {I V : V is a paravaluation ring of T containing R} as well as every unit of T in {I V : V is a valuation ring of T containing R}. Proof. Suppose that x ∈ T is not integral over I . Use the notation as in Proposition 4.4.17. It follows from that proposition that x −1 I extends to a proper ideal of R[x −1 I ]. Let p be a prime ideal of R[x −1 I ] containing x −1 I . By Proposition 3.10.1 there exists a valuation pair (V , q) of T [x −1 ] dominating (R[x −1 I ], p), say, with associated valuation v and value group . Let v 0 be the restriction of v to T . Then v 0 is a paravaluation on the ring T . Define v0 : T −→  ∪ {∞} via v0 (x) = v 0 (x) for all x ∈ K . Then v0 is a paravaluation on T , and therefore Tv0 is a paravaluation ring of T containing R. Moreover, if x ∈ I Tv0 , then x ∈ I V ⊆ x p V and therefore 1 ∈ p V ⊆ q, which is a contradiction. Therefore x ∈ / I Tv0 . This proves the first

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4 Semiprime, Star, and Semistar Operations …

claim of the theorem. To prove the second claim, observe that if x is a unit of T , then  T = T = T [x −1 ] and therefore v0 = v is valuation on T . Definition 4.4.19. Let R be a ring and T an R-algebra, and let I be an R-submodule of T . Let  T IaR = {I V : V is a paravaluation ring of T containing R} T (R)

and let I a = I a R = I a R . Corollary 4.4.20. Let R be a ring and T an R-algebra, and let I an R-submodule of T T T . Then a TR is a nucleus on Mod R (T ) with a TR  b TR . Moreover, Ra R = Rb R is equal to the integral closure of R in T . A Kaplansky fractional ideal I of a ring R is said to be dense if I has trivial annihilator (that is, (0 : R I ) = 0), and I is said to be semiregular if I contains a finitely generated dense J ∈ K(R). One has the following irreversible implications (equivalent for Noetherian rings): regularly generated ⇒ regular ⇒ semiregular ⇒ dense. Rings in which every semiregular ideal is regular are said to be McCoy or satisfy Property A. As we have seen, rings in which every regular ideal is regularly generated are said to be Marot. Theorem 4.4.21. If R is a Marot and McCoy ring, then I a = I b for all I ∈ K(R) such that I b (or I ) is regular, and therefore a b . Proof. Suppose that R is Marot and McCoy and let x ∈ I b be regular. By Proposition 4.4.6, there exist elements b1 , . . . , bn of K such that x J ⊆ I J , where J = (b1 , . . . , bn ) ∈ K(R) and a J = 0 implies ax n = 0 for some n, which in turn implies that a = 0 since x is regular, for all a ∈ R. Thus J is finitely generated and dense. Therefore, since R is McCoy, J is regular. Let V be any paravaluation overring of R. Since R is Marot, V is r-Bézout and therefore J V is principal, say, J V = yV , where y ∈ K is regular. Then x yV = x J V ⊆ I J V = y I V , and therefore x V ⊆ I V and x ∈ I V . Therefore x ∈ I a . Thus every regular element of I b lies in I a , so, since I b is regularly generated, one has I b ⊆ I a . The reverse containment follows from Corollary 4.4.20.  Corollary 4.4.22. If R is a Marot and McCoy ring (e.g., any integral domain or any overring of a Noetherian ring), then Ib =

 {I V : V is a valuation overring of R}

for all regular I ∈ K(R) and

4.4 Integral Closure of Ideals and Submodules

I =

335

 {I V ∩ R : V is a valuation overring of R}

for all regular ideals I of R. In 1954, Chevalley proved the following result, whose proof can also be found in [178, Theorem 6.4.3]. Proposition 4.4.23 ([33]). Let D be a Noetherian domain, and let p be a nonzero prime ideal of R. Then there exists a DVR overring of D whose unique maximal ideal lies over p. Theorem 4.4.24. Let D be a Noetherian domain. Then  {I V : V is a DVR overring of D} Ib = for all finitely generated I ∈ K(D). Consequently, one has I =



{I V ∩ D : V is a DVR overring of D}

for all ideals I of D. Proof. Let R = D. The containment ⊆ for I b follows immediately from Corollary 4.4.22. The proof of the reverse containment is similar to the proof of Theorem 4.4.18, where we apply Proposition 4.4.23 in the proof instead of Proposition 3.10.1. For this purpose, observe that, if I is finitely generated and x ∈ T (R) is nonzero then x is a unit of T (R) and the ring R[x −1 I ] = R[x −1 I ] as used in the proof of Theorem 4.4.18 is finitely generated as an R-algebra and is therefore Noetherian.  Note that equality for I b in Theorem 4.4.24 need not hold if I is not finitely generated. Definition 4.4.25. For any ring R, let Min(R) denote the set of minimal prime ideals of R. The minimal primes provide another tool for studying integral closure, often allowing one to reduce to the integral domain case. Proposition 4.4.26. Let R be a ring and T an R-algebra, and let I be an Rsubmodule of T . An element x of T is integral over I in T if and only if x is integral over (I + q)/q in T /q for every minimal prime q of T . Proof. The forward implication is clear. To prove the reverse implication, let W be the set of all elements of T of the form x n + a1 x n−1 + · · · + an with n a positive integer and ai ∈ I i for all i. Then W is a multiplicative subset of T , and x is integral over I if and only if 0 ∈ W . Suppose that x is not integral over I . Then there exists a prime ideal q of T disjoint from W , and, since any prime contains a minimal prime, we may assume that q is minimal. Since no element of W lies in q, no element of W is zero in T /q, and therefore x is not integral over (I + q)/q in T /q. 

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Let p be any non-regular prime of a ring R. There is a commutative square R

T (R)

R/p

T (R/p)

mapping the ring homomorphism R −→ T (R) to the homomorphism R/p −→ T (R/p) of integral domains. If I is a Kaplansky fractional ideal of R, so that I is an R-submodule of T (R), then we let I (R/p) denote the R/p-submodule of T (R/p) generated by the image of I in T (R/p), which is a Kaplansky fractional ideal of the domain R/p. Thus there is a natural map K(R) −→ K(R/p) acting by I −→ I (R/p). Since the minimal primes of T (R) are in one-to-one correspondence with the minimal primes of R, we obtain from Proposition 4.4.26 the following corollaries. Corollary 4.4.27. Let R be a ring, I a Kaplansky fractional ideal of R, and x an element of T (R). Then x ∈ I b if and only if x ∈ (I (R/p))b for all p ∈ Min(R). Corollary 4.4.28 ([178, Proposition 1.1.5]). Let R be a ring, I an ideal of R, and x an element of R. Then x ∈ I if and only if x lies in the integral closure of (I + p)/p in R/p for all p ∈ Min(R). Combining Proposition 4.4.22 for integral domains and Theorem 4.4.24 and Corollary 4.4.27, we obtain the following. Proposition 4.4.29. Let R be a ring and I ∈ K(R). Then Ib =



{I V ∩ T (R) : V is a valuation overring of R/p for some p ∈ Min(R)},

where I V ∩ T (R) denotes the preimage of I V under the map T (R) −→ T (R/p) (which is well-defined because every regular element of R is a unit in T (R/p)). Moreover, if R is Noetherian and I is fractional, then Ib =

 {I V ∩ T (R) : V is a DVR overring of R/p for some p ∈ Min(R)}.

Corollary 4.4.30. Let R be a ring and I an ideal of R. Then I =

 {I V ∩ R : V is a valuation overring of R/p for some p ∈ Min(R)},

where I V ∩ R denotes the preimage of I V in R. Moreover, if R is Noetherian, then I =



{I V ∩ R : V is a DVR overring of R/p for some p ∈ Min(R)}.

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337

One might try to extend the analysis above to the following definition. Let R be a ring and T an R-algebra, and let I be an R-submodule of T . Let T

I cR =

 {I V : V is a valuation ring of T containing R} T (R)

and let I c = I c R = I c R . Although a TR  c TR , the operations c TR and b TR are not necessarily -comparable, even for T = T (R). For instance, there are examples of integrally closed rings R where Rc is strictly larger than Rb = R, so c ł b [90]. However, T T by Theorem 4.4.18, at least every unit in T of I c R lies in I b R for all I ∈ Mod R (T ), and thus every regular element of I c lies in I b for all I ∈ K(R). Obviously for rings that are Marot and McCoy one has a b c . For general rings, the author admits to holding the biased position that the three semistar operations a , b , c should be ranked b , a , c in order of importance, since a and c apparently need not be of finite type and Rc need not equal R, while the operation b generalizes integral closure of ideals and has connections to several previously defined notions as in the following proposition, whose proof is left as an exercise. (The proof uses two nontrivial results, namely, Theorems 3.8.1 and 3.8.8.) Proposition 4.4.31. Let R be a ring. (1) R is integrally closed if and only if b t , if and only if b  t . (2) R is a PVMR if and only if b t w . (3) R is t -Prüfer if and only if b t w , for any unital semistar operation  on R. (4) R is Prüfer if and only if b d , if and only if b t , if and only if R is integrally closed and b -Prüfer, if and only if R is FJS b -Prüfer.  (5) If R is Prüfer, then a = c = d and therefore I = {I V : V is a valuation overring of R} for all I ∈ K(R). Corollary 4.4.32. A ring R is Prüfer if and only if every finitely generated regular ideal (or every regular ideal) of R is integrally closed. Note also that a ring R is integrally closed if and only if every regular principal ideal of R is integrally closed. (See Exercise 4.4.11.) Example 4.4.33. Let D and V be domains as in Example 4.2.41 and Example 4.2.64(1), and let  = {V } : I −→ I V . Suppose further that V is finitely generated as a D-module and that the maximal ideal m of D is finitely generated. (See [97, Example 3.6], for example.) Since V m ⊆ m, by Proposition 4.4.6 we have V ⊆ D, whence V = D and V is the smallest valuation overring of D. Therefore, by Corollary 4.4.22, the semistar operation  = {V } coincides with the integral closure operation b . It follows that D is strictly b -Dedekind but not FJS b -Prüfer.

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4 Semiprime, Star, and Semistar Operations …

4.5 Complete Integral Closure and Related Operations on Ideals In this section, we use the theory of semistar operations to study the complete integral closure operation on ideals. Definition 4.5.1. Let R be a ring. Then R reg denotes the set of all regular elements of R and R o denotes the complement of the union of the minimal primes of R, that is, the set of all elements of R that are not in any minimal prime of R. The following result is straightforward to check. Proposition 4.5.2. Let R be a ring. One has R reg = {a ∈ R : (0 : R a) = 0} and

√ √ R o = {a ∈ R : ( 0 : R a) = 0}.

In particular, R reg and R o are multiplicatively closed subsets of R with R reg ⊆ R o , and equality holds if R is reduced. Example 4.5.3. Let k be a field. (1) The ring R = k[X, Y, Z ]/(X 2 Z , Y 2 Z ) is non-reduced ring with R o = R reg = R − ((x, y) ∪ (z)), where x and y denote, respectively, the image of X and Y in R. (2) The ring R = k[X, Y ]/(X 2 , X Y ) is a non-reduced ring with R o = R − (x) and R reg = R − (x, y)  R o , where x and y denote, respectively, the image of X and Y in R. Remark 4.5.4. Let R be a Noetherian ring, or more generally a ring in which the zero ideal has a (finite) primary decomposition. Then R reg is the complement of the union of the associated primes of the zero ideal, and R o = R reg if and only if the zero ideal of R has no embedded associated primes. Definition 4.5.5. Let V be paravaluation overring of a ring R with paravaluation v : T (R) −→  ∪ {∞}. For any ideal I of R, we write v(I ) = inf{v(x) : x ∈ I } if the infimum exists. Lemma 4.5.6. Let R be a ring, and let V be a discrete rank one valuation overring of R with valuation v : K −→ Z ∪ {∞}. Then v(I ) exists, v(I ) = v(I ), and v(I n ) = nv(I ) for any ideal I of R and any nonnegative integer n. Proof. Let I be an ideal of R. Since the natural numbers are well-ordered, it is clear that v(I ) = min{v(x) : x ∈ I } exists and v(I n ) = nv(I ) for all n. Since I ⊇ I , it suffices to show that v(x)  v(I ) for all x ∈ I . Suppose to obtain a contradiction that

4.5 Complete Integral Closure and Related Operations on Ideals

339

v(x) < v(I ) for some x ∈ I . Since x ∈ I , we can write x n + a1 x n−1 + · · · + an = 0 with ai ∈ I i for all i. Note that v(ai )  iv(I ) > iv(x), and therefore v(ai x n−i ) = v(ai ) + (n − i)v(x) > iv(x) + (n − i)v(x) = nv(x), for all i. Thus, all of the terms ai x n−i of the sum a1 x n−1 + · · · + an = −x n have valuation greater than nv(x). It follows that v(−x n ) = v(a1 x n−1 + · · · + an ) > nv(x) = v(−x n ), which is a contradiction.  Proposition 4.5.7 (cf. [178, Corollaries 6.8.11 and 6.8.12]). Let I be an ideal of a Noetherian ring R, and let x ∈ R. The following conditions are equivalent. (1) (2) (3) (4) (5) (6)

x ∈ I. There exists an integer N such that x n ∈ I n−N for all n  N . There exists a c ∈ R o such that cx n ∈ I n for all n  0. There exists a c ∈ R o such that cx n ∈ I n for infinitely many n. There exists a c ∈ R o such that cx n ∈ I n for infinitely many n. x ∈ I V for all minimal primes p of R and for all DVR overrings V of R/p.

Proof. Suppose that (1) holds. Then I is a reduction of I + x R, so that (I + x R) N +1 = I (I + x R) N for some N . Then for all n  N we have (I + x R)n = I n−N (I + x R) N ⊆ I n−N , so that x n ∈ I n−N . Thus (2) holds. Suppose that (2) holds. Let p1 , . . . , pl be the minimal primes of R. By relabelling the pi we may suppose that I ⊆ p1 ∩ · · · ∩ pk and I Ę pk+1 , . . . , pl . Then I Ę pk+1 ∪ · · · ∪ pl , so we may choose d ∈ I − (pk+1 ∪ · · · ∪ pl ). Then we have d N ∈ I N − (pk+1 ∪ · · · ∪ pl ). Likewise, we may choose e ∈ (pk+1 ∩ · · · ∩ pl ) − (p1 ∪ · · · ∪ pk ). Since nilrad(R) = p1 ∩ . . . ∩ pl is finitely generated, there exists a positive integer M such that (p1 ∩ . . . ∩ pl ) M = 0. Then we have e M ∈ (pk+1 ∩ · · · ∩ pl ) M − (p1 ∪ · · · ∪ pk ) and e M (p1 ∩ · · · ∩ pk ) M = 0. Let c = d N + e M . Since by construction d N and e M lie in mutually exclusive minimal primes of R, it follows that c ∈ R o . We claim that, for all n  M + N , one has cx n ∈ I n . Let n  M + N . Since d ∈ I , one has d N x n ∈ I N I n−N = I n . Moreover, one has e M x n ∈ e M I n−N ⊆ e M I M ⊆ e M (p1 ∩ · · · ∩ pk ) M = 0. Therefore cx n = (d N + e M )x n = d N x n ∈ I n . Thus (3) holds. The implications (3) ⇒ (4) ⇒ (5) are trivial. Suppose that (5) holds. Let p be a minimal prime of R and V a DVR overring of R/p with corresponding valuation v. Then v(c) > 0, and for infinitely many n one has cx n ∈ I n V = I n V . This implies that v(c) + nv(x)  v(I n V ) = v(I n ) = nv(I ), so that v(I ) − v(x)  v(c)/n, for infinitely many n. Necessarily, then, one must have v(I )  v(x), so x ∈ I V . Thus (6) holds. Finally, we have (6) ⇒ (1) by Corollary 4.4.28 and Theorem 4.4.24.  The following example shows that for an ideal I of a Noetherian ring R the containment I ⊇ {x ∈ R : ∃c ∈ R o such that cx n ∈ I n for all n} may be proper, even if I is nonzero and principal, R is a total quotient ring, and R reg = R o .

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4 Semiprime, Star, and Semistar Operations …

Example 4.5.8. Let k be a field, let R = k[X, Y ]/(X 2 , Y 2 ), and let x and y denote, respectively, the image of X and Y in R. Then (x, y) is the only prime ideal of R, so R is a local total quotient ring and R reg = R o = R − (x, y). Since y 2 = x 2 , one has y ∈ (x). In fact, every ideal of R/(x, y) ∼ = k is integrally closed, so by Corollary 4.4.28 one has (x) = (x, y). Suppose that f ∈ R has the property that there exists a regular g ∈ R such that g f n ∈ (x)n for all n (not just for n  0), so, in particular, g f ∈ (x). Then g f x = 0, so f x = 0, which is easily seen to imply f ∈ (x). Indeed, one has f = F(x, y) for some F ∈ k[X, Y ], and then f x = 0 means F X ∈ (X 2 , Y 2 ), which implies that F ∈ (X, Y 2 ) and therefore f ∈ (x, y 2 ) = (x). Thus y ∈ (x) = (x, y) does not have the property required of f . Proposition 4.5.9. Let R be a ring with total quotient ring K , let I be a fractional ideal of R, and let x ∈ K . If R is Noetherian, then the following conditions are equivalent. (1) (2) (3) (4)

x ∈ Ib. There exists a c ∈ R o such that cx n ∈ I n for all n  0. There exists a c ∈ R o such that cx n ∈ I n for infinitely many n. There exists a c ∈ R o such that cx n ∈ (I n )b for infinitely many n.

Moreover, if I is regular or R is Noetherian and reduced, then condition (1) implies the following. (5) There exists a c ∈ R reg such that cx n ∈ I n for all n. (6) There exists a c ∈ R o such that cx n ∈ I n for all n. Proof. By replacing I with d I for some regular d ∈ R such that d I ⊆ R, we may assume without loss of generality that I is an ideal of R. Let x = a/b with a, b ∈ R and b regular. Then x ∈ I b if and only if a ∈ bI b = (bI )b , if and only if a ∈ bI . Likewise, cx n ∈ I n if and only if ca n ∈ (bI )n , and cx n ∈ (I n )b if and only if ca n ∈ (bI )n . Therefore the equivalence of conditions (1)–(4) follows from Proposition 4.5.7. Suppose that I is regular and condition (1) holds, that is, x ∈ I b . Then there is an equation x n+1 + a1 x n + · · · + an+1 = 0 of integral dependence of x over I . Write x = r/s with r, s ∈ R and s regular. Let a ∈ R be any regular element of I , so a n ∈ I i ∩ R for all i  n and c = s n a n ∈ R is regular. Then cx i = s n a n (r/s)i = s n−i r i a n ∈ I i for all i  n. Now, we have cx n+1 = −a1 cx n − · · · − can+1 ∈ I I n + · · · + I n+1 = I n+1 , so cx n+1 ∈ I n+1 . Then we also have cx n+2 = −a1 cx n+1 − · · · − an+1 cx ∈ I I n+1 + · · · + I n+1 I = I n+2 , so cx n+2 ∈ I n+2 . By induction, then, we see that cx i ∈ I i for all positive integers i. Therefore conditions (5) and (6) hold. Suppose, on the other hand, that R is Noetherian and reduced and condition (1) holds. Note first that R o = R reg and, since R is Noetherian and reduced, K is a finite direct product of fields. Therefore J K is generated by an idempotent and so J n K = (J K )n = J K for all n, for all J ∈ K(R). Suppose that condition (2) holds, so that there exists a c ∈ R o = R reg and a positive integer N such that cx n ∈ I n for all n  N . Let n < N . Then cx n K = cx K = cx N K ⊆ I NK = I K = I n K , so there exists a cn ∈ R reg such that cn cx n ∈ I n . Letting d = c n 0. Let x = r/s ∈ I m+n be nonzero, where r, s ∈ R are nonzero. By Proposition 4.4.5, there exists a positive integer l such that for all k  l one has (I m+n + x R)k+l = I (m+n)k (I m+n + x R)l ⊆ I (m+n)k . If I = (a1 , . . . , an ), then we claim that I (m+n)k ⊆ (a1k , . . . , ank )m+1 . To prove this  claim, it suffices to prove that a1b1 a2b2 · · · anbn , for all bi with i bi = (m + n)k, lies in k k m+1 , . . . , a ) . Let c = b /k. Then c + 1 > b /k  c , so that (a i i i i i n i (ci + 1) >   1 b /k = m + n and so c = (c + 1) − n  m + 1. It follows that i i i i i i a1b1 a2b2 · · · anbn ∈ a1kc1 a2kc2 · · · ankcn R ⊆ (a1k , . . . , ank )m+1 . This proves the claim. From the claim it follows that (I m+n + x R)k+l ⊆ (a1k , . . . , ank )m+1 . Set c = r l ∈ R − {0} = R o . Then for k = q  l one has cx q = r q+l /s q ∈ (x R)q+l ⊆ (I m+n + x R)q+l ⊆ (a1 , . . . , an )m+1 = (I [q] )m+1 = (I m+1 )[q] q

and therefore x ∈ (I m+1 )T . This completes the proof.

q



Corollary 4.6.14 (Generalized Briançon-Skoda theorem [102, Theorem 5.4]). Let R be a Noetherian ring of prime characteristic and I an ideal of R generated by at most n elements of R. Then for all nonnegative integers m one has I m+n ⊆ (I m+1 )∗ . The following corollaries of the generalized Briançon-Skoda theorem are wellknown. Corollary 4.6.15. Let R be a Noetherian ring of prime characteristic. Then I ∗ = I for every principal ideal I of R.

350

4 Semiprime, Star, and Semistar Operations …

Corollary 4.6.16. Let R be a weakly F-regular ring and I an ideal of R generated by at most n elements of R. Then for all nonnegative integers m one has I m+n ⊆ I m+1 . In particular, every principal ideal of R is integrally closed. Next, we discuss tight closure in non-Noetherian rings. First, note that the preclosure T on K(R) of Definition 4.6.1 is defined even if R is not Noetherian. However, in that case there is no reason to expect that the prenucleus pclT : I −→ I T ∩ R is a semiprime operation. Nevertheless, as we did with complete integral closure in the previous section, we may apply Proposition 4.3.20 to obtain a smallest semiprime operation ∗ = (pclT )∞ on R greater than pclT . Then I ∈ I(R) is ∗-closed if and only if I = I T ∩ R, and ∗ is given by I −→ I ∗ =

 {J ∈ I(R) : J ⊇ I and J = J T ∩ R}.

By Proposition 4.6.5, the semiprime operation ∗ coincides with tight closure if R is Noetherian. As of yet, there is no good extension of tight closure theory to nonNoetherian rings, and it is unlikely that this particular semiprime operation ∗ or even its associated finite type semiprime operation ∗t can satisfy this need. Nevertheless, its more elementary properties are worth pursuing. Instead of applying Proposition 4.3.20 to the preclosure I −→ I T ∩ R on I(R), we may apply the analogous proposition for semistar operations, Proposition 4.3.19, to the preclosure I −→ I T on K(R), to obtain the smallest semistar operation T∞ greater than or equal to T. One might ask whether or not the semiprime operation ∞ sprimT∞ : I −→ I T ∩ R so defined coincides with the tight closure operation ∗ = (pclT )∞ on R, at least if R is Noetherian. We show in Corollary 4.6.22 below that the answer is affirmative provided that the integral closure of R is finitely generated as an R-module. Lemma 4.6.17. Let R be a ring with total quotient ring K , and let x ∈ K . Suppose that there is an element c of R such that cx n ∈ R for infinitely many positive integers  n. Then cx n ∈ R for all positive integers n. In particular, if c ∈ R reg , then x ∈ R , o and if c ∈ R o , then x ∈ R . Proof. We show that for every positive integer m we have cx m ∈ R. Let m be a positive integer. Since cx n ∈ R for infinitely many n, we can choose n > m such that cx n ∈ R. Then we have (cx n )m ∈ R, so cn−m (cx n )m ∈ R, which equals (cx m )n . Therefore, since (cx m )n ∈ R, the element cx m of K is a root of the monic polynomial X n − (cx m )n ∈ R[X ], and therefore cx m ∈ R.  Proposition 4.6.18. Let R be a ring of prime characteristic. Then one has R o ⊆ ∞ ∞ o R T ⊆ R and R T ⊆ R T ⊆ (Ro )T = Ro . Consequently, if (R o )o = R o , then ∞ R T = R o = R o . Proof. Let x ∈ R o . Then there exists c ∈ R o such that cx n ∈ R for all n, and therefore cx q ∈ R [q] = R for all q = p e . Therefore x ∈ R T . Thus we have R o ⊆ R T .

4.6 Tight Closure

351

Next, let x ∈ R T . Then there exists c ∈ R o such that cx q ∈ R [q] = R for all q = p  0, and in particular, cx n ∈ R for infinitely many n. Therefore, by the lemma o o we have x ∈ R . Therefore we have R T ⊆ R . Now, let x ∈ (Ro )T . Then there exists a c ∈ R o such that cx n ∈ Ro for infinitely many n. Therefore, by the lemma we have x ∈ (Ro )o = Ro . Therefore (Ro )T = ∞ ∞ ∞  Ro , so that (Ro )T = Ro , and, finally, R T ⊆ (Ro )T = Ro . e

∞

Corollary 4.6.19. Let R be a Noetherian ring of prime characteristic. Then R T = R T = R. Example 4.6.20. Let k be a finite field of prime characteristic p, and let R be the sub∞ 2 ring k[X, X Y, X Y p , X Y p , . . .] of k[X, Y ]. Then R T = R T = (R  ) = k[X, Y ],   o T∞ but Y ∈ / R , whence R = R  R . By Propositions 4.6.18 and 4.3.40, we have the following. Proposition 4.6.21. Let R be a ring of prime characteristic p, and let ∗ = (pclT )∞ denote the smallest semiprime operation larger than the prenucleus pclT : I −→ {x ∈ R : ∃c ∈ R o such that cx q ∈ I [q] for all q = p e  0} on I(R). ∞

(1) If R is totally integrally closed, then I ∗ = I T ∩ R for all ideals I of R. ∞ (2) Suppose that R o is a fractional ideal of R. Then R T = R T = Ro = R o , and ∞ I −→ I T ∩ R is the smallest rigid semiprime operation greater than or equal to ∗. Corollary 4.6.22. Let R be a Noetherian ring of prime characteristic such that the ∞ integral closure of R is finitely generated as an R-module. Then I ∗ = I T ∩ R for all ideals I of R, and T = (T∞ )t . Thus, if R is a Noetherian ring of prime characteristic, then the operations T and (T∞ )t are finite type semistar operations on R with T  (T∞ )t , and equality holds if the integral closure of R is finitely generated as an R-module. Open problem 4.6.23. If R is an arbitrary Noetherian ring of prime characteristic, ∞ then must one have T = (T∞ )t ? Equivalently, must one have I ∗ = I T ∩ R for all ideals I of R?

Exercises Section 4.1 1. Prove Proposition 4.1.4. 2. Let ϕ : R −→ S be a ring homomorphism, and let s be a semiprime operation on S. Show that the map I −→ ϕ−1 ((ϕ(I )S)s ) on I(R) is a semiprime operation on R.

352

4 Semiprime, Star, and Semistar Operations …

3. Let S be an overring of a ring R. Prove that the following conditions are equivalent. 1) S is a fractional ideal of R. 2) Freg (S) is a subsemigroup of Freg (R). 3) The association ∗ : I −→ I S for I ∈ Freg (R) defines a nonunital star operation on R. 4. Prove Corollary 4.1.25. 5. Prove Proposition 4.1.15. 6. Prove statements (2) and (3) of Proposition 4.1.17. 7. Let ∗ be a nonunital star operation on a ring R. Show that the map I −→ I ∗ ∩ I v on Freg (R) is the largest star operation on R that is less than or equal to ∗. 8. Let  be a semistar operation on a ring R. Recall that sprim denotes the semiprime operation I −→ I  ∩ R induced by . Prove that the following conditions are equivalent. 1)  is unital, that is, R  = R. 2) Every sprimv -closed ideal of R is sprim -closed. 3) I sprim ⊆ I sprimv for every ideal I of R. 4) Every invertible ideal of R is sprim -closed. 5) Every regular principal ideal of R is sprim -closed. Moreover, if the conditions above hold, then (J I )sprim = J I sprim and (a I )sprim = a I sprim for all ideals I, J of R with J invertible and for all regular a ∈ R. 9. Let D be an integral domain. Prove the following. a) For every finite type star operation on D, there is a unique finite type unital semistar operation on D that restricts to ∗. b) There is an isomorphism between the poset of all finite type star operations on D and the poset of all finite type unital semistar operations on D. c) If R is a ring, then there exists an embedding of the poset of all finite type star operations on R into the poset of all finite type unital semistar operations on R. d) There exists a total quotient ring R such that the poset of all finite type star operations on R is not isomorphic to the poset of all finite type unital semistar operations on R. 10. Let R be a ring. For any overring S of R, let NUStar(R, S) denote the poset of all nonunital star operations ∗ on R with R ∗ = S. Prove the following. a) Let ∗ be a nonunital star operation on R. Then Freg (R ∗ ) is subsemigroup of Freg (R) and ∗ restricts to a star operation on R ∗ . b) The association in part (a) provides for any overring S of R a map NUStar(R, S) −→ Star(S). c) Let ∗ be a star operation on an overring S of R. Then I −→ (I S)∗ defines a nonunital star operation on R if and only if S is a fractional ideal of R. d) The association in part (c) provides for any overring S of R that is a fractional ideal of R a map Star(S) −→ NUStar(R, S). e) If S is an overring of R that is a fractional ideal of R, then the two maps constructed in parts (b) and (d) are inverses of each other.

Exercises

11.

12.

13.

14.

15.

353

 f) There is a bijection between NUStar(R) and S∈F(R)∩O(R) Star(S), where  denotes disjoint union and O(R) denotes the set of all overrings of R. g) O(R) = {I ∈ K(R) : I ⊇ R and I 2 = I }. h) The finite type nonunital star operations  in NUStar(R) correspond precisely to the finite type star operations in S∈F(R)∩O(R) Star(S) under the given bijection. Let k be a field. Prove the following. a) There is a unique nonunital star operation on k[[X ]]. b) There is a unique star operation on k[[X 2 , X 3 ]]. c) There are exactly two nonunital star operations on k[[X 2 , X 3 ]]. Let R be a ring and J ∈ Freg (R). Prove the following. a) v(J ) : I −→ I v(J ) = (J : K (J : K I )) is a nonunital star operation on R. b) J ∗ = J if and only if ∗  v(J ), for all nonunital star operations ∗ on R. c) v(I ) is a star operation on R for all I ∈ Freg (R) if and only if R is completely integrally closed. Let  be a nonempty set of nonunital  star operations on a ring R. a) Show that the association I −→ {I ∗ : ∗ ∈ } for I ∈ Freg (R) defines a nonunital star operation on R, equal to the infimum inf  of  in the poset NUStar(R) of all nonunital star operations on R. Conclude that the poset NUStar(R) is bounded complete. b) Suppose that   is bounded above. Show that the supremum sup  of  is given by I −→ {J ∈ Freg (R) : J ⊇ I and J ∗ = J for all I ∈ }. c) Show that ∗ = inf{v(J ) : J ∈ Freg (R)∗ } for all nonunital star operations ∗ on R. d) Show that every nonunital star operation on R is a star operation on R if and only if v(J ) is a star operation on R for all J ∈ Freg (R). Let R be a ring.  a) Show that there is a natural  bijection between Semistar(R) and S∈O(R) USemistar(S), where denotes disjoint union and O(R) denotes the set of all overrings of R. (Hint: Follow a similar method to that outlined in Exercise 10.) b) Show that the finite type semistar operations in Semistar(R) correspond precisely to the finite type star operations in S∈O(R) USemistar(S) under the given bijection. Let V be a valuation domain. Using the previous exercise and Exercises 3.1.12, 3.1.20, and 3.1.21, prove the following. a) Semistar(V ) = {Vp : p ∈ Spec(V )} ∪ {vv (Vp ) : p ∈ Spec(V ), p2 = p}, where  R : I −→ I R for all I ∈ K(V ) for any overring R of V . b) If V is finite dimensional, then |Semistar(V )| = 1 + dim V + |{p ∈ Spec(V ) : p2 = p}|.

Section 4.2 1. Let D be an integral domain with quotient field K . Show that e : I −→ K is the only semistar operation on D that is not reduced.

354

4 Semiprime, Star, and Semistar Operations …

2. Let  be a semistar operation on a ring R, and let S be an overring of R. Prove the following. a)  S : I −→ I S is the smallest semistar operation  on R such that R  = S. b) R  = S if and only  S    v (S). c) If I ∈ K(R) is invertible, then I  = I R  . d) If R is Dedekind, then   R  v (R  ). 3. Prove Propositions 4.2.8 and 4.2.9. 4. Prove Proposition 4.2.10. 5. Prove Proposition 4.2.11 and Corollary 4.2.12. 6. Verify the equivalences in Definition 4.2.14(1). 7. Prove Lemma 4.2.16. 8. Prove Lemmas 4.2.17 and 4.2.20. 9. Prove statements (1) and (2) of Theorem 4.2.25. 10. Prove Corollary 4.2.26. 11. Prove Theorem 4.2.30. 12. Prove Theorem 4.2.31. 13. Let  be a semistar operation on a ring R. Show that Inv (R) is a group under -multiplication. 14. Let R be a ring. For all  ⊆ Spec R, let  = {q ∈ Spec R : q ⊆ p for some p ∈ } denote the generization of . One says that  is closed under generization if  = . Prove the following. a) For all ,   ⊆ Spec R, one has []  [ ] if and only if  ⊇   . b) The map  −→ [] is an anti-isomorphism from the poset of all subsets of Spec R closed under generization to the poset of all spectral semistar operations, with inverse  −→ -Spec(R). 15. Prove Theorems 4.2.60 and 4.2.61. 16. Let R be a ring with total quotient ring K , and let J, H ∈ K(R) with J ⊆ H . Prove the following. a) There exists a smallest closure operation  on K(R) such that J  = H , and  acts by   : I −→

I + H if J ⊆ I I otherwise.

b) There exists a smallest semiprime operation  = u (J, H ) on M such that J  ⊇ H . For all I ∈ K(R), one has I u (J,H ) =

 {L ∈ K(R) : L ⊇ I and (L : K J ) ⊆ (L : K H )}.

Equivalently, I u (J,H ) is the smallest L ∈ Mod R (M) containing I such that H ⊆ J v (L) . c) The following conditions are equivalent. 1) There exists a semiprime operation  on M such that J  = H . 2) (H : K J ) ⊆ (H : K H ).

Exercises

17. 18. 19. 20. 21. 22. 23.

24.

25.

26.

27.

28.

355

3) J v (H ) = H . 4) J u (J,H ) = H . 5) u (J, H ) is the smallest semiprime operation  on M such that J  = H . 6) v (H ) is the largest semiprime operation  on M such that J  = H . Prove Proposition 4.2.32 by generalizing the proof of [176, Proposition 3.13]. Verify all of the equivalences in Definition 4.2.38. Prove Lemma 4.2.45. Let  be a semistar operation on a ring R. Show that if  is of finite type, then ˙ is of finite type, and if  is stable, then ˙ is stable. Prove Lemma 4.2.48. Prove Proposition 4.2.49. Let  be a semistar operation on a ring R. Show that, if Rp is Prüfer for every t -maximal ideal p of R, then R is w -Prüfer. Show that the converse holds if R is an integral domain. Let R be a ring, and let  and  be semistar operations on R with   . Prove the following. a) If I ∈ K(R) is FJS  -invertible (resp., strictly  -invertible,  -invertible, quasi  -invertible), then I is FJS -invertible (resp., strictly -invertible, -invertible, quasi -invertible). b) If R is FJS  -Prüfer (resp., strictly  -Prüfer,  -Prüfer, quasi  -Prüfer), then R is FJS -Prüfer (resp., strictly -Prüfer, -Prüfer, quasi -Prüfer). Let R be a ring and R  an overring of R. Prove the following. a) Let  be a semistar operation on R, so that  = |K(R  ) is a semistar operation on R  . If R is quasi -Prüfer, then R  is quasi  -Prüfer.  b) Let  be a semistar operation on R  . Then  : I −→ (I R  ) is a semistar operation on R, and R  is quasi  -Prüfer if and only if R is quasi -Prüfer. Let  be a set of semistar operations on a ring R. Prove the following. a) I ∈ K(R) is FJS inf -invertible if and only if I is FJS -invertible for all  ∈ . b) R is FJS inf -Prüfer if and only if R is FJS -Prüfer for all  ∈ . c) R is v -Prüfer if and only if R is v -Prüfer and -Prüfer. d) R is (v )t -Prüfer if and only if R is a PVMR and w -Prüfer. Prove that the following conditions are equivalent for any integral domain D with quotient field K = D. 1) D is a DVR. 2) The only semistar operations on D are d , e , and e red . 3) v (J ) = d for all J ∈ K(D) with J = K and J = (0). 4) (J : K (J : K I )) = I for all I, J ∈ K(D) with I, J = K and I, J = (0). Let D be a PID with exactly two maximal ideals p1 and p2 . For i = 1, 2, let I i = I Dpi for all I ∈ K(D). Show that Semistar(D) = {ee , e red , v , 1 , 2 , 1 ∧ v , 2 ∧ v , d }, with Hasse diagram given as below.

356

4 Semiprime, Star, and Semistar Operations … e

e red

1

v

2

1 ∧ v

2 ∧ v

d

29. Let D be a Dedekind domain with quotient field K . For all I ∈ K(D), let I D = {p ∈ Max(D) : I Dp = K }. Provethe following. a) For all ∈ K(D) one has I = p∈I / D I Dp . b) There is a poset embedding of the poset Clos(2Max(D) ) of all closure operations on 2Max(D) into the poset RSemistar(D) of all reduced semistar operations on D. Explicitly, the map Clos(2Max(D) ) −→ RSemistar(D) acting by ∗ −→  is an embedding, where I =



I Dp

p∈(I / D )∗

and (I  ) D = (I D )∗ for all I ∈ K(D). c) The given embedding in part (b) has an order-preserving left inverse acting by  −→ ∗, where ∗ is the largest closure operation on 2Max(D) such that I D is ∗-closed for all I ∈ K(D) . d) If Max(D) is finite, then the given embedding Clos(2Max(D) ) −→ RSemistar(D) is an isomorphism. e) Use part (d) to complete Exercise 28. 30. Let  be a semistar operation on a ring R. Prove the following. a) R is quasi -Dedekind if and only if R  is ˙ -Dedekind. b) The following conditions are equivalent for all I ∈ K(R). 1) I is strictly w -invertible. 2) I is w -invertible. 3) I is quasi w -invertible. c) The following conditions are equivalent. 1) R is strictly w -Dedekind. 2) R is w -Dedekind. 3) R is quasi w -Dedekind. 4) R w is (˙w )-Dedekind. d) (∗∗) If R is w -Dedekind, then must R be FJS w -Dedekind?

Exercises

357

Section 4.3 1. Verify examples (1)–(11) of Example 4.3.1. 2. Give an example of a ring on which the radical semiprime operation is not standard. 3. Prove Proposition 4.3.4 and Corollary 4.3.6. 4. Prove Proposition 4.3.9. 5. Prove Proposition 4.3.12. 6. Prove Proposition 4.3.13. 7. Prove Proposition 4.3.14. 8. Prove Proposition 4.3.19. 9. Prove Proposition 4.3.20. 10. Prove Proposition 4.3.22. 11. Prove Lemma 4.3.25. Deduce that, for any ring R, there is an isomorphism between the poset of all finite type fractional star operations on R and poset of all finite type unital semistar operations on R. 12. Verify that the map ∗ : F(R) −→ F(R) defined in the proof of Theorem 4.3.26 is a well-defined fractional star operation on R that restricts to s. 13. Complete the proof of Proposition 4.3.38 by showing that pst for any prenucleus p on a ring R is the smallest standard preclosure on R larger than p. 14. Prove statement (2) of Theorem 4.3.32. 15. Prove Proposition 4.3.39. 16. (∗∗) For which rings R is the radical semiprime operation regular? For which R is it standard? 17. (∗∗) Must a standard semiprime operation be rigid? Prove or give a counterexample. 18. (∗∗) Generalize Theorem 4.3.32 by finding necessary and sufficient conditions for a general semiprime operation on a ring to be rigid. Section 4.4 1. Let R be a ring and I an ideal of R. Prove the following. a) If I is a radical ideal, then I is integrally closed. b) Let Rred = R/ nilrad(R). One has I Rred = I Rred . c) I = R if and only if I = R. 2. Use Exercise 2.1.8 to give an alternative proof of Proposition 4.4.7(5). 3. Let a and b be elements of a ring R. Show that (a n , bn ) = (a, b)n for all positive integers n. Use this to provide a proof of Corollary 4.4.32 that does not use Proposition 4.4.31. 4. Show that (X, Y ) is integrally closed in k[X, Y ], while (X 2 , Y 2 ) = (X, Y )2 and therefore (X 2 , Y 2 ) is not integrally closed in k[X, Y ], for any field k. 5. Let R be a ring. An ideal I of R is said to be integral over an ideal J of R if I contains J and every element of I is integral over J . Let I, J, I  , J  , H be ideals of R. Prove the following. a) J is the largest ideal of R that is integral over J . b) If I is integral over J and J is integral over H , then I is integral over H .

358

6.

7.

8.

9.

10.

11.

12. 13.

4 Semiprime, Star, and Semistar Operations …

c) If I is integral over J and I  is integral over J  , then I + I  is integral over J + J  and I I  is integral over J J  . d) If I is finitely generated and integral over J , then J I . e) If I is finitely generated, then I I . f) If I is finitely generated and regular, then I is integral over J if and only if J I. Let R be a subring of a ring T , and let I and J be R-submodules of T . Suppose that T T I = J + H for some finitely generated R-submodule H of T and that I b R = J b R . Show that J is a reduction of I . Let ϕ : R −→ S be a ring homomorphism. Prove the following. a) ϕ(I )S ⊆ ϕ(I )S and I ⊆ ϕ−1 (ϕ(I )S)) for every ideal I of R. b) ϕ−1 (J ) ⊆ ϕ−1 (J ), and, in particular, if J is integrally closed in S, then ϕ−1 (J ) is integrally closed in R, for every ideal J of S. c) Suppose that I = ϕ−1 (ϕ(I )S) for every ideal I of R. (This holds if the ring homomorphism ϕ : R −→ S is faithfully flat.) Then I = ϕ−1 (ϕ(I )S)) for every ideal I of R. Let S be an R-torsion-free extension of a ring R, so that the total quotient ring of R is a subring of the total quotient ring of S. Prove that I b S ⊆ (I S)b for all I ∈ K(R). Let R be a ring and U a multiplicative subset of R. Prove the following. a) U −1 I = U −1 I for every ideal I of R. b) An ideal I of R is integrally closed if and only if I Rm is integrally closed over Rm for every maximal ideal m of R. Let R be a ring and U a multiplicative subset of R. Prove the following. a) U −1 I b = (U −1 I )b for all I ∈ K(R). b) A Kaplansky fractional ideal I of R is b -closed if and only if I Rm is b -closed in Rm for every maximal ideal m of R. Let R be a ring. Prove that the following conditions are equivalent. 1) R is integrally closed. 2) Every t -closed ideal of R is integrally closed (or equivalently I ⊆ I t for every ideal I of R). 3) Every v -closed ideal of R is integrally closed (or equivalently I ⊆ I v for every ideal I of R). 4) Every invertible ideal of R is integrally closed. 5) Every regular principal ideal of R is integrally closed. Moreover, if the conditions above hold, then J I = J I and a I = a I for all ideals I, J of R with J invertible and for all regular a ∈ R. Let I be an ideal of a ring R. Show that I is integrally closed in R if and only if ideal I R[X ] + X R[X ] is integrally closed in R[X ]. Let R be a ring. a) Show that if I is an ideal of R, then I b = Rb if and only if I = R. b) Show that b w = d , that is, the largest stable semistar operation on R less than or equal to b is trivial.

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c) Let I, J, H ∈ K(R). Show that if I H ⊆ J H and H is finitely generated and dense then I b ⊆ J b . d) Prove Proposition 4.4.31. 14. Let I be an ideal of a ring R and r an element of R. Prove the following. a) r ∈ I if and only if there exists a finitely generated R-module M √ such that r M ⊆ I M and whenever a M = 0 for some a ∈ R one has r ∈ (0 : R a). b) If I is finitely generated and regular, then r ∈ I if and only if there exists a finitely generated faithful R-module M such that r M ∈ I M. c) If S is an integral extension of R, then I = I S ∩ R. (Hint: Reduce to the case where S is finitely generated as an R-module and then apply part (a).) Section 4.5 1. Let R be a ring. Show that (0) = (0) = (0) = (0)o = (0)o = nilrad R. 2. Let R be a ring such that R is Noetherian. Show that R = R o = Ro . 3. Let R be a ring with total quotient ring K . Prove the following. a) If R is a fractional ideal of R, then I −→ I b defines a nonunital star operation on R. b) If R  is a fractional ideal of R, then R is a fractional ideal of R, the map  I −→ I  is a nonunital star operation on R, the map I −→ I v(R ) = (R : K (R : K I )) is the largest nonunital star operation on R, and one has b    v(R ). 4. Let S be an R-torsion-free extension of a ring R, so that the total quotient ring of R is a subring of the total quotient ring of S. Prove that I  S ⊆ (I S) and I  S ⊆ (I S) for all I ∈ K(R). 5. Let R be a ring. Let I c = I  ∩ R for all ideals I of R. Let us say that an ideal I of R is completely integrally closed if I c = I . Prove that the following conditions are equivalent. 1) R is completely integrally closed. 2) Every v -closed ideal of R is completely integrally closed. 3) I c ⊆ I v for every ideal I of R. 4) I ⊆ I v for every ideal I of R. 5) Every invertible ideal of R is completely integrally closed. 6) Every regular principal ideal of R is completely integrally closed. J I = J I for all Moreover, if the conditions above hold, then (J I )c = J I c and  ideals I, J of R with J invertible. 6. Prove Proposition 4.5.2. 7. Verify Example 4.5.3. 8. Verify Remark 4.5.4. 9. Let R be a Noetherian ring. Prove the following. a) I  ⊆ I o ⊆ I b for any fractional ideal I of R, and equalities hold if R is reduced or I is regular. I ⊆ Io ⊆ I for any ideal I of R, and equalities b) I  ∩ R ⊆ I o ∩ R ⊆ I and hold if R is reduced or I is regular. Io ⊆ I for an ideal I may be proper, even if R c) The inclusions I o ⊆ I b and is a total quotient ring and I is principal.

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4 Semiprime, Star, and Semistar Operations …

10. Show that localization of a completely integrally closed domain need not be completely integrally closed. 11. (∗∗) If I is a regular ideal of a ring R, then must I  be b -closed? Section 4.6 1. Let R be a ring of prime characteristic p, and let I ∈ K(R). Show that (I T )[q] ⊆ ∞ ∞ (I [q] )T and (I T )[q] ⊆ (I [q] )T for all q = p e . 2. Let R be a Noetherian ring of prime characteristic, and let S be an extension ring of R. Suppose there exists an R-module homomorphism ϕ : S −→ R such that ϕ(1) ∈ R o . Prove that I S ∩ R ⊆ I ∗ for every ideal I of R. 3. Let R be a ring of prime characteristic, and let a be a regular element of R. Prove the following. o a) a R o ⊆ (a R)T ⊆ a R . b) If R is Noetherian, then (a R)∗ = a R ∩ R = a R. 4. Verify Lemma 4.6.2. 5. Prove Lemma 4.6.8. 6. Verify Example 4.6.20. 7. Let D be an integral domain with quotient field K , and let D + denote the integral closure of D in an algebraic closure of K . For all I ∈ K(D), we let I s + = I D + ∩ K , which we call the semistar plus closure of I . For any ideal I of D, the ideal I + = I s + ∩ D = I D + ∩ D is known as the plus closure of I . a) Show that the operation I −→ I s + on K(D) is a finite type semistar operation on D, and the operation I −→ I + on I(D) is a finite type semiprime operation on D. b) It is well-known that I + ⊆ I ∗ for all ideals I of D if D is a Noetherian domain of prime characteristic [109, p. 15]. Generalize this by showing that ∞ I s + ⊆ I (T )t for all I ∈ K(D), and if D is Noetherian then I s + ⊆ I T for all I ∈ K(D). 8. Let R be a Noetherian ring of prime characteristic. a) Show that, for any ideal I of R, one has I ∗ = R if and only if I = R. b) For which Noetherian rings R of prime characteristic is the tight closure semiprime operation on R stable? 9. (with N. Epstein) Let R be a Noetherian ring of prime characteristic. Prove that the following conditions are equivalent. 1) R is weakly F-regular. 2) R is reduced and every regular ideal of R is tightly closed. 3) R is isomorphic to a finite direct product of weakly F-regular integral domains. 10. (with N. Epstein) Let us say that an ideal I of a ring R is o-regular if I contains an element of R o . Since R reg ⊆ R o , every regular ideal is o-regular. Let R be a Noetherian ring of prime characteristic. Prove that the following conditions are equivalent. 1) R/ nilrad(R) is weakly F-regular. 2) Every o-regular ideal of R containing nilrad(R) is tightly closed.

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3) R/ nilrad(R) is isomorphic to a finite direct product of weakly F-regular integral domains. 11.  (with N. Epstein) Let R be a ring. Let n : I −→ I + nilrad(R) and p : I −→ of R, and denote by b : I −→ I the integral p∈Min(R) (I + p) for every ideal I √ closure operation and rad : I −→ I the radical operation on I(R). Prove the following. a) n, p, b, and rad are semiprime operations on R with n  p  b  rad. b) R is reduced if and only if n = d, and R is von Neumann regular if and only if rad = d. c) There exists a reduced Noetherian ring R of prime characteristic on which n < p < b < rad. d) Let R be a Noetherian ring of prime characteristic. i) n  p  ∗  b  rad.  ii) ∗ = p (that is, I ∗ = p∈Min(R) (I + p) for every ideal I of R) if and only if R/p is weakly F-regular for every minimal prime p of R. iii) If R/ nilrad(R) is weakly F-regular, then R/p is weakly F-regular for every minimal prime p of R. iv) Let R = k[[X, Y ]]/(X Y ), where k is a field. Then R is reduced and (X + Y )∗ = (X, Y ), so R = R/ nilrad(R) is not weakly F-regular. However, R/p is weakly F-regular for every minimal prime p of R. 12. (∗∗) For which Noetherian rings of prime characteristic does the tight closure semiprime operation coincide with the integral closure semiprime operation?

Chapter 5

Noncommutative Rings and Closure Operations on Submodules

Radicals and torsion theories, first introduced, respectively, by Maranda in 1962 [142] and Dickson and Lambek in 1966 [39, 129], are important tools for studying rings and their modules. In this chapter, we show that both are equivalent to certain functorial closure operations on submodules, and any “cohereditary” radical is completely determined by the closure operation it induces on left ideals. This allows us to prove the equivalence of the following categories, for any ring R: (1) the category of all cohereditary radicals on the category of left R-modules; (2) the category of all closure operations on submodules of left R-modules that are minimal in a certain sense; and (3) the poset of all semiprime closure operations on the left ideals of R. These equivalences motivate a natural extension of the theory of closure operations on ideals and submodules in commutative algebra to modules over noncommutative rings. Convention 5.0.1. All rings R in this chapter are assumed unital, but not necessarily commutative, and all R-modules are left R-modules, unless otherwise stated. All of the results herein have analogues for right R-modules. For any ring R, we denote by R-Mod the category of all left R-modules and Mod-R the category of all right R-modules. All sets of sets by convention are assumed to be ordered by the subset relation, and all posets are also considered to be categories in the usual way. For proofs of some elementary results regarding preradicals, torsion theories, injective envelopes, Gabriel filters, and localization at Gabriel filters, we refer the reader to [177, Chapters VI and IX].

5.1 Chapter Summary Let R be a ring. A preradical on R-Mod is a subfunctor of the identity functor on R-Mod, that is, it is a (class) function r : R-Mod −→ R-Mod satisfying the following conditions: (1) r(M) is an R-submodule of M for every R-module M, © Springer Nature Switzerland AG 2019 J. Elliott, Rings, Modules, and Closure Operations, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-24401-9_5

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and (2) f (r(M)) ⊆ r(N ) for every R-module homomorphism f : M −→ N . A preradical r on R-Mod is idempotent if r(r(M)) = r(M) for every R-module M. A preradical r on R-Mod is a radical if r(M/r(M)) = 0 for every R-module M. It is well-known that an idempotent radical on R-Mod is categorically equivalent to a torsion theory on R-Mod [39], and a left exact radical on R-Mod is equivalent to a hereditary torsion theory on R-Mod, which in turn is equivalent to a Gabriel topology on R [39, 75, 129, 177]. One of the main goals of this chapter is to establish natural equivalences between various ordered classes of radicals, torsion theories, and closure operations on ideals and submodules, as listed in Table 5.1. For any ring R, we say that a (left) semiprime operation on R is a closure operation on the poset Il (R) of all left ideals of R such that I s a ⊆ (I a)s for all I ∈ Il (R) and all a ∈ R. The notion extends naturally to any R-module M: a semiprime operation on M is a closure operation  on the poset Mod R (M) of all R-submodules of M such that f (L  ) ⊆ f (L) for all L ∈ Mod R (M) and all R-endomorphisms f of M. (A left semiprime operation on a ring R is equivalently a semiprime operation on the left R-module R.) If r is M a radical on R-Mod, then for any R-module M the operation L −→ L r , where M L r /L = r(M/L) for all L ∈ Mod R (M), is a semiprime operation on M. Moreover, for any semiprime operation s on R, there is a unique cohereditary radical r on R-Mod such that s = r R , where we say that a preradical r is cohereditary if r(Rx) = 0 for all x ∈ M implies r(M) = 0 for any R-module M. Thus, the map r −→ r R is an isomorphism from the poset of all cohereditary radicals on R-Mod to the poset of all semiprime operations on R. This chapter advances a theory of semiprime operations on modules, which include both semiprime operations and semistar operations, as studied in Chapters 2–4, as special cases. This chapter also serves to extend the theories of closure operations in commutative algebra to noncommutative rings by locating them, along with the theories of radicals and torsion theories, within the common framework of preradicals and, more generally, functorial systems of operations, on module categories. The ordered classes across each row of Table 5.1 are isomorphic, for any ring. The isomorphisms written in italics are new, and all of the relevant terms are defined in Sections 5.2–5.6. The containment lattice for the various classes in Table 5.1 is shown in Figure 5.1. Note that the intersection of any two of the classes in the containment lattice is the class lying immediately below them. Unfortunately, however, we are unable to show that the classes (9) and (10) are distinct, that is, that a cohereditary idempotent radical need not be left exact. We leave this as an open conjecture. A primary tool we introduce in Section 5.2 to prove the equivalences for the class (7) in Table 5.1 is a natural Galois connection between the poset of all closure operations on Mod R (M) and the poset of all closure operations on Mod R (N ) for any R-modules M and N . Taking M to be the left R-module R yields a method for lifting any closure operation on Il (R) to a closure operation on Mod R (N ) for all R-modules N . This provides a novel way of lifting semiprime operations, including tight closure and integral closure, to operations on submodules. (See Corollary 5.2.34 an Example 5.5.15.) A rigid closure system of operations on R-Mod is precisely a

5.1 Chapter Summary

365

Table 5.1 Isomorphisms of ordered classes for any ring R system of operations on endofunctor class of R-modules R-Mod of R-Mod 1 2 3

4

5

6

7

functorial functorial closure functorial right exact preclosure, or functorial left exact preinterior functorial right exact closure, or functorial left exact right semiexact preinterior functorial right exact left semiexact preclosure, or functorial left exact interior functorial right exact left semiexact closure, or functorial left exact right semiexact interior rigid closure

8

exact preclosure, or functorial left exact basic interior

9

rigid left semiexact closure

10

exact closure, or stable closure, or functorial left exact right semiexact basic interior

alternative realization

preradical

radical

pretorsion-free class

idempotent preradical

pretorsion class

idempotent radical

torsion class, or torsion-free class

torsion theory

cohereditary radical left exact preradical

strongly hereditary pretorsion-free class uniform left ideal filter

cohereditary idempotent radical left exact radical

strongly hereditary torsion-free class

semiprime operation, or left ideal system linear topology, or stable intersectionpreserving presemiprime operation torsion semiprime operation

Gabriel filter, or hereditary torsion class, or torsion-free class closed under essential extensions

Gabriel topology, or stable semiprime operation

class of closure operations, one on Mod R (M) for each R-module M, that can all be obtained from a single semiprime operation on R in this way. We remark that both preradicals and functorial systems of preclosure operations have been studied in the more general setting of abelian and Grothendieck categories [40, 112]. For the sake of concreteness, we have chosen to work primarily with the category R-Mod of all (left) R-modules, although many of our definitions and results can be generalized to the setting of abelian or Grothendieck categories or subcategories of R-Mod. Further work may strengthen the connections to algebraic and noncommutative algebraic geometry. This chapter is organized as follows. Section 5.2 develops the theory of closure operations on ideals and submodules, and, together with Sections 5.3–5.6, proves the equivalences described in Table 5.1. Section 5.3 studies the stable and the

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Fig. 5.1 Containment lattice for the classes in Table 5.1

finite type semiprime operations, and Section 5.4 introduces so-called divisorial semiprime operations, which in a precise sense generate all semiprime operations, as well as their “dual,” the codivisorial semiprime operations. As a corollary, we determine the poset of all semiprime operations on any Dedekind domain. To minimize prerequisites, in Sections 5.5 and 5.6 we have collected some results from [177] that are well-known to specialists, and we have incorporated them in our main results in that section. Finally, the last section, Section 5.7, studies semiprime operations on algebras, including semistar operations, and provides an application to the canonical topology and w-envelopes.

5.2 Closure Operations on Ideals and Submodules We begin with some basic notation and terminology. Definition 5.2.1. Let S and T be posets. (1) A monotone Galois connection between S and T is a pair of order-preserving maps f : S −→ T and g : T −→ S such that f (x)  y if and only if x  g(y) for all x ∈ S and all y ∈ T . The pair is a monotone Galois injection (resp. monotone Galois surjection) if also f is injective (resp., surjective), or, equivalently, if g ◦ f is the identity on S (resp., f ◦ g is the identity on T ). (2) An antitone Galois connection between S and T is a pair of order-reversing maps f : S −→ T and g : T −→ S such that y  f (x) if and only if x  g(y) for all x ∈ S and all y ∈ T . The pair is an antitone Galois injection (resp. antitone Galois surjection) if also f is injective (resp., surjective), or, equivalently, if g ◦ f is the identity on S (resp., f ◦ g is the identity on T ). Definition 5.2.2. Let R be a ring. (1) I(R) (resp., Il (R), Ir (R)) denotes the ordered semigroup of all two-sided ideals (resp., left ideals, right ideals) of R under multiplication.

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367

(2) R-Mod (resp., Mod-R) denotes the category of all left R-modules (resp., right R-modules). (3) If M is a left R-module (resp., right R-module), then Mod R (M) (resp., Mod(M) R) denotes the poset of all R-submodules of M. Definition 5.2.3. Let R be a ring and M a (left) R-module. (1) For any subset X of M and any left ideal I of R, we let IX =



Ix

x∈X

as an R-submodule of M, where I x = {ax : a ∈ I } for all x ∈ M. (2) For any R-submodule N of M and any subset X of M, we let (N : R X ) = {a ∈ R : a X ⊆ N }. Note that (N : R X ) is the largest left ideal I of R such that I X ⊆ N , and if X is an R-submodule of M then (N : R X ) is a (two-sided) ideal of R. (3) We write (N : R x) = (N : R {x}) for all x ∈ M. (4) For any R-submodule N of M and any left ideal I of R, we let (I \ M N ) = {x ∈ M : I x ⊆ N }. Note that (I \ M N ) is the largest subset X of M such that I X ⊆ N , and if I is an ideal of R then (I \ M N ) is an R-submodule of M. If R is commutative, then we also denote (I \ M N ) by (N : M I ). Definition 5.2.4. Let R be a ring. A left semiprime operation on R is a closure operation s on the poset Il (R) such that I s a ⊆ (I a)s for all I ∈ Il (R) and all a ∈ R. A right semiprime operation on R is a closure operation s on the poset Ir (R) such that a I s ⊆ (a I )s for all I ∈ Ir (R) and all a ∈ R. A left semiprime operation on a commutative ring is equivalently a right semiprime operation and is the same as a semiprime operation. Semiprime operations on commutative rings are studied in Chapter 4. In this chapter, by a semiprime operation on a ring we always mean, by convention, a left semiprime operation. Any left semiprime operation s on a ring R is a right nucleus on the ordered semigroup Il (R), that is, s is a closure operation on Il (R) such that I s J ⊆ (I J )s for all I, J ∈ Il (R). The converse is true if R is commutative but is not true in general. Generally, a left semiprime operation on a ring R is equivalently a nucleus on the right quantale R • -module Il (R): see Exercise 5.2.15. Next, we consider closure operations on submodules. Lemma 5.2.5. Let S be a complete lattice. Let  be a closure operation on S, and let X be a subset of S.

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(1) (sup X ) = (sup(X  )) . (2) inf(X  ) = (inf(X  )) . (3) The infimum of a collection of -closed elements of S is -closed. Proposition 5.2.6. Let R be a ring and M an R-module. Let  be a closure operation on Mod R (M), and let {L λ : λ ∈ } be a subset of Mod R (M).   (1) ( λ∈ L λ ) = ( λ∈ L λ ) .     (2) λ∈ L λ = ( λ∈ L λ ) . (3) The intersection of a collection of -closed R-submodules of M is -closed. Definition 5.2.7. Let R be a ring, let M and N be R-modules, and let  be an order-preserving operation on Mod R (M) and  an order-preserving operation  on Mod R (N ). We say that  is compatible with  if f (L  ) ⊆ f (L) for all L ∈ Mod R (M) and all R-module homomorphisms f : M −→ N . Proposition 5.2.8. Let R be a ring, let M and N be R-modules, let  be an order-preserving operation on Mod R (M) and  an order-preserving operation on Mod R (N ), and let f ∈ Hom R (M, N ). The following conditions are equivalent. (1) (2) (3) (4)



f (L  ) ⊆ f (L) for all L ∈ Mod R (M).  L  ⊆ f −1 ( f (L) ) for all L ∈ Mod R (M).  f −1 (L) ⊆ f −1 (L  ) for all L ∈ Mod R (N ).  f ( f −1 (L) ) ⊆ L  for all L ∈ Mod R (N ).

If  is a preclosure operation and  is a closure operation, then the conditions above are also equivalent to the following. 

(5) L  = L implies f −1 (L) = f −1 (L) for all L ∈ Mod R (N ). Moreover,  is compatible with  if and only if the equivalent conditions (1)–(4) hold for all f ∈ Hom R (M, N ). Remark 5.2.9. The equivalent conditions of Proposition 5.2.8 for an R-module homomorphism f are analogous to the definition of a continuous map of topological spaces: a map f : X −→ Y of topological spaces is continuous if and only if f (A) ⊆ f (A) for all A ⊆ X , if and only if A ⊆ f −1 ( f (A)) for all A ⊆ X , and if and only if the preimage of a closed subset of Y is a closed subset of X . Following terminology in [40], we say that f ∈ Hom R (M, N ) is (,  )-continuous if the equivalent conditions of Proposition 5.2.8 hold. Then  is compatible with  if and only if every f ∈ Hom R (M, N ) is (,  )-continuous. These notions can be relativized to any subcategory of R-Mod. Since the R-module homomorphisms R −→ M for any R-module M are precisely those of the form a −→ ax for some fixed x ∈ M, we have the following. Corollary 5.2.10. Let R be a ring and M an R-module, let s be an order-preserving operation on Il (R), and let  be an order-preserving operation on Mod R (M). The following conditions are equivalent.

5.2 Closure Operations on Ideals and Submodules

(1) (2) (3) (4) (5) (6) (7)

369

 is compatible with s. I s x ⊆ (I x) for all I ∈ Il (R) and all x ∈ M. I s X ⊆ (I X ) for all I ∈ Il (R) and all subsets X of M. I s ⊆ ((I x) : R x) for all I ∈ Il (R) and all x ∈ M. I s ⊆ ((I X ) : R X ) for all I ∈ Il (R) and all subsets X of M. (L : R x)s ⊆ (L  : R x) for all L ∈ Mod R (M) and all x ∈ M. (L : R X )s ⊆ (L  : R X ) for all L ∈ Mod R (M) and all subsets X of M.

Moreover, if s is a preclosure operation and  is a closure operation, then the conditions above are also equivalent to the following. (8) L  = L implies (L : R x)s = (L : R x) for all L ∈ Mod R (M) and all x ∈ M. (9) L  = L implies (L : R X )s = (L : R X ) for all L ∈ Mod R (M) and all subsets X of M. Corollary 5.2.11. Let R be a ring, and let s be a closure operation on Il (R). The following conditions are equivalent. (1) s is a left semiprime operation on R. (2) s is compatible with s. (3) f (I s ) ⊆ f (I )s for all I ∈ Il (R) and all R-module endomorphisms f of the left R-module R. (4) I s X ⊆ (I X )s for all I ∈ Il (R) and all subsets X of R. (5) I s ⊆ ((I a)s : R a) for all I ∈ Il (R) and all a ∈ R. (6) I s ⊆ ((I X )s : R X ) for all I ∈ Il (R) and all subsets X of R. (7) (I : R a)s ⊆ (I s : R a) for all I ∈ Il (R) and all a ∈ R. (8) (I : R X )s ⊆ (I s : R X ) for all I ∈ Il (R) and all subsets X of R. (9) I s = I implies (I : R a)s = (I : R a) for all I ∈ Il (R) and all a ∈ R. (10) I s = I implies (I : R X )s = (I : R X ) for all I ∈ Il (R) and all subsets X of R. Example 5.2.12. The various inequivalent versions of integral closure of submodules [56] [178, Chapter 16] are all compatible with integral closure of ideals, and tight closure of submodules, when defined, is compatible with tight closure of ideals [102, Section 8]. Remark 5.2.13. Let R be a ring and M an R-module. Let s be an order-preserving operation on Il (R) and  a closure operation on Mod R (M). If  is compatible with s, then I s N ⊆ (I N ) for all I ∈ Il (R) and all N ∈ Mod R (M). Moreover, the converse holds if R is commutative (but not necessarily if R is noncommutative). Remark 5.2.14. Let R be a ring, and let f : M −→ N be a homomorphism of R-modules. If  is an order-preserving operation on Mod R (M), then the map f () : L −→ f ( f −1 (L) ) is an order-preserving operation on Mod R (N ). Likewise, if  is an order-preserving operation on Mod R (N ), then the map f −1 ( ) : L −→  f −1 ( f (L) ) is an order-preserving operation on Mod R (M). Moreover, if  is a closure operation, then so is f −1 ( ). By Proposition 5.2.8, one has f ()   if and only if   f −1 ( ). In particular, f induces a monotone Galois connection

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between the poset of all order-preserving operations on Mod R (M) and the poset of all order-preserving operations on Mod R (N ), given by the pair of maps  −→ f () and  −→ f −1 ( ). By Lemma 5.2.24 below, any operation  on a complete lattice S is less than or equal to a smallest closure operation clos on S. Consequently, any f ∈ Hom R (M, N ) induces a monotone Galois connection between the poset Clos(Mod R (M)) of all closure operations on Mod R (M) and the poset Clos(Mod R (N )) of all closure operations on Mod R (N ), given by the pair of maps  −→ f ()clos and  −→ f −1 ( ). Moreover, these associations are functorial: there is a covariant functor from RMod to the category of lattices acting by M −→ Clos(Mod R (M)) on objects and f −→ ( −→ f ()clos ) on morphisms, and likewise there is a contravariant functor from R-Mod to the category of lattices acting by M −→ Clos(Mod R (M)) on objects and f −→ ( −→ f −1 ( )) on morphisms. Proposition 5.2.8 and Corollary 5.2.11 motivate the following definition. Definition 5.2.15. Let R be a ring and M an R-module. A semiprime operation on M is a closure operation  on Mod R (M) such that  is compatible with , that is, such that f (L  ) ⊆ f (L) for all L ∈ Mod R (M) and all f ∈ End R (M) (or equivalently, such that f −1 (L) is -closed for all -closed L ∈ Mod R (N ) and all f ∈ End R (M)). A semiprime operation on an R-module M is equivalently a nucleus on the right quantale End R (M)• -module Mod R (M): see Exercise 5.2.15. Example 5.2.16. Let R be a ring and M an R-module. (1) A left semiprime (resp., right semiprime) operation on R is equivalently a semiprime operation on the left R-module (resp., right R-module) R. (2) If R is an integral domain with quotient field K , then a semistar operation on R in the sense of [62, 161] is equivalently a semiprime operation on the R-module K other than the constant operation I −→ K . (3) If R is commutative, then a semistar operation on R in the sense of Definition 4.1.2 is equivalently a semiprime operation on the R-module T (R). (4) A closure operation  on Mod R (M) is modular if f −1 (L  ) = f −1 (L) for all L ∈ Mod R (M) and all f ∈ End R (M) [34]. Any modular closure operation on Mod R (M) is a semiprime operation on M. Proposition 5.2.17. Let R be a ring and  a semiprime operation on an R-module M. An order-preserving operation  on Mod R (M) is compatible with  if and only if    . Any closure operation on a poset S is completely determined by the set of all closed elements in S. Moreover, if S is a complete lattice, then a nonempty subset C of S is the set of all closed elements of some closure operation on S if and only if  X ∈ C for all subsets X of C. In other words, Lemma 5.2.19 below holds. Definition 5.2.18. Let S be a complete lattice. A closure system on S, or Moore family in S, is a nonempty subset C of S such that, if X ⊆ C, then inf X ∈ C.

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371

Lemma 5.2.19. Let S be a complete lattice. If  is a closure operation on S, then S  is a closure system on S. Conversely, if C is closure system on S, then C : x −→ inf{y ∈ C : y  x} is closure operation on S. The association  −→ S  defines an anti-isomorphism from the poset of all closure operations on S to the poset of all closure systems on S, with inverse C −→ C . Definition 5.2.20. Let R be a ring. A left ideal system R is a nonempty set C of left ideals of R such that the following conditions hold.  (1) If J ⊆ C, then J ∈ C. (2) If I ∈ C, then (I : R a) ∈ C for all a ∈ R. Condition (2) may be replaced with the following. (2 ) If I ∈ C, then (I : R X ) ∈ C for all subsets X of R. Proposition 5.2.21. Let R be a ring. If s is a left semiprime operation on R, then Il (R)s = {I s : I ∈ Il (R)} = {I ∈ Il (R) : I s = I } is a left ideal system on R. Conversely, if C is a left ideal system on R, then sprimC : I −→ {J ∈ C : J ⊇ I } is a left semiprime operation on R. The association s −→ Il (R)s defines a poset antiisomorphism from the poset of all left semiprime operations on R to the poset of all left ideal systems on R, with inverse C −→ sprimC . Proof. This follows from Lemma 5.2.19 and the fact that, by statements (9) and (10) of Corollary 5.2.11, condition (2) (or (2 )) in the definition of a left ideal system is precisely what is needed for a closure system C on the poset Il (R) to yield a  semiprime operation sprimC on R. Definition 5.2.22. Let R be a ring and M an R-module. A nonempty set C of R-submodules  of M is a submodule closure system on M if for all subsets N of C one has N ∈ C. Let N be an R-module. If C is a submodule closure system on M, then a submodule closure system on N over C is a submodule closure system D on N such that, if L ∈ D and f ∈ Hom R (M, N ), then f −1 (L) ∈ C. Proposition 5.2.23. Let R be a ring and M an R-module. (1) If  is a closure operation on Mod R (M), then Mod R (M) is a submodule closure system on M.  (2) Conversely, if C is a submodule closure system on M, then C : L −→ {L  ∈ C : L  ⊇ L} is a closure operation on Mod R (M). (3) The association  −→ Mod R (M) defines an anti-isomorphism from the poset of all closure operations on Mod R (M) to the poset of all submodule closure systems on M, with inverse C −→ C . (4) Let  be a closure operation on Mod R (M) and N an R-module. The association   −→ Mod R (N ) restricts to an anti-isomorphism from the poset of all closure operations on Mod R (N ) compatible with  to the poset of all submodule closure systems on N over Mod R (M) .

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Proof. The proof is similar to that of Proposition 5.2.21.



The following lemma is easily verified. Lemma 5.2.24. Let S be a complete lattice. (1) The poset S S of all operations on S is a complete lattice. For any subset  of S S , one has x sup  = sup{x  :  ∈ }, x inf  = inf{x  :  ∈ }, for all x ∈ S. Moreover, if  is a set of order-preserving operations on S, then sup  and inf  are order-preserving. (2) The poset Clos(S) of all closure operations on S is a complete lattice. For any subset  of Clos(S), one has x inf  = inf{x  :  ∈ }, x sup  = inf{y ∈ S : y  x and y  = y for all  ∈ }, for all x ∈ S. Moreover, sup  is the unique closure operation  on S such that  for all x ∈ S one has x  = x if and only if x  = x for all  ∈ . (3) The poset Inter(S) of all interior operations on S is a complete lattice. For any subset  of Inter(S), one has x sup  = sup{x  :  ∈ }, x inf  = sup{y ∈ S : y  x and y  = y for all  ∈ }, for all x ∈ S. Moreover, inf  is the unique interior operation  on S such that  for all x ∈ S one has x  = x if and only if x  = x for all  ∈ . Proposition 5.2.25. Let R be a ring and M an R-module. (1) The poset of all operations on Mod R (M) is a complete lattice. For any set  of operations on Mod R (M), one has L sup  =

 {L  :  ∈ },

L inf  =

 {L  :  ∈ },

for all L ∈ Mod R (M). Moreover, if  is a set of order-preserving operations on Mod R (M), then sup  and inf  are order-preserving. (2) The poset of all closure operations on Mod R (M) is a complete lattice. For any set  of closure operations on Mod R (M), one has

5.2 Closure Operations on Ideals and Submodules

L inf  = L sup  =

373

 {L  :  ∈ },

 {N ∈ Mod R (M) : N ⊇ L and N  = N for all  ∈ },

for all L ∈ Mod R (M). Moreover, sup  is the unique closure operation  on  Mod R (M) such that for all L ∈ Mod R (M) one has L  = L if and only if L  = L for all  ∈ . (3) If  is a set of semiprime operations on M, then inf  and sup  are semiprime operations on M. Proof. Statements (1) and (2) follow immediately from Lemma 5.2.24. Moreover, statement (3) follows readily from Propostions 5.2.27 and 5.2.28 below. Indeed, let  be a set of semiprime operations on M, and let  = sup . Then  is compatible with  for all  ∈ , whence    is compatible with  for all  ∈ , so that  is compatible with  = sup  by Proposition 5.2.28 (2). Therefore sup  is a semiprime operation on M. The proof that inf  is a semiprime operation on M is similar.  The following definition plays a crucial role in the theory. Definition 5.2.26. Let R be a ring, and let M and N be R-modules. (1) For any closure operation  on Mod R (M), we let  N denote the smallest closure operation on Mod R (N ) that is compatible with . (2) For any closure operation  on Mod R (N ), we let  M denote the largest closure operation  on Mod R (M) such that  is compatible with . The next two propositions show that the closure operations  N and  M always exist. Proposition 5.2.27. Let R be a ring, let M and N be R-modules, and let  be a closure operation on Mod R (M). (1) If 1 and 2 are closure operations on Mod R (N ) such that 1  2 and 1 is compatible with , then 2 is compatible with . (2) If  is a set of closure operations on Mod R (N ) that are compatible with , then inf  is compatible with . (3) There exists a smallest closure operation  N on Mod R (N ) that is compatible with . (4) (sup ) N = sup{ N :  ∈ } for any set  of closure operations on Mod R (M). Proof. Statements (1)–(3) are easily verified. For all L ∈ Mod R (N ), one has

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L (sup ) N = L ⇔ ∀ f ∈ Hom R (M, N ) ( f −1 (L)sup  = f −1 (L)) ⇔ ∀ f ∈ Hom R (M, N ) ∀ ∈  ( f −1 (L) = f −1 (L)) ⇔ ∀ ∈  ∀ f ∈ Hom R (M, N ) ( f −1 (L) = f −1 (L)) ⇔ ∀ ∈  (L  N = L) ⇔ L sup{ N :∈} = L . Statement (4) follows.



Proposition 5.2.28. Let R be a ring, let M and N be R-modules, and let  be a closure operation on Mod R (N ). (1) If 1 and 2 are closure operations on Mod R (M) such that 2  1 and  is compatible with 1 , then  is compatible with 2 . (2) If  is a set of closure operations on Mod R (M) such that  is compatible with  for all  ∈ , then  is compatible with sup . (3) There exists a largest closure operation  = M on Mod R (N ) such that  is compatible with . (4) (inf )M = inf{M :  ∈ } for any set  of closure operations on Mod R (M). Proof. Statement (1) is clear. To prove (2), observe that, by Proposition 5.2.27, since  N   for all  ∈ , one has (sup ) N = sup{ N :  ∈ }   , so that  is compatible with sup . Statements (3) and (4) are then easily verified.  The next two theorems determine explicit formulas for the closure operations  N and  M and show that they are always semiprime operations. Theorem 5.2.29. Let R be a ring, let M and N be R-modules, and let  be a closure operation on Mod R (M). (1)  N is a semiprime operation on N . (2) L  N for all L ∈ Mod R (N ) is the smallest H ∈ Mod R (N ) containing L such that f −1 (H ) = f −1 (H ) for all f ∈ Hom R (M, N ). (3) One has Mod R (N ) N = {H ∈ Mod R (N ) : f −1 (H ) = f −1 (H ) for all f ∈ Hom R (M, N )}

and therefore L  N =

 {H ∈ Mod R (N ) : H ⊇ L , ∀ f ∈ Hom R (M, N ) ( f −1 (H ) = f −1 (H ))}

for all L ∈ Mod R (M). all f ∈ Hom R (M, N )}. Proof. Let C = {H ∈ Mod R (N ) : f −1 (H ) = f −1 (H ) for Since inverse images respect arbitrary intersections, one has N ∈ C for all N ⊆ C. Therefore C is a submodule closure system on N . By Proposition 5.2.23, we may let

5.2 Closure Operations on Ideals and Submodules

375 

 be the closure operation on Mod R (N ) corresponding to C, so that Mod R (N ) = C.  We claim that  is compatible with . Let L ∈ Mod R (N ) . Then f −1 (L) = f −1 (L)  for all f ∈ Hom R (M, N ). Therefore  is compatible with , by Proposition 5.2.8. Let 1 be any closure operation on Mod R (N ) compatible with . Let L ∈ Mod R (N ) be 1 -closed. Then f −1 (L) = f −1 (L) for all f ∈ Hom R (M, N ), again by Proposition 5.2.8, so L is  -closed. Thus   1 . It follows, then, that  =  N . This proves (2) and (3). To prove (1), let L ∈ Mod R (N ) be  N -closed, and let f ∈ End R (N ). For all g ∈ Hom R (M, N ) one has f ◦ g ∈ Hom R (M, N ) and therefore g −1 ( f −1 (L)) = ( f ◦ g)−1 (L) = ( f ◦ g)−1 (L) = g −1 ( f −1 (L)), so f −1 (L) is  N -closed. Therefore  N is a semiprime operation on N .  Theorem 5.2.30. Let R be a ring, let M and N be R-modules, and let  be a closure operation on Mod R (N ). (1)  M is a semiprime operation on M.  (2) L  M for all L ∈ Mod R (M) is the largest H ∈ Mod R (M) such that f (H ) ⊆  f (L) for all f ∈ Hom R (M, N ). (3) One has 



Mod R (M) M = {L ∈ Mod R (M) : ∀x ∈ M ((∀ f ∈ Hom R (M, N ) ( f (x) ∈ f (L) )) ⇒ x ∈ L)} 

= {L ∈ Mod R (M) : ∀H ⊆ M ((∀ f ∈ Hom R (M, N ) ( f (H ) ⊆ f (L) )) ⇒ H ⊆ L)}

and therefore 

L  M =

  { f −1 ( f (L) ) : f ∈ Hom R (M, N )} 

= {x ∈ M : ∀ f ∈ Hom R (M, N ) ( f (x) ∈ f (L) )}   = {H ∈ Mod R (M) : ∀ f ∈ Hom R (M, N ) ( f (H ) ⊆ f (L) )} for all L ∈ Mod R (M). Proof. Note first that, for any f ∈ Hom R (M, N ), the map f −1 ( ) : L −→  f −1 ( f (L) ) is a closure operation on Mod R (M). Let  = inf{ f −1 ( ) : f ∈ Hom R (M, N )}, so that  is also closure operation on Mod R (M). We claim that −1     =  M . First, since f (L  ) ⊆ f (L f ( ) ) = f ( f −1 ( f (L) )) ⊆ f (L) for all L ∈ Mod R (M) and all f ∈ Hom R (M, N ), the operation  is compatible with . Moreover, if 1 is any closure operation on Mod R (M) such that  is compatible with 1 , then for all L ∈ Mod R (M) and all f ∈ Hom R (M, N ) one has L 1 ⊆  f −1 ( f (L) ), whence L 1 ⊆ L  , so that 1  . It follows, then, that  =  M . This proves statements (2) and (3). To prove (1), let  =  M , and let L ∈ Mod R (M) and f ∈ End R (M). For  any g ∈ Hom R (M, N ), one has g( f (L  )) = (g ◦ f )(L  ) ⊆ ((g ◦ f )(L)) =   (g( f (L)) , and therefore f (L  ) ⊆ g −1 (g( f (L)) ). Since this holds for all g ∈ Hom R (M, N ), one has f (L  ) ⊆ f (L) . Therefore  is a semiprime operation on M. 

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5 Noncommutative Rings and Closure Operations on Submodules

Proposition 5.2.25 justifies the following definitions. Definition 5.2.31. Let R be a ring, let M be an R-module, and let  be a closure operation on Mod R (M). (1)  denotes the smallest semiprime operation on M greater than or equal to . (2)  denotes the largest semiprime operation on M less than or equal to . Corollary 5.2.32. Let R be a ring, let M be an R-module, and let  a closure operation on Mod R (M). (1) One has  =  M . In particular,  is a semiprime operation on M if and only if  M  . (2) One has  = M . In particular,  is a semiprime operation on M if and only if   M . Corollary 5.2.33. Let R be a ring, let M and N be R-modules, let  be a closure operation on Mod R (M), and let  be a closure operation on Mod R (N ). The following conditions are equivalent. (1) (2) (3) (4)

 is compatible with .  is compatible with  .  is compatible with .  is compatible with  .

In particular, one has  N =  N and  M =  M . Proof. If  is compatible with , then    M , so    M since  M is a semiprime operation, so  is compatible with  . The converse is clear since   . Thus (1) ⇔ (2) and (3) ⇔ (4). A similar argument shows that (1) ⇔ (3). One easily checks the last statement of the proposition.  Corollary 5.2.32 invokes the special case of Theorems 5.2.29 and 5.2.30 where M = N . Another special case is where M = R. Corollary 5.2.34. Let R be a ring and M an R-module, and let s be a closure operation on Il (R). One has L s M =

 {H ∈ Mod R (M) : H ⊇ L and (H : R x)s = (H : R x) for all x ∈ M}

for all L ∈ Mod R (M) and Mod R (M)s M = {H ∈ Mod R (M) : (H : R x)s = (H : R x) for all x ∈ M}. Corollary 5.2.35. Let R be a ring and s a closure operation on Il (R). Then Il (R) s

is the largest left ideal system on R contained in Il (R)s , and one has Il (R) s = {I ∈ Il (R) : (I : R a)s = (I : R a) for all a ∈ R}.

5.2 Closure Operations on Ideals and Submodules

377

Corollary 5.2.36. Let R be a ring and M an R-module, and let  be a closure operation on Mod R (M). One has I R =

 {((I x) : R x) : x ∈ M}

= {a ∈ R : ∀x ∈ M (ax ∈ (I x) )}  = {J ∈ Il (R) : ∀x ∈ M (J x ⊆ (I x) )} for all I ∈ Il (R) and Il (R)R = {I ∈ Il (R) : ∀a ∈ R ((∀x ∈ M (ax ⊆ (I x) )) ⇒ a ∈ I )} = {I ∈ Il (R) : ∀J ∈ Il (R) ((∀x ∈ M (J x ⊆ (I x) )) ⇒ J ⊆ I )} Corollary 5.2.37. Let R be a ring and s a closure operation on Il (R). Then Il (R)s is the smallest left ideal system containing Il (R)s , and one has Il (R)s = {I ∈ Il (R) : ∀a ∈ R ((∀b ∈ R (ab ⊆ (I b)s )) ⇒ a ∈ I )} = {I ∈ Il (R) : ∀J ∈ Il (R) ((∀b ∈ R (J b ⊆ (I b)s )) ⇒ J ⊆ I )} Definition 5.2.38. Let R be a ring, and let M and N be R-modules. (1) An operation  on Mod R (M) is an N -subprime operation on M, or N -subprime, if  =  M for some closure operation  on Mod R (N ). (2) An operation  on Mod R (N ) is an M-surprime operation on N , or M-surprime, if  =  N for some closure operation  on Mod R (M). Note that any N -subprime or M-surprime operation is a semiprime operation. Example 5.2.39. (1) A semiprime operation on an R-module M is equivalently an M-subprime operation on M, or equivalently an M-surprime operation on M. (2) A rigid semiprime operation on a commutative ring R, in the sense of Definition 4.3.15, is equivalently a T (R)-subprime operation on R. Theorem 5.2.48 provides equivalent characterizations of the rigid semiprime operations on a commutative ring. Combining Propositions 5.2.27 and 5.2.28, we obtain the following. Theorem 5.2.40. Let R be a ring, and let M and N be R-modules. (1) There is a monotone Galois connection between the poset Clos(Mod R (M)) of all closure operations on Mod R (M) and the poset Clos(Mod R (N )) of all closure operations on Mod R (N ) given by the pair of maps  −→  N and  −→  M . For any closure operation  on Mod R (M) and any closure operation  on Mod R (N ), one has  N   if and only if    M , if and only if  is compatible with .

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5 Noncommutative Rings and Closure Operations on Submodules

(2) For any closure operation  on Mod R (M), the closure operation  N is an M-surprime operation on N , and  N M is the smallest N -subprime operation on M greater than or equal to . (3) For any closure operation  on Mod R (N ), the closure operation  M is an N -subprime operation on M, and  M N is the largest M-surprime operation on N less than or equal to  . (4) The map  −→  N , with inverse  −→  M , is an isomorphism from the poset of all N -subprime operations on M to the poset of all M-surprime operations on N . (5) If  is a set of closure operations on Mod R (N ), then (inf )M = inf{ M :  ∈ }. Consequently, the infimum of a set of N -subprime operations on M is N -subprime. (6) If  is a set of closure operations on Mod R (M), then (sup ) N = sup{ N :  ∈ }. Consequently, the supremum of a set of M-surprime operations on N is M-surprime. Corollary 5.2.41. Let R be a ring, and let M and N be R-modules. For any closure operation  on Mod R (M), the following conditions are equivalent.  is N -subprime.  =  M for some semiprime operation  on N .  =  N M .    N M . L  = {x ∈ M : ∀ f ∈ Hom R (M, N ) ( f (x) ∈ f (L) N )} for any L ∈ Mod R (M). L  for any L ∈ Mod R (M) is the largest H ∈ Mod R (M) such that for all f ∈ Hom R (M, N ) one has f (H ) ⊆ P for all P ∈ Mod R (M) such that f (L) ⊆ P and g −1 (P) = g −1 (P) for all g ∈ Hom R (M, N ). (7) L  = L (if and) only if for all x ∈ M one has x ∈ L provided that f (x) ∈ f (L) N for all f ∈ Hom R (M, N ), for any L ∈ Mod R (M). (8) L  = L (if and) only if for all x ∈ M one has x ∈ L provided that f (x) ∈ P for all f ∈ Hom R (M, N ) and all P ∈ Mod R (N ) such that f (L) ⊆ P and g −1 (P) = g −1 (P) for all g ∈ Hom R (M, N ), for any L ∈ Mod R (M).

(1) (2) (3) (4) (5) (6)

Corollary 5.2.42. Let R be a ring, and let M and N be R-modules. For any closure operation  on Mod R (N ), the following conditions are equivalent.  is M-surprime.  =  N for some semiprime operation  on M.  =  M N .    M N .  L  for any L ∈ Mod R (N ) is the smallest P ∈ Mod R (N ) containing L such that  f −1 (P) M = f −1 (P) for all f ∈ Hom R (M, N ).  (6) L  for any L ∈ Mod R (N ) is the smallest P ∈ Mod R (N ) containing L such  that for all f ∈ Hom R (M, N ) one has f (x) ∈ P if g(x) ∈ g( f −1 (P)) for all g ∈ Hom R (M, N ).   (7) L  = L if (and only if) f −1 (L) M = f −1 (L) for all f ∈ Hom R (M, N ), for any L ∈ Mod R (N ). (1) (2) (3) (4) (5)

5.2 Closure Operations on Ideals and Submodules

379



(8) L  = L if (and only if) f −1 (L) for all f ∈ Hom R (M, N ) is the largest H ∈  Mod R (M) such that g(H ) ⊆ g( f −1 (L)) for all g ∈ Hom R (M, N ), for any L ∈ Mod R (N ). Again, important special cases are where M = N or M = R. Corollary 5.2.43. Let R be a ring and M an R-module. (1) There is a monotone Galois connection between the poset Clos(Mod R (M)) and itself given by the pair of maps  −→  and  −→  . (2) There is a monotone Galois connection between the poset Clos(Il (R)) and the poset Clos(Mod R (M)) given by the pair of maps s −→ s M and  −→ R . (3) The map s −→ s M , with inverse  −→ R , is an isomorphism from the poset of all M-subprime operations on R to the poset of all R-surprime operations on M. Corollary 5.2.44. Let R be a ring and M an R-module. For any closure operation s on Il (R), the following conditions are equivalent. (1) (2) (3) (4) (5) (6)

s is M-subprime. s = R for some semiprime operation  on M. s = s M R . s  s M R . I s = {a ∈ R : ∀x ∈ M (ax ∈ (I x)s M )} for any I ∈ Il (R). I s for any I ∈ Il (R) is the largest J ∈ Il (R) such that J x ⊆ P for all x ∈ M and all P ∈ Mod R (M) such that I x ⊆ P and (P : R y)s = (P : R y) for all y ∈ M. (7) I s = I (if and) only if for all a ∈ R one has a ∈ I provided that ax ∈ (I x)s M

for all x ∈ M, for any I ∈ Il (R). (8) I s = I (if and) only if for all a ∈ R one has a ∈ I provided that ax ∈ P for all x ∈ M and all P ∈ Mod R (M) such that I x ⊆ P and (P : R y)s = (P : R y) for all y ∈ M, for any I ∈ Il (R). Corollary 5.2.45. Let R be a ring and M an R-module. For any closure operation  on Mod R (M), the following conditions are equivalent.

 is R-surprime.  = s M for some semiprime operation s on R.  = R M .   R M . L  for any L ∈ Mod R (M) is the smallest P ∈ Mod R (N ) containing L such that (P : R x)R = (P : R x) for all x ∈ N . (6) L  for any L ∈ Mod R (M) is the smallest P ∈ Mod R (N ) containing L such that  (P : R x) for all x ∈ N is the largest J ∈ Il (R) such that J y ⊆ ((P : R x)y) for all y ∈ N . (7) L  = L if (and only if) (L : R x)R = (L : R x) for all x ∈ M, for any L ∈ Mod R (M). (8) L  = L if (and only if), for all a ∈ R and all x ∈ M, one has ax ∈ L provided that ay ∈ ((L : R x)y) for all y ∈ M, for any L ∈ Mod R (M). (1) (2) (3) (4) (5)

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Example 5.2.46. Let R be a ring and F a free R-module. Any semiprime operation on R is F-subprime. Consequently, the map s −→ s F is an isomorphism from the poset of all semiprime operations on R to the poset of all R-surprime operations on F. Recall from Definition 4.3.15 that a semiprime operation s on a commutative ring R is rigid if s = sprim for some semistar operation  on R. As an application of the two corollaries above, we provide some equivalent characterizations of the rigid semiprime operations. Definition 5.2.47. A semistar operation on a commutative ring R is rigid if it is an R-surprime operation on T (R). Theorem 5.2.48. Let R be a commutative ring. (1) A semistar operation on R is equivalently a semiprime operation on the Rmodule T (R). Moreover, for any semistar operation  on R, one has R = sprim , that is, I R = I  ∩ R for all I ∈ I(R). (2) For any closure operation s on I(R), the following conditions are equivalent. (a) s is a rigid semiprime operation on R. (b) s is T (R)-subprime. (c) I s = I s T (R) ∩ R for all I ∈ I(R). (d) I s = I (if and) only if I = J ∩ R for some J ∈ K(R) such that (J : R x)s = (J : R x) for all x ∈ T (R), for all I ∈ I(R). (e) I s = I (if and) only if I = J ∩ R for some J ∈ K(R) such that (a J ∩ R)s = a J ∩ R for all regular a ∈ R, for all I ∈ I(R). (3) For any closure operation  on K(R), the following conditions are equivalent. (a)  is a rigid semistar operation on R. (b)  is of the form s T (R) for some semiprime operation s on R. (c) I  = I R T (R) for all I ∈ K(R). (d) I  = I if (and only if) (I : R x) = (I : R x) ∩ R for all x ∈ T (R), for all I ∈ K(R). (e) I  = I if (and only if) for all x ∈ T (R) one has (I : R x) = J  ∩ R for some J ∈ K(R), for all I ∈ K(R). (f) I  = I if (and only if) a I ∩ R = (a I ∩ R) ∩ R for all regular a ∈ R, for all I ∈ K(R). (4) For any closure operation s on I(R), the operation s T (R) is the smallest semistar operation  = d s such that s  sprim , and the operation sprims T (R)

is the largest rigid semiprime operation on R greater than or equal to s. (5) For any closure operation  on K(R), the operation sprim  T (R) is the smallest rigid semistar operation on R less than or equal to . (6) The association s −→ s T (R) , with inverse  −→ sprim , defines an isomorphism from the poset of all rigid semiprime operations on R to the poset of all rigid semistar operations on R. (7) The association s −→ s T (R) , with inverse  −→ sprim , defines an isomorphism from the poset of all finite type rigid semiprime operations on R to the poset of all finite type rigid semistar operations on R.

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By Theorem 4.3.32, we also have the following. Theorem 5.2.49. Let R be a commutative ring. (1) A finite type semiprime operation on R is rigid if and only if it is standard. (2) Any finite type semistar operation on R is rigid. (3) The association s −→ s T (R) , with inverse  −→ sprim , defines an isomorphism from the poset of all finite type standard semiprime operations on R to the poset of all finite type semistar operations on R.

5.3 Stable and Finite Type Closure Operations In this section, we study the finite type and the stable closure operations, and we introduce an analogue for semiprime operations of the divisorial semistar operations. Definition 5.3.1. Let R be a ring, let M and N be R-modules, let  be an orderpreserving operation on Mod R (M), and let  be an order-preserving operation on  Mod R (N ). We say that  is -stable if f −1 (L  ) = f −1 (L) for all L ∈ Mod R (N ) and all f ∈ Hom R (M, N ). If  is -stable, then  is compatible with . Thus the stability condition can be thought of as a strengthening of the compatibility condition. Proposition 5.3.2. Let R be a ring, let M and N be R-modules, let , 1 , and 2 be order-preserving operations on Mod R (M), and let  , 1 , and 2 be order-preserving operations on Mod R (N ). (1) (2) (3) (4) (5)

If  is -stable, then  is compatible with . If  is 1 -stable, and if 1  2 and  is compatible with 2 , then  is 2 -stable. If 1 is -stable, and if 1  2 and 2 is compatible with , then 2 is -stable. If  is -stable, then  is  M -stable and  -stable. If  is -stable, then  N and  are -stable.

Proof. Statements (1)–(3) follow easily from the definitions of -stability and compatibility. Statement (4) follows easily from (2) and Corollary 5.2.33. Likewise, (5) follows easily from (3).  Thus, a necessary condition for an order-preserving operation  on Mod R (N ) to be -stable, where  is an order-preserving operation on Mod R (M), is that  be  M -stable and  N be -stable. A case of stability discussed in the literature is where M = N . A closure operation  on Mod R (M) is said to be modular if f −1 (L  ) = f −1 (L) for all L ∈ Mod R (M) and all f ∈ End R (M) [34], that is, if  is -stable. Clearly, every modular closure operation on Mod R (M) is a semiprime operation on M.

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Proposition 5.3.3. Let R be a ring, let M be an R-module, and let  and  be closure operations on Mod R (M). Then  is -stable if and only if  is a modular semiprime operation on M and  = . The following case of stability is much more general. Proposition 5.3.4. Let R be a ring, let M and N be R-modules, and let  be a closure operation on Mod R (M). The following conditions are equivalent.  N is -stable.  is -stable for some semiprime operation  on N .  is -stable for some closure operation  on Mod R (N ). f −1 (L  N ) = f −1 (L) for all L ∈ Mod R (N ) and all f ∈ Hom R (M, N ). f −1 (L  N ) ⊆ f −1 (L) for all L ∈ Mod R (N ) and all f ∈ Hom R (M, N ). For all L ∈ Mod R (N ) and all x ∈ M, and for all f ∈ Hom R (M, N ), if f (x) ∈ L  N , then x ∈ f −1 (L) . (7) For all L ∈ Mod R (N ), for all x ∈ M, and for all f ∈ Hom R (M, N ), if f (x) ∈ H for every H ∈ Mod R (N ) containing L such that g −1 (L) = g −1 (L) for all g ∈ Hom R (M, N ), then x ∈ f −1 (L) .

(1) (2) (3) (4) (5) (6)

Moreover, if the conditions above hold, then  N is the smallest closure operation on Mod R (N ) that is -stable. For us an important case of stability is where M = R. Definition 5.3.5. Let R be a ring and M an R-module. A closure operation  on Mod R (M) is stable if  is R -stable. A semiprime operation s on a ring R is stable if and only if s is s-stable, if and only if s is modular. More generally, a closure operation s on Il (R) is stable if and only if s is s stable, if and only if s = s is a stable semiprime operation on R. Corollary 5.3.6. Let R be a ring, let M be an R-module, and let s be a closure operation on Il (R). The following conditions are equivalent. (1) (2) (3) (4) (5) (6) (7)

s M is s-stable.  is s-stable for some semiprime operation  on M.  is s-stable for some closure operation  on M. (L s M : R x) = (L : R x)s for all L ∈ Mod R (M) and all x ∈ M. s L s M = {x  ∈ MM: (L : R x) = R} fors all L ∈ Mod R (M). s M

= {(J \ L) : J ∈ Il (R), J = R} for all L ∈ Mod R (M). L For every L ∈ Mod R (M), if for all x ∈ M one has x ∈ L whenever (L : R x)s = R, then (L : R x)s = (L : R x) for all x ∈ M.

Consequently, if there exists an order-preserving operation  on Mod R (M) that is s-stable, then  = s M . Corollary 5.3.7. Let R be a ring and s a closure operation Il (R). The following conditions are equivalent.

5.3 Stable and Finite Type Closure Operations

(1) (2) (3) (4)

383

s is a stable semiprime operation on R. s is s-stable. (I s : R a) = (I : R a)s for all I ∈ Il (R) and all a ∈ R. I s = {a ∈ R : (I : R a)s = R} for all I ∈ Il (R).

Moreover, if the conditions above hold, then s M for any R-module M is the unique s-stable closure operation on Mod R (M). Corollary 5.3.8. Let R be a ring and s a closure operation Il (R). Let  be an s-stable closure operation on Mod R (M).  (1) L  = {x ∈ M : (L : R x)s = R} = {(J \ M L) : J ∈ Il (R), J s = R} for all L ∈ Mod R (M). (2)  = s M . (3)  is stable and -stable. (4) (H ∩ L) = H  ∩ L  for all H, L ∈ Mod R (M). Corollary 5.3.9. Let R be a ring and M an R-module, and let  be a closure operation on Mod R (M). If  is stable, then  is R-surprime and -stable and is therefore a semiprime operation on M. Moreover, the following conditions are equivalent. (1) (2) (3) (4)

 and R are stable.  is R-surprime and R is stable.  is s-stable for some stable semiprime operation  s on R.  acts by L −→ {x ∈ M : (L : R x)s = R} = {(J \ M L) : J ∈ Il (R), J s = R} for some stable semiprime operation s on R.

Proof. If  is stable, that is, R -stable, then R M is R -stable by Proposition 5.3.2(5), and therefore  = R M by Corollary 5.3.8. Therefore  is R-surprime. Moreover,  is -stable by Corollary 5.3.8. Finally, one easily verifies that conditions (1)–(4) are equivalent.  Open Problem 5.3.10. Let R be a ring and M an R-module, and let  be a closure operation on Mod R (M). If  is R-surprime and -stable, then must  be stable? The following proposition and corollary are dual to Proposition 5.3.4 and Corollary 5.3.6, respectively. Proposition 5.3.11. Let R be a ring, let M and N be R-modules, and let  be a closure operation on Mod R (N ). The following conditions are equivalent. (1) (2) (3) (4) (5) (6)

 is  M -stable.  is -stable for some semiprime operation  on M.  is -stable for some closure operation  on Mod R (M).   f −1 (L  ) = f −1 (L) M for all L ∈ Mod R (N ) and all f ∈ Hom R (M, N ).  −1  −1 f (L ) ⊆ g (g( f −1 (L)) ) for all L ∈ Mod R (N ) and all f ∈ Hom R (M, N ). −1  −1  g( f (L ) ⊆ g( f (L)) for all L ∈ Mod R (N ) and all f, g ∈ Hom R (M, N ).

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Corollary 5.3.12. Let R be a ring and M an R-module, and let  be a closure operation on Mod R (M). The following conditions are equivalent. (1) (2) (3) (4) (5) (6) (7) (8)

 is stable.  is s-stable for some semiprime operation s on R.  is s-stable for some closure operation s on Il (R). (L  : R x) = (L : R x)R for all L ∈ Mod R (M) and all x ∈ M. (L  : R x) ⊆ (((L : R x)y) : R y) for all L ∈ Mod R (M) and all x, y ∈ M. (L  : R x)y ⊆ ((L : R x)y) for all L ∈ Mod R (M) and all x, y ∈ M. L  = {x ∈ M : (L : R x)R = R)} for all L ∈ Mod R (M). L  = {x ∈ M : ∀y ∈ M (y ∈ ((L : R x)y) )} for all L ∈ Mod R (M).

Example 5.3.13. Let  be a semistar operation on a commutative ring R. One can show that the following conditions are equivalent. (1)  is a stable semistar operation, in the sense of Definition 4.2.7. (2)  is stable as a closure operation on Mod R (T (R)). (3)  is sprim -stable, where sprim : I −→ I  ∩ R is the semiprime operation on R induced by . (4) (I  : R x) = (I : R x) ∩ R for all I ∈ K(R) and all x ∈ T (R). (5) (I  : R J ) = (I : R J ) ∩ R for all I, J ∈ K(R) with J finitely generated. Remark 5.3.14. Let R be a commutative ring and M an R-module. A stable semiprime operation on M is equivalently a componental operation on Mod R (M) in the sense of Kirby [119]. Next, we focus in particular on the stable semiprime operations on a ring (i.e., the case M = N = R). The following definition is due to Gabriel [75]. Definition 5.3.15. Let R be a ring. A nonempty set L of left ideals of R is a (left) ideal filter on R if the following two conditions hold. (1) If I ∈ L and J is a left ideal of R containing I , then J ∈ L. (2) I ∩ J ∈ L for all I, J ∈ L. An ideal filter L on R is said to be uniform, or topologizing, if the following condition holds. (3) If I ∈ L and a ∈ R, then (I : R a) ∈ L. A uniform ideal filter L on R is said to be a (left) Gabriel filter on R if the following condition holds. (4) If I ∈ L and J is a left ideal of R such that (J : R a) ∈ L for all a ∈ I , then J ∈ L. Remark 5.3.16. Let R be a ring. (1) Conditions (3) and (4) of Definition 5.3.15 imply conditions (1) and (2), and therefore they suffice for L to be a Gabriel filter on R [177, Lemma VI.5.2].

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(2) By [177, Lemma VI.5.3], for any Gabriel filter L on R, one has R ∈ L and I J ∈ L for all I, J ∈ L. (3) If L is a uniform ideal filter on R, then the smallest Gabriel filter on R containing L is the set {I ∈ Il (R) : ∀J ∈ Il (R) (J ⊇ I, J = R ⇒ ∃a ∈ R − J ((J : R a) ∈ L))} [177, Proposition VI.5.4]. (4) Suppose that R is commutative. Then condition (3) of Definition 5.3.15 follows from condition (1), that is, every ideal filter on R is uniform. A Gabriel filter on R is also called a localizing system on R. If condition (4) holds, then {I ∈ I(R) : I ⊇ J for some J ∈ L} is the smallest Gabriel filter containing L. Remark 5.3.17 ([75, Chapter V]). Let L be an ideal filter on a ring R. An R-module M is L-torsion if ann R (x) ∈ L (or equivalently I x = 0 for some I ∈ L) for all x ∈ M. The Gabriel composition L  L of ideal filters L and L on R is given by L  L = {I ∈ Il (R) : J/I is L-torsion for some J ∈ L }. The ideal filter L  L is the unique ideal filter L such that an R-module M is L -torsion if and only if 0 −→ N −→ M −→ P −→ 0 is exact for some L-torsion R-module N and some L -torsion R-module P. Moreover, L is uniform if and only if L ⊆ L  {R}, and a uniform ideal filter L is a Gabriel filter if and only if L  L = L. For this reason, Gabriel filters are also known as idempotent (topologizing) filters. Remark 5.3.18. A left Gabriel filter on a ring R is equivalently a D-system of left ideals of R in the sense of [34]. Definition 5.3.19. A ring topology on a ring R is a topology on R such that the operations of addition, subtraction, and multiplication are continuous maps from R × R to R. A ring topology on a ring R is said to be (left) linear if R has a fundamental system of neighborhoods of 0 consisting of left ideals of R. The following proposition and theorem are provedin [177, Chapter VI Section 4]. Proposition 5.3.20 ([177, Chapter VI Section 4]). Let R be a ring. The set of all open left ideals of any linear topology on R is a uniform ideal filter on R. Conversely, any uniform ideal filter on R is the set of all open left ideals of R with respect to a unique linear topology on R. Thus, there is a one-to-one correspondence between the set of all uniform ideal filters on R and the set of all linear topologies on R. Moreover, this one-to-one correspondence is an isomorphism of posets. By Proposition 5.3.20, any uniform ideal filter L on a ring R forms a fundamental system of neighborhoods of 0 of a unique (linear) ring topology on R. If L is a Gabriel filter on R, then the corresponding linear topology on R is called the Gabriel topology on R associated to L.

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Definition 5.3.21. Let R be a ring. (1) A presemiprime operation on R is a preclosure operation s on Il (R) such that I s a ⊆ (I a)s for all I ∈ Il (R) and all a ∈ R. (2) An order-preserving operation s on Il (R) is filtered if (1) R s = R and (2) I s = R and J s = R implies (I ∩ J )s = R for all I, J ∈ Il (R). (3) An order-preserving operation s on Il (R) is intersection-preserving if (I ∩ J )s = I s ∩ J s for all I, J ∈ Il (R). Lemma 5.3.22. Let R be a ring. (1) Any semiprime operation on R is filtered, and any stable semiprime operation on R is intersection-preserving. (2) A stable presemiprime operation on R is filtered if and only if it is intersectionpreserving. (3) If s is any presemiprime operation on R, then the idempotent hull s∞ of s is a semiprime operation on R. (4) If s is any stable preclosure on Il (R), then s is a presemiprime operation on R, and s∞ is a stable semiprime operation on R. Theorem 5.3.23. Let R be a ring. (1) If s is any order-preserving operation on R, then Filts = {I ∈ Il (R) : I s = R} is an ideal filter on R if and only if s is filtered. (2) If s is any stable intersection-preserving presemiprime operation on R, then Filts is a uniform ideal filter on R. (3) If L is any uniform ideal filter on R, then the operation SIPL on Il (R) acting by I SIPL : I −→ {a ∈ R : (I : R a) ∈ L} =



{(J \ R I ) : J ∈ L}

for all I ∈ Il (R) is a stable intersection-preserving presemiprime operation on R. (4) The association s −→ Filts , with inverse L −→ SIPL , defines an isomorphism from the poset of all stable intersection-preserving presemiprime operations on R to the poset of all uniform ideal filters on R. (5) For any pair (s, L) with s = SIPL and L = Filts , the following conditions are equivalent. (a) L is a Gabriel filter on R. (b) (I s )s = R implies I s = R for all I ∈ Il (R). (c) s is idempotent. (d) s is a (stable) semiprime operation on R. (6) The association s −→ Filts , with inverse L −→ SIPL , defines an isomorphism from the poset of all stable semiprime operations on R to the poset of all Gabriel filters on R.

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Example 5.3.24. Let R be a commutative ring. (1) Let S be a commutative R-algebra. The map I −→ I S ∩ R on I(R) is a semiprime operation on R. The given semiprime operation is stable if S is a flat R-algebra, and the corresponding Gabriel filter is the set {I ∈ I(R) : I S = S}. (2) The ordered monoid Ireg (R) of all regular ideals of R is a Gabriel  filter on R. The corresponding semiprime operation r 1 acts by r 1 : I −→ J ∈Ireg (R) (I : R J ) = I K ∩ R, where K is the total quotient ring of R. Theorem 5.3.25. Let R be a ring. (1) If L is any ideal filter on R, then L = {I ∈ Il (R) : (I : R r ) ∈ L for all r ∈ R} is the largest uniform ideal filter on R contained in L. (2) Let s be any filtered presemiprime operation on R. Then Filts = {I ∈ Il (R) : (I : R r )s = R for all r ∈ R} is the largest uniform ideal filter on R contained in Filts . Moreover, the operation s = SIPFilts on Il (R) acts by s : I −→ {a ∈ R : (I : R ra)s = R for all r ∈ R}  = {(J \ R I ) : (J : R r )s = R for all r ∈ R}} for all I ∈ Il (R) and is the largest stable intersection-preserving presemiprime operation on R less than or equal to s. (3) Let s be any semiprime operation on R. Then s is the largest stable semiprime operation less than or equal to s. The theorem justifies the following definition. Definition 5.3.26. Let R be a ring and s a semiprime operation on R. Then s denotes the largest stable semiprime operation on R less than or equal to s. The following proposition provides a useful condition under which the construction of s is much simpler. Proposition 5.3.27. Let R be a ring and s a semiprime operation on R. Let Filts = {I ∈ Il (R) : I s = R}. Then Filts is an ideal filter on R that is closed under multiplication and satisfies conditions (1), (2), and (4) of Definition 5.3.15. Moreover, the following conditions are equivalent. (1) Filts is a Gabriel filter on R. (2) Filts is a uniform ideal filter on R. (3) I s = R implies (I : R r )s = R for all I ∈ Il (R) and all r ∈ R.

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(4) I s = R if (and only if) I s = R for all I ∈ Il (R). s (5) I s = {a  ∈ R R: (I : R a) = R} fors all I ∈ Il (R). s (6) I = {(J \ I ) : J ∈ Il (R), J = R} for all I ∈ Il (R). Definition 5.3.28. A semiprime operation s on a ring R is semistable if it satisfies the equivalent conditions of Proposition 5.3.27. Clearly, any stable semiprime operation is semistable. Corollary 5.3.29. Let s be a semiprime operation on a commutative ring R. Then s  is semistable, and one has I s = {a ∈ R : (I : R a)s = R} = {(I : R J ) : J ∈ I(R), J s = R} for all ideals I of R. We note that the radical, integral closure, and tight closure semiprime operations are very much non-stable, in the following sense. Definition 5.3.30. A semiprime operation s on a ring R is antistable if it satisfies the equivalent conditions of Proposition 5.3.31 below. Proposition 5.3.31. Let R be a ring and s a semiprime operation on R. The following conditions are equivalent. (1) (2) (3) (4)

d : I −→ I is the only stable semiprime operation less than or equal to s. s is equal to the trivial semiprime operation d : I −→ I on R. I s = R if and only if I = R for all I ∈ Il (R). R is the only left ideal I of R such that (I : R r )s = R for all r ∈ R.

Moreover, s is antistable and semistable if and only if R is the only left ideal I of R such that I s = R. Example 5.3.32. Let R be a ring. (1) The trivial operation d is the only semiprime operation on R that is both stable and antistable. (2) Any semiprime operation less than or equal to an antistable semiprime operation is antistable. √ (3) Suppose that R is commutative. The radical semiprime operation I −→ I on R is antistable√(and semistable). Moreover, since the integral closure I of I is contained in I and the tight closure I ∗ of I (when defined) is contained in I , the integral closure and tight closure semiprime operations are antistable as well. We now turn our attention to the finite type semiprime operations. Definition 5.3.33. Let R be a ring and M an R-module, and let  be an orderpreserving operation on Mod R (M). The operation  is of finite type if N =

 {L  : L ∈ Mod R (M) is f.g. and L ⊆ N }

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for all N ∈ Mod R (M). We let t denote the operation on Mod R (M) acting by t : N −→ N t =

 {L  : L ∈ Mod R (M) is f.g. and L ⊆ N }

for all N ∈ Mod R (M). The unions in the definition above can be replaced with sums. An order-preserving operation  on Mod R (M) is of finite type if and only if the map  is continuous when Mod R (M) is endowed with the Scott topology. Proposition 5.3.34. Let R be a ring, let M and N be R-modules, let  be an orderpreserving operation on Mod R (M), and let  be an order-preserving operation on Mod R (N ). (1) t is the largest finite type order-preserving operation on Mod R (M) less than or equal to . (2)  is of finite type if and only if  = t . (3) If  is a closure operation (resp., preclosure operation, interior operation, preinterior operation), then so is t . (4) If  is compatible with , then t is compatible with t . (5) If  is a semiprime operation on M, then so is t . (6) t N   N t ; equivalently, if  is of finite type, then  N is of finite type. (7)  M t  t M . Definition 5.3.35. Let R be a ring. (1) A left ideal filter L on R is of finite type if every element of L contains a finitely generated left ideal in L. (2) A linear topology on R is of finite type if it has a fundamental system of neighborhoods of 0 consisting of finitely generated left ideals of R. Proposition 5.3.36. Let R be a ring and s a semistable semiprime operation on R, and let L = {I ∈ Il (R) : I s = R} be the Gabriel filter corresponding to s. The following conditions are equivalent. (1) (2) (3) (4)

s is of finite type. L is of finite type. The Gabriel topology associated to L is of finite type.  I s = {(J \ R I ) : J ∈ L, J f.g.} = {a ∈ R : J a ⊆ I for some f.g. J ∈ L}.

Moreover, if s is of finite type, then s is also of finite type. Proof. Since s is semistable, by Proposition 5.3.27 one has I s = {a ∈ R : (I : R a)s = R} for all I ∈ Il (R) and L = {I ∈ Il (R) : I s = R}. From this it is trivial to verify that the four conditions of the proposition are equivalent. Suppose now that s is of finite type. Let I ∈ L, so that I s = R. Since 1 ∈ I s , it follows that 1 ∈ J s , hence J ∈ L, for some finitely generated left ideal J ⊆ I . Therefore L is of finite  type, whence s is of finite type.

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Unfortunately, we are unable to show, when s is not semistable, that s is of finite type whenever s is of finite type. This prevents us from providing an explicit construction of the largest finite type stable semiprime operation sw less than or equal to s, for any semiprime operation s. If R is commutative, then sw = st , but for general rings the construction is not so simple. What will allow us to construct sw for general rings is Lemma 5.2.24(3). Indeed, observe that the maps s : s −→ s, t : s −→ st are interior operations on the complete lattice Semiprime(R) of all semiprime operations on R, so by Lemma 5.2.24(3) we may let w = inf{s, t} : s −→ sw as interior operations on Semiprime(R). Then, since w = inf{s, t}, one has sw = s if and only if ss = s and st = s, if and only if s is stable and of finite type. Consequently, sw = sw is the largest finite type stable semiprime operation less than or equal to s. Thus we have the following. Proposition 5.3.37. Let R be a ring and s a semiprime operation on R. Then sw = sw is the largest finite type stable semiprime operation less than or equal to s. Moreover, the Gabriel filter corresponding to sw is the largest finite type Gabriel filter contained in {I ∈ Il (R) : I s = R}. Furthermore, if s is semistable or R is commutative, then sw = st . We also note the following. Proposition 5.3.38. Let  be a set of semiprime operations on a ring R. One has the following. (1) inf  = inf{s : s ∈ }. (2) If  is finite, then (inf )t = inf{st : s ∈ } and (inf )w = inf{sw : s ∈ }.

5.4 Divisorial and Codivisorial Semiprime Operations The divisorial closure operations, defined and constructed in Theorem 5.4.2 below, are useful for providing an explicit construction of all of the N -surprime operations on M and all of the M-subprime operations on N for any R-modules M and N . Definition 5.4.1. Let M and N be sets, let X be a subset of M and Y a subset of N , and let H ⊆ M N . We let  Y ·H= { f (Y ) : f ∈ H}

5.4 Divisorial and Codivisorial Semiprime Operations

391

denote the smallest subset of N containing the image of Y under f for all f ∈ H. We also let  {K ⊆ H : Y · K ⊆ X } (Y \H X ) = = { f ∈ H : f (Y ) ⊆ X } denote the largest subset K of H such that Y · K ⊆ X , and we let (X : H) =

 {Z ⊆ N : Z · H ⊆ X }

= {z ∈ N : ∀ f ∈ H ( f (z) ∈ X )}  = { f −1 (X ) : f ∈ H} denote the largest subset Z of N such that Z · H ⊆ X . Theorem 5.4.2. Let R be a ring, let M and N be R-modules, and let P be an R-submodule of M. (1) The operation

 M

= P} vM N (P) = sup{ ∈ Clos(N ) : P

on N is the largest closure operation and largest semiprime operation  on N such that P  M = P. (2) For all L ∈ Mod R (N ), one has L v N (P) = (P : (L \Hom R (N ,M) P))  = {H ∈ Mod R (N ) : (L \Hom R (N ,M) P) ⊆ (H \Hom R (N ,M) P)}  = {H ∈ Mod R (N ) : ∀ f ∈ Hom R (N , M) ( f (L) ⊆ P ⇒ f (H ) ⊆ P)}, M

and L v N (P) is the largest H ∈ Mod R (N ) such that f (H ) ⊆ P for all f ∈ Hom R (N , M) such that f (L) ⊆ P. Moreover, L ∈ Mod R (N ) is v M N (P)-closed if and only if L = (P : H) for some subset H of Hom R (N , M). M Let M ⊆ Mod R (M). The operation v M N (M) = inf{v N (P) : P ∈ M} is the largest closure operation and largest semiprime operation  on N such that P  M = P for all P ∈ M. For any closure operation  on N , one has    M

= P for all P ∈ M. vM N (M) if and only if P M One has  = v N (Mod R (M) ) M for any N -surprime operation  on M. Thus, every N -surprime operation  on M is of the form v M N (M) M for some M ⊆ Mod R (M).  M

) M N for any M-subprime operation  on One has  = v M N (Mod R (M) N . Thus, every M-subprime operation  on N is of the form v M N (M) M N for some M ⊆ Mod R (M).  One has  = v M M (Mod R (M) ) for any semiprime operation  on M. Thus, every semiprime operation  on M is of the form v M M (M) for some M ⊆ Mod R (M). M

(3)

(4)

(5)

(6)

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Proof. First, observe that, by Corollary 5.2.25(2) and Proposition 5.2.27(4), one has P v N (P) M = P (sup{∈Clos(N ): P M

 M

=P}) M

= P sup{ M : ∈Clos(N ), P

 M

=P}

= P. v N (P) M

= P. Therefore v M N (P) is the largest closure operation on N such that P Next, note that, as is easy to verify,for any R-submodule L of N , the R (N ,M) P)) and {H ∈ Mod R (N ) : (L \Hom R (N ,M) P) ⊆ R-submodules (P : (L \Hom  Hom R (N ,M) P)} and {H ∈ Mod R (N ) : ∀ f ∈ Hom R (N , M) ( f (L) ⊆ P ⇒ (H \ f (H ) ⊆ P)} of N are each equal to the largest H ∈ Mod R (N ) such that f (L) ⊆ P implies f (H ) ⊆ P for all f ∈ Hom R (N , M). We check that  = (P : (− \Hom R (N ,M) P)) is a semiprime operation on N such that P  M = P. For any f ∈ Hom R (N , M), it is clear that f −1 (P) is the largest H ∈ Mod R (N ) such that f ( f −1 (P)) ⊆ P implies f (H ) ⊆ P. Therefore one has f −1 (P) = f −1 (P) for all f ∈ Hom R (N , M). It follows from Theorem 5.2.29, then, that P  M = P. To show that  is a semiprime operation on N , let g ∈ End R (N ). We wish to show that g(L  ) ⊆ g(L) . Let f ∈ Hom R (N , M). Then f ◦ g ∈ Hom R (N , M), so that M

f (g(L)) = ( f ◦ g)(L) ⊆ P implies f (g(L  )) = ( f ◦ g)(L  ) ⊆ P by definition of . Therefore g(L  ) ⊆ g(L) , again by definition of . Thus,  is a semiprime operation on N .  Next, let  be any closure operation on N such that P  M = P. Then f (L) ⊆    P implies f (L  ) ⊆ f (L) M ⊆ P  M = P for all f ∈ Hom R (N , M), whence  L  ⊆ L  , for all L ∈ Mod R (N ), and therefore    . This proves statements (1) and (2) of the theorem. To prove (3), first observe that v M N (M) is a semiprime operation on N and that M M vM P N (M) M ⊆ P v N (P) M = P, whence P v N (M) M = P, for all P ∈ M. Let  be  M

= P for all P ∈ M, then   v M any closure operation on N . If P N (P) for M (M), then P  M ⊆ all P ∈ M, whence   v N (M); and, conversely, if   v M N M M P v N (M) M ⊆ P v N (P) M = P, whence P  M = P, for all P ∈ M. Next, we prove (4). Let  be any N -surprime operation on M, so that  = N M . By (3) it is clear that  M  N  inf{v M N (P) : P ∈ Mod R (M) } = v N (Mod R (M) ).  Let  be any semiprime operation on M such that   v M N (P) for all P ∈ Mod R (M) .  M

   Then P = P for all P ∈ Mod R (M) , so that  M   and therefore    M N  N . Therefore  M  N = inf{v M N (P) : P ∈ Mod R (M) } = v N (Mod R (M) ),

5.4 Divisorial and Codivisorial Semiprime Operations

whence

393

  = N M = v M N (Mod R (M) ) M .

This proves (4). Finally, statements (5) and (6) follow from statement (4) and Theorem 5.2.40(4).  Corollary 5.4.3. Let R be a ring. (1) Let J ∈ Il (R). The semiprime operation v(J ) = sup{s ∈ Semiprime(R) : J s = J } on R is the largest semiprime operation s on R such that J s = J . (2) For all I ∈ Il (R), one has I v(J ) = (J : R (I \ R J ))  = {H ∈ Il (R) : ∀a ∈ R (I a ⊆ J ⇒ H a ⊆ J )}  = {H ∈ Il (R) : (I \ R J ) ⊆ (H \ R J )}, and I v(J ) is the largest H ∈ Il (R) such that H a ⊆ J for all a ∈ R such that I a ⊆ J . Moreover, I ∈ Il (R) is v(J )-closed if and only if I = (J : R X ) for some subset X of R. (3) Let J ⊆ Il (R). The operation v(J ) = inf{v(J ) : J ∈ J } is the largest semiprime operation s on R such that J s = J for all J ∈ J . For any semiprime operation s on R, one has s  v(J ) if and only if J s = J for all J ∈ J . Moreover, Il (R)v(J ) is the smallest left ideal system on R containing J . (4) One has s = v(Il (R)s ) for any semiprime operation s on R. Thus, every semiprime operation s on R is of the form v(J ) for some J ⊆ Il (R). In the commutative case, statement (4) of the corollary was first observed by Kirby [119, p. 284]. Definition 5.4.4. Let R be a ring, let M and N be R-modules, and let M be a subset of Mod R (M). The semiprime operation v M N (M) on N is the largest closure operation  on N such that P  M = P for all P ∈ M and is called divisorial closure in N with respect to M in M. If M = {P}, then we replace M with P in the definition above. Definition 5.4.5. Let R be a ring and J a subset of Il (R). The semiprime operation v(J ) is the largest closure operation on R such that J s = J for all J ∈ J and is called divisorial closure with respect to J . If J = {J }, then we replace J with J in the definition above.

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Remark 5.4.6. Let R be a ring. Let J, J  ∈ Il (R). One has 0v(J ) = J . Consequently, if v(J )  v(J  ), then J ⊆ J  . (However, the converse is false in general.) Moreover, since s J : I −→ I + J is the smallest semiprime operation s on R with 0s = J , one has s J  v(J ). Definition 5.4.7. Let R be a ring and X a subset of R. We let annl (X ) = (0 : R X ) = {a ∈ R : a X = 0}, annr (X ) = {a ∈ R : Xa = 0}. Note that annl (X ) is a left ideal of R and, if I is a left ideal of R, then annr (I ) = (I \ R 0). Definition 5.4.8. A map f : S −→ T of posets is reduced, or grounded, if inf S does not exist or else f (inf S) = inf T . A closure operation  on Mod R (M) is reduced if and only if 0 = 0. Corollary 5.4.9. Let R be a ring. Then r = v((0)) is the largest reduced semiprime operation on R and acts by r : I −→ (0 : R (I \ R 0)) = annl (annr (I )) for all I ∈ Il (R). Moreover, for any semiprime operation s on R, the semiprime operation sred = inf{s, r} is the largest reduced semiprime operation on R less than or equal to s. Example 5.4.10. Let R be a commutative ring. (1) The largest reduced semiprime operation on R is the semiprime operation r : I −→ ann R (ann R (I )). (2) The largest reduced on R is the semiprime oper stable semiprime operation  ation r : I −→ J ∈I(R):J r =R (I : R J ) = J ∈I(R):ann R (J )=0 (I : R J ). The corresponding Gabriel filter is the set D(R) of all dense ideals of R, where an ideal I of R is said to be dense if ann R (I ) = 0, or equivalently, if I r = R. The corresponding Gabriel topology on R is called  the dense topology on R. The semiprime operation r also acts by I −→ J ∈D(R) (I : R J ). (3) Let M be an R-module. The set D(M) = {I ∈ I(R) : (0 : M I ) = 0} is a Gabriel filter  on R. The corresponding stable semiprime operation on R acts by I −→ J ∈D(M) (I : R J ) for all I ∈ I(R). (4) An ideal of R is essential if its intersection with any nonzero ideal of R is nonzero. (Unfortunately, this terminology conflicts with the notion of an essential prime as defined in Chapter 3.) For example, any dense ideal is essential. The set G(R) of all essential ideals of R is a Gabriel filter on R, and the corresponding Gabriel topology on R is called the Goldie topology on R. If g denotes the corresponding stable semiprime operation on R, then g  r, and one has g = r if and only if g

5.4 Divisorial and Codivisorial Semiprime Operations

395

is reduced, if and only if every essential ideal of R is dense, and if and only if the ring R is reduced. (5) An ideal of R is semiregular if it contains a finitely generated dense ideal of R. The set Isreg (R) of all semiregular ideals of R is a Gabriel filter on R. The corresponding stable semiprime operation on R is the largest reduced finite type  stable semiprime operation r w on R, acting by I −→ J ∈Isreg (R) (I : R J ) for all I ∈ I(R). (6) One has inclusions Ireg (R) ⊆ Isreg (R) ⊆ D(R) ⊆ G(R) of Gabriel filters, with corresponding stable semiprime operations r 1  r w  r  g . One has r w = r if and only if r is of finite type, if and only if every dense ideal of R is semiregular. Similarly, one has r w = r 1 if and only if every semiregular ideal of R is regular. Such a commutative ring R is said to be McCoy, or satisfy Property A [107]. As an application of the divisorial semiprime operations, we determine the structure of the poset of all semiprime operations on any Dedekind domain, in a manner different than [181, Proposition 3.6]. Let R be a Dedekind domain. Let I, J, J  be ideals of R with J, J  = 0. Note the following. (1) (2) (3) (4) (5) (6) (7)

(J : R I ) = J (I + J )−1 . v(J ) : I −→ I + J . I v(J ) = I if and only if J ⊆ I . v(J )  v(J  ) if and only if J ⊆ J  . sup{v(J ), v(J  )} = v(J + J  ). inf{v(J ), v(J  )} = v(J ∩ J  ). r = v((0)) : I −→ R for I = 0, and r : 0 −→ 0. Hence, if s is a semiprime operation on R, then I s = I sred if I = 0.

By Corollary 5.4.3, every semiprime operation on R is an infimum of semiprime operations of the form v(J ) for J ∈ I(R). By statement (6), a infimum of finitely many such divisorial closure operations is again such an operation, but to construct all semiprime operations we need to be able to describe infima of arbitrary sets of such operations. Let Max(R) denote the set of all maximal ideals  R. Consider the ordered commutative monoid S(R) = (Z0 ∪ {∞})Max(R) = p∈Max(R) (Z0 ∪ {∞}) under componentwise addition, where (n p )p∈Max(R)  (m p )p∈Max(R) ifn p  m p for all p ∈ Max(R). We denote the element I = (n p )p∈Max(R) of S(R) as p∈Max(R) pn p , and we write νp (I ) = n p for all p ∈ Max(R), so that νp : S(R) −→ Z0 ∪ {∞}  is the canon0 ical projection onto the pth coordinate. The identity element (1) =  p∈Max(R) p of S(R) is the largest element of S(R), and the smallest element (0) = p∈Max(R) p∞ of S(R) is an annihilator of S(R) in the sense that (0)I = (0) for all I ∈ S(R). The poset S(R) is a complete lattice, with sup X =

p∈Max(R)

pinf{νp (X )} ,

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5 Noncommutative Rings and Closure Operations on Submodules

inf X =



psup{νp (X )}

p∈Max(R)

for any subset X of S(R). (In fact, like I(R) and K(R), the ordered monoid S(R) is a coherent multiplicative lattice.) Elements of S(Z) are known supernatural numbers, so we call elements of S(R) superideals of R. Let νp : I(R) −→ Z0 ∪ {∞} also denote the p-adic valuation on I(R). There is a ν : I(R) −→ S(R) of ordered monoids acting by ν : I −→  homomorphism νp (I ) p that is an isomorphism onto its image, which is equal to p∈Max(R)

 p∈Max(R) Z0 ∪ {∞}. Thus, we may consider I(R) to be an ordered submonoid of S(R), and I(R) − {0} is upward-closed in S(R) in the sense that if I ∈ I(R) − {0} and I  J ∈ S(R), then J ∈ I(R) − {0}. Moreover, every element of S(R) is the infimum of some subset of I(R). (In fact, the elements of I(R) are precisely the compact elements of S(R), in the sense of Definition 6.4.11.) One has the following. (8) For all J ∈ I(R), the association v(J ) : I −→ (J : R (J : R I )) for all ideals I defines a semiprime operation v(J ) on R with (0)v(J ) = J . (9) For all J ∈ S(R), the association v (J ) : I −→ sup{I, J } for all ideals I = (0), and v (x) : (0) −→ (0), defines a semiprime operation v (J ) on R with  (0)v (J ) = (0). (10) e = v((1)) and d = v ((0)). (11) r = v((0)) = v ((1)). (12) v (J ) = v(J )red < v(J ) for all J ∈ I(R) − {(0)}.  (13) I v (J ) = I if and only if J  I , for all I ∈ I(R) − {(0)} and all J ∈ S(R). (14) v (J )  v (J  ) if and only if J  J  , for all J, J  ∈ S(R). (15) v(J )  v(J  ) if and only if J  J  , for all J, J  ∈ I(R) − {(0)}. (16) For every subset X of S(R), one has inf{v (J ) : J ∈ X } = v (inf X ). (17) For every subset X of I(R), one has inf{v(J ) : J ∈ X } = v(inf X ) if inf X ∈ I(R), and inf{v(J ) : J ∈ X } = v (inf X ) otherwise. By Corollary 5.4.3 and statement (17), every semiprime operation s on R has the form v (J ) for some J ∈ S(R) (when s is reduced) or v(J ) for some J ∈ I(R) − {(0)} (when s is not reduced). Thus Semiprime(R) is equal to the disjoint union Semiprime(R) = v (S(R))

v(I(R) − {(0)})

of v (S(R)) and v(I(R) − {(0)}). If S and T are subposets of some poset X , then we let S  T = (S × {1}) ∪ (T × {2}) ⊆ X × {1, 2} be endowed with the product order, so that (a, i)  (b, j) if and only if a  b and i  j, for all a, b ∈ X and all i, j ∈ {1, 2}. (As a set the poset

S  T equals the disjoint union S T of S and T , but the disjoint union of S and T as a poset is usually endowed with the ordering defined so that (a, i)  (b, j) if and only if a  b and i = j.) From Corollary 5.4.3 and statements (8)–(17), we obtain the following.

5.4 Divisorial and Codivisorial Semiprime Operations

397

 Proposition 5.4.11. Let R be a Dedekind domain, and let S(R) = p∈Max(R) (Z0 ∪ {∞}). The association v : x −→ v (x) defines an embedding of the poset S(R) into the poset Semiprime(R) of all semiprime operations on R. The association v : J −→ v(J ) defines an embedding of the poset I(R) − {(0)} into the poset Semiprime(R). One has Semiprime(R) = v (S(R))

v(I(R) − {(0)})

as posets, where the disjoint union is subject to the additional relations v (J1 ) < v(J2 ) for all J1 ∈ S(R) and J2 ∈ I(R) − {(0)} with J1  J2 . Thus the embeddings v and v induce an isomorphism Semiprime(R) ∼ = S(R)  (I(R) − {(0)}) of posets. Moreover, one has v (S(R)) = RSemiprime(R) and v(I(R) − {(0)}) = Semiprime(R) − RSemiprime(R), where RSemiprime(R) denotes the poset of all reduced semiprime operations on R. Now we turn our attention to the codivisorial semiprime operations. The proofs are left as exercises. Definition 5.4.12. Let R be a ring and M an R-module. A presemiprime operation on M is a preclosure operation  on Mod R (M) such that f (L  ) ⊆ f (L) for all L ∈ Mod R (M) and all f ∈ End R (M). Proposition 5.4.13. Let R be a ring, let M and N be R-modules, and let  be a preclosure operation on Mod R (N ). (1) There is a largest preclosure operation  =  M on M such that  is compatible with . Moreover,  M is a presemiprime operation on M.  (2) L  M for all L ∈ Mod R (M) is the largest H ∈ Mod R (M) such that f (H ) ⊆  f (L) for all f ∈ Hom R (M, N ). (3) One has 

L  M =

  { f −1 ( f (L) ) : f ∈ Hom R (M, N )} 

= {x ∈ M : ∀ f ∈ Hom R (M, N ) ( f (x) ∈ f (L) )}   = {H ∈ Mod R (M) : ∀ f ∈ Hom R (M, N ) ( f (H ) ⊆ f (L) )} for all L ∈ Mod R (M).

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5 Noncommutative Rings and Closure Operations on Submodules

Theorem 5.4.14. Let R be a ring, let M and N be R-modules, and let P be an R-submodule of M. N (1) There is a smallest preclosure operation  = p M (P) on M such that P N = N . N Moreover, p M (P) is a presemiprime operation on M. (2) For all L ∈ Mod R (M), one has

L p M (P) = L + N · (P \Hom R (N ,M) L)  =L+ { f (N ) : f ∈ Hom R (N , M), f (P) ⊆ L}, N

and L p M (P) is the smallest H ∈ Mod R (M) containing L such that f (N ) ⊆ H for all f ∈ Hom R (N , M) such that f (P) ⊆ L. N N (P) = p M (P)∞ is the smallest closure operation  on M such that P N = (3) u M N N , and u M (P) is a semiprime operation on M. Let L ∈ Mod R (M). Then N L p M (P) is the smallest H ∈ Mod R (M) containing L such that f (N ) ⊆ H for all f ∈ Hom R (N , M) such that f (P) ⊆ H . Moreover, the following conditions are equivalent. N (P)-closed. (a) L is u M N (P)-closed. (b) L is p M (c) L = L + N · (P \Hom R (N ,M) L). (d) L ⊇ N · (P \Hom R (N ,M) L) (e) f (N ) ⊆ L for all f ∈ Hom R (N , M) such that f (P) ⊆ L. N (N ) = sup{u M (4) Let N ⊆ Mod R (N ). The operation u M N (P) : P ∈ N } is the smallest closure operation  on M such that P N = N for all P ∈ N , and it is a semiprime operation on M. For any closure operation  on N , one has N (N )   if and only if P N = N for all P ∈ N . uM N

Open Problem 5.4.15. Let R be a ring, and let M and N be R-modules. Characterize N the semiprime operations on N that are of the form u M (N )N for some N ⊆ Mod R (N ).

5.5 Preradicals, Pretorsion Theories, and Systems of Closure Operations In this section and the next, we prove the equivalences in Table 5.1, assuming results from [177]. Definition 5.5.1. Let R be a ring. A preradical on R-Mod is a subfunctor of the identity functor on R-Mod, that is, it is a (class) function r : R-Mod −→ R-Mod satisfying the following conditions. (1) r(M) is an R-submodule of M for every R-module M. (2) f (r(M)) ⊆ r(N ) for every R-module homomorphism f : M −→ N .

5.5 Preradicals, Pretorsion Theories, and Systems of Closure Operations

399

The class of all preradicals on R-Mod is partially ordered by the relation  defined by r  r if r(M) ⊆ r (M) for all R-modules M. Definition 5.5.2. Let R be a ring, and let r be a preradical on R-Mod. (1) Let

r−1 (M) = M/r(M)

for every R-module M. (2) Let Tr = {M ∈ R-Mod : r(M) = M} = {M ∈ R-Mod : r−1 (M) = 0}. The R-modules in Tr are said to be r-torsion. (3) Let Fr = {M ∈ R-Mod : r(M) = 0} = {M ∈ R-Mod : r−1 (M) = M}. The R-modules in Fr are said to be r-torsion-free. Definition 5.5.3. Let R be a ring, and let r be a preradical on R-Mod. (1) r is idempotent if r(r(M)) = r(M) for every R-module M. (2) r is a radical if r(M/r(M)) = 0 for every R-module M. (3) r is hereditary if for every R-module M one has r(M) = M (if and) only if r(N ) = N for all R-submodules N of M. (4) r is cohereditary if for every R-module M one has r(M) = 0 if (and only if) r(Rx) = 0 for all x ∈ M. (5) r is left exact if it is left exact as an endofunctor of R-Mod. Example 5.5.4. If R is a commutative ring and M is an R-module, then let Tors M = {m ∈ M : am = 0 for some a ∈ R reg } denote the R-torsion submodule of M. The association Tors : M −→ Tors M defines a preradical on R-Mod that satisfies all five of the conditions defined above. Note that a preradical r is idempotent if and only if r−1 (r(M)) = 0 for every R-module M, while r is a radical if and only if r(r−1 (M)) = 0 for every R-module M, if and only if r−1 (r−1 (M)) = r−1 (M) for every R-module M. In this sense, idempotent and radical are dual notions. By the following proposition, hereditary and cohereditary are also dual notions, but in a different sense. Proposition 5.5.5. Let R be a ring. A preradical r on R-Mod is hereditary if and only if for every R-module M one has r(M) = M (if and) only if r(Rx) = Rx for all x ∈ M. A proof of the proposition is outlined in Exercise 5.5.2. Proposition 5.5.6. Let R be a ring and r a preradical on R-Mod. The following conditions are equivalent.

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5 Noncommutative Rings and Closure Operations on Submodules

(1) (2) (3) (4)

r is left exact. r is idempotent and hereditary. r(N ) = r(M) ∩ N for all R-modules M and N with N ⊆ M. For all R-modules M and all x ∈ M, one has x ∈ r(M) (if and) only if r(Rx) = Rx.

Proof. Conditions (1)–(3) are equivalent by [177, Proposition VI.1.7], and conditions (3) and (4) are easily seen to be equivalent.  Corollary 5.5.7. Let R be a ring. Any left exact preradical on R-Mod is both hereditary and cohereditary. Proof. By Proposition 5.5.5, if a preradical r satisfies condition (4) of Proposition 5.5.6, then r is both hereditary and cohereditary.  Because of the proposition and corollary above, one has the full implication lattice below among various types of preradicals. Unfortunately, we do not know if all of the arrows in the diagram are irreversible. (In the diagram, “rad” is short for “radical,” “her” is short for “hereditary,” “coher” is short for “cohereditary,” and “idem” is short for “idempotent.”) Remark 5.5.8. Let R be a ring. Let us say that a preradical r on R-Mod is strongly cohereditary if, for any R-module M and any M ⊆ Mod R (M), one has  r( N ∈M N ) = 0 if (and only if) r(N ) = 0 for all N ∈ M. Clearly any cohereditary preradical is strongly cohereditary. The notions of cohereditary and strongly cohereditary are both dual to the notion of hereditary, but the former notion for us is more useful since it relates more closely to the notion of a semiprime operation. left exact rad

coher idem rad

her coher rad

her rad

idem rad

left exact

coher rad

rad

her coher

her

coher idem

idem

coher

preradical

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401

Proposition 5.5.9. Let R be a ring. (1) The ordered class of all preradicals on R-Mod is complete: if  is a set of preradicals on R-Mod, then the supremum sup  acts by M  − → r∈ r(M)  and the infimum inf  acts by M −→ r∈ r(M). (2) If  is a set of radicals, then inf  is a radical. (3) If  is a set of hereditary preradicals, then inf  is a hereditary preradical. (4) If  is a set of hereditary radicals, then inf  is a hereditary radical. (5) If  is a set of left exact preradicals, then inf  is a left exact preradical. (6) If  is a set of left exact radicals, then inf  is a left exact radical. (7) If  is a set of cohereditary preradicals, then sup  is a cohereditary preradical. (8) If  is a set of idempotent preradicals, then sup  is an idempotent preradical. (9) If  is a set of cohereditary idempotent preradicals, then sup  is a cohereditary idempotent preradical. Definition 5.5.10 (cf. [55, Definition 7.0.2]). Let R be a ring and M a subcategory of R-Mod. A system of (order-preserving) operations  on M is an indexed collection  = { M : M ∈ M}, where  M is an order-preserving operation on Mod R (M) for all R-modules M in M. The class of all systems of operations on M is partially ordered by the relation , where    if  M  M for all R-modules M in M. Let  be a system of operations on M. (1)  is a closure system (resp., preclosure system) on M if  M is a closure operation (resp., preclosure operation) on Mod R (M) for all M ∈ M. (2)  is an interior system (resp., preinterior system) on M if  M is an interior operation (resp., preinterior operation) on Mod R (M) for all M ∈ M. (3)  is compatible with s, where s is an order-preserving operation on Il (R), if  M is compatible with s for all M ∈ M. (4)  is functorial if f (L  M ) ⊆ f (L) N for all homomorphisms f : M −→ N in M and all L ∈ Mod R (M). (5)  is right functorial if f (L  M ) ⊆ f (L) N for all surjections f : M −→ N in M and all L ∈ Mod R (M). (6)  is left functorial if f (L  M ) ⊆ f (L) N for all injections f : M −→ N in M and all L ∈ Mod R (M). (7)  is invariant if f (L  M ) = f (L) N for all isomorphisms f : M −→ N in M and all L ∈ Mod R (M). (8)  is right exact, or residual, if L  N = f ( f −1 (L) M ) for all surjections f : M −→ N in M and all L ∈ Mod R (N ). (9)  is left exact, or hereditary, if L  M = f −1 ( f (L) N ) for all injections f : M −→ N in M and all L ∈ Mod R (M). (10)  is exact if  is left exact and right exact. (11)  is right semiexact if f is invariant and (M/M  M )(M/M  M ) = 0 for all M ∈ M such that the projection M −→ M/M  M is in M. (12)  is left semiexact if f is invariant and 0(0 M ) = 0 M for all M ∈ M such that the inclusion is 0 M −→ M is in M.

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(13)  is a basic closure system on M if for all M ∈ M there exists an Mod R (M) such that L  M = L + H for all L ∈ Mod R (M) (in which case 0 M ). (14)  is a basic interior system on M if for all M ∈ M there exists an Mod R (M) such that L  M = L ∩ H for all L ∈ Mod R (M) (in which case M  M ).

H∈ H= H∈ H=

Lemma 5.5.11. Let R be a ring, let M be a subcategory of R-Mod, and let  be a system of operations on M. (1) If  is functorial, then  is right functorial and left functorial, and the converse holds if M is a full and replete subcategory of R-Mod and is closed under quotient modules. (2) If M is a full subcategory of R-Mod, then  is functorial if and only if  N is compatible with  M for all M, N ∈ M, in which case  M is a semiprime operation on M for all M ∈ M. (3) If  is right functorial or left functorial, then  is invariant. (4) If  is right exact, then  is right functorial and right semiexact. (5) If  is left exact, then  is left functorial and left semiexact. (6) If M is a full and replete subcategory of R-Mod, then  is right exact if and only if  is invariant and (M/L) N /L = (M  N + L)/L for all R-modules L ⊆ M ⊆ N with N , N /L ∈ M. (7) If M is a full and replete subcategory of R-Mod, then  is left exact if and only if  is invariant and L  M = L  N ∩ M for all R-modules L ⊆ M ⊆ N with M, N ∈ M. Remark 5.5.12. Let R be a ring and M a subcategory of R-Mod. (1) The terms left exact and right exact for systems of operations are here used merely in loose analogy with the same terms for functors between abelian categories. (2) With regard to Definition 5.5.10, we are almost exclusively interested in the case where M is the category R-Mod or its full subcategory R-Modfin of all finitely generated R-modules. For other categories of R-modules, it is prudent to alter the definitions by replacing injections and surjections with monomorphisms and epimorphisms, respectively. (3) A functorial preclosure system on R-Mod is equivalently a closure operator of R-Mod with respect to the class of all inclusions in R-Mod, in the sense of [40]. (4) Assume the notation of Remark 5.2.14. Let f : M −→ N be a homomorphism of R-modules. A system of operations  on M is right functorial (resp., left functorial) if and only if  M  f −1 ( N ), or equivalently f ( M )   N , for all surjections (resp., all injections) f : M −→ N in M. Moreover,  is right exact if and only if f ( M ) =  N for all surjections f : M −→ N in M; dually,  is left exact if and only if  M = f −1 ( N ) for all injections f : M −→ N in M. (5) Let C be a class of R-modules and  M an order-preserving operation on Mod R (M) for all M ∈ C. There exists a largest subcategory M of R-Mod with objects C such that the system of operations { M : M ∈ M} on M is functorial.

5.5 Preradicals, Pretorsion Theories, and Systems of Closure Operations

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An R-module homomorphism f : M −→ N lies in M if and only if M, N ∈ C and f (L  M ) ⊆ f (L) N for all L ∈ Mod R (M). Moreover, M is a full subcategory of R-Mod if and only if  N is compatible with  M for all M, N ∈ C. (6) Suppose that R is commutative, M is a full subcategory of R-Mod, and  is a functorial system of operations on M. As observed in [55, Remark 7.0.3], for any R-module M in M one has I N  M ⊆ (I N ) M (and consequently I  R N  M ⊆ (I N ) M ) for all ideals I of R and all N ∈ Mod R (M). This is proved by considering the endomorphisms x −→ ax of M for a ∈ R and observing then that a N  M ⊆ (a N ) M for all a ∈ R. However, the inclusion I N  M ⊆ (I N ) M does not hold generally if R is not assumed commutative, even for M = R. Nevertheless, closure operations  on Mod R (M) such that I N  ⊆ (I N ) for all I and N are of interest when R is commutative and were first studied by Kirby [119]. Definition 5.5.13. Let R be a ring. A closure system  on R-Mod is rigid if  is the smallest closure system on R-Mod compatible with  R . Theorem 5.5.14. Let R be a ring and  a closure system on R-Mod. (1)  is rigid if and only if  = { R M : M ∈ R-Mod}. (2) If  is rigid, then  is functorial and right exact. (3) If  is functorial if and only if  N is compatible with  M for all R-modules M and N . (4)  is compatible with  R if and only if { R M : M ∈ R-Mod} is the largest rigid closure system on R-Mod less than or equal to . (5) The association s −→ {s M : M ∈ R-Mod} defines an isomorphism, with inverse  −→  R , from the poset of all left semiprime operations on R to the poset of all rigid closure systems on R-Mod. Proof. Statement (1) is true by the definition of rigid and of the closure operations s M . To prove (2), let s be a left semiprime operation on R. We must show that {s M : M ∈ R-Mod} is functorial and right exact. Right exactness, hence right functoriality and invariance, is easily verified. To prove left functoriality, let M ⊆ P be R-modules and L an R-submodule of M. One has  {N  ∈ Mod R (M) : N  ⊇ L , (N  : R x)s = (N  : R x) for all x ∈ M} L s M = and L s P =

 {N ∈ Mod R (P) : N ⊇ L , (N : R x)s = (N : R x) for all x ∈ P}.

Let N ∈ Mod R (P) with N ⊇ L and (N : R x)s = (N : R x) for all x ∈ P, and let N  = N ∩ M. Then N  ∈ Mod R (M) and N  ⊇ L, and for all x ∈ M one has (N  : R x)s = (N : R x)s = (N : R x) = (N  : R x), and therefore L s M ⊆ N  ⊆ N . Since this holds for all N ∈ Mod R (M) as chosen, L s M is contained in the intersection of all such N , which is L s P . Therefore L s M ⊆ L s P . Since also {s M : M ∈ R-Mod} is invariant, this shows that it is also left functorial.

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Statement (3) is clear, statement (4) follows from the fact that  R M   M if  and only if  M is compatible with  R , and (5) follows easily from (1). Example 5.5.15. Let R be a commutative ring. (1) Let rad denote the smallest closure system on R-Mod compatible with the radical √ semiprime operation I −→ √I . For all R-modules M, an R-submodule L of M is rad M -closed if and only if (L : R x) = (L : R x) for all x ∈ M, and therefore L rad M =

  {N ∈ Mod R (M) : N ⊇ L and (N : R x) = (N : R x) for all x ∈ M}.

(2) By Theorem 5.5.14, any semiprime operation, including the integral closure, tight closure, plus closure, Frobenius closure, solid closure, and basically full closure semiprime operations on R, when they are defined, each extends uniquely to a rigid closure system on R-Mod. For each of these semiprime operations s, the corresponding rigid closure system is {s M : M ∈ R-Mod} and is the smallest closure system on R-Mod compatible with s. The standard ways of extending these semiprime operations to closure systems yield functorial and right exact closure systems on R-Mod or R-Modfin [55, Remark 7.0.6]. (Integral closure is known to extend in several nonequivalent ways [56] [178, Chapter 16].) Therefore, they are each greater than or equal to the respective closure system {s M : M ∈ R-Mod} or {s M : M ∈ R-Modfin }. This provides an alternative, but likely inequivalent, way of extending these semiprime operations on ideals to closure operations on Mod R (M) that works the same for any ring R and any R-module M. For each of these semiprime operations s on R and related closure system , we do not know for which rings R and R-modules M one has  M = s M , or ( M )t = s M . (3) For example, suppose that R is Noetherian of prime characteristic. For any R-module M, let ∗ M denote the tight closure operation on Mod R (M) [102, Section 8]. If M is finitely generated, then L ∗ M ⊇ L ∗ M =

 {N ∈ Mod R (M) : N ⊇ L and (N : R x)∗ = (N : R x) for all x ∈ M}.

We do not know for which R and M one has ∗ M = ∗ M , or (∗ M )t = ∗ M . However, it follows at least from Theorem 4.6.6 that T = (∗ K )t = ∗ K , where K is the total quotient ring of R and where T denotes semistar tight closure on R. (4) Let J be a finitely generated  ideal of R. For any R-module M and any L ∈ n n Mod R (M), let (L : M J ∞ ) = ∞ n=0 (L : M J ) = {x ∈ M : J x ⊆ L for some 0 ∞ n ∈ Z0 }. One has (L : M J )/L = H J (M/L). Moreover, (− : M J ∞ ) is the smallest closure system compatible with the semiprime operation (− : R J ∞ ). Lemma 5.5.16. Let R be a ring, let  be a system of operations on R-Mod, let {Mi : i ∈ } be a set of R-modules, and let L i be an R-submodule of Mi for all i ∈ .

5.5 Preradicals, Pretorsion Theories, and Systems of Closure Operations

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   M (1) If  is right functorial, then ( i∈ L i ) i∈ Mi ⊆ i∈ L i i .     M (2) If  is left functorial, then ( i∈ L i ) i∈ Mi ⊇ i∈ L i i .     M (3) If  is functorial, then ( i∈ L i ) i∈ Mi = i∈ L i i .   Proof. Suppose that  is right functorial. Let M = i∈ Mi and L = i∈ L i , and for all i let πi : M −→ Mi be the canonical  projection. By right functoriality one M M has πi (L  M ) ⊆ L i i for all i, whence L  M ⊆ i∈ L i i .  Suppose, on the other hand, that  is left functorial. Let M = i∈ Mi and  L = i∈ L i , and for all i let ιi : Mi −→ M be the canonical inclusion. By left  M M functoriality one has ιi (L i i ) ⊆ L  M for all i, whence i∈ L i i ⊆ L  M . Finally, suppose that  is functorial. Then one has  

i∈ Mi ⊆

Li



i∈

i∈ Mi ⊆

Li

i∈



M

Li i .

i∈

Therefore, one also has  

i∈ Mi Li

i∈

whence equalities hold.

⊆

 i∈

 M Li i

∩

  i∈

 Mi

=



M

Li i ,

i∈



The following proposition is a straightforward generalization of [55, Lemma 7.0.5] to noncommutative rings. Proposition 5.5.17 (cf. [55, Lemma 7.0.5]). Let R be a ring, and let (F) be a closure operation on Mod R (F) for every free R-module F. (1) There exists a (unique) functorial and right exact closure system  on R-Mod with  F = (F) for every free R-module F if and only if (L 1 ⊕ L 2 )(F1 ⊕F2 ) = 1) 2) ⊕ L (F for all free R-modules F1 and F2 and all R-submodules L 1 of F1 L (F 1 2 and L 2 of F2 . (2) There exists a (unique) functorial and right exact closure system  on R-Modfin with  F = (F) for every f.g. free R-module F if and only if (L 1 ⊕ L 2 )(F1 ⊕F2 ) = 1) 2) ⊕ L (F for all f.g. free R-modules F1 and F2 and all R-submodules L 1 L (F 1 2 of F1 and L 2 of F2 . Proof. We prove (1); the proof of (2) is similar. By the lemma, the given condition on direct sums is necessary. Suppose that (F) is a closure operation on Mod R (F) for all free R-modules F satisfying the given condition on direct sums. Let M be an R-module, and let π1 : F1 −→ M be any homomorphism onto M from a free R-module F1 . If  is any right exact closure system on R-Mod such that  F = (F) for all free R-modules F, then one must have L  M = f 1 ( f 1−1 (L) F1 ) for all Rsubmodules L of M, and therefore such a  is unique. We must also show that  exists and is both left functorial and right exact. To prove existence, we show

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5 Noncommutative Rings and Closure Operations on Submodules

that the equation L  M = f 1 ( f 1−1 (L)(F1 ) ) doesn’t depend on choice of F1 and f 1 . Let f 2 : F2 −→ M be any homomorphism onto M from a free R-module F2 . Let F = F1 ⊕ F2 , and let f = f 1 ⊕ f 2 : F −→ M be the unique homomorphism such that f = f 1 ◦ π1 = f 2 ◦ π2 . Then for all L ∈ Mod R (M) one has f −1 (L)(F) = ( f 1−1 (L) ⊕ f 2−1 (L))(F) = f 1−1 (L)(F1 ) ⊕ f 2−1 (L)(F2 ) and therefore f ( f −1 (L)(F) ) = f 1 ( f 1−1 (L)(F1 ) ) = f 2 ( f 2−1 (L)(F2 ) ). This shows that  is well-defined, and the proof that  is left functorial and right exact is straightforward.  Next we relate preradicals to functorial systems of operations. Definition 5.5.18. Let R be a ring and  a system of operations on R-Mod. (1) Let r : R-Mod −→ R-Mod act by r : M −→ 0 M . (2) Let r  : R-Mod −→ R-Mod act by r  : M −→ M  M . Lemma 5.5.19. Let R be a ring and  a functorial system of operationson R-Mod. Then r : R-Mod −→ R-Mod and r  : R-Mod −→ R-Mod are preradicals on R-Mod with r  r  . Definition 5.5.20. Let R be a ring, let r be a preradical on R-Mod, and let M be an R-module. (1) Let r M : Mod R (M) −→ Mod R (M) act by L −→ L r M = r(L) for all L ∈ Mod R (M). M (2) Let r M : Mod R (M) −→ Mod R (M) be the unique map such that L r /L = r(M/L) for all L ∈ Mod R (M). Lemma 5.5.21. Let R be a ring and r a preradical on R-Mod. (1) {r M : M ∈ R-Mod} is the smallest functorial (or smallest left functorial, or unique left exact) system of operations (or preinterior system)  on R-Mod such that r   r, that is, such that M  M ⊇ r(M) for all R-modules M. (2) {r M : M ∈ R-Mod} is an interior system (resp., right semiexact, a basic interior system) if and only if r is idempotent (resp., a radical, left exact). (3) {r M : M ∈ R-Mod} is the largest functorial (or largest right functorial, or unique right exact) system of operations (or preclosure system)  on R-Mod such that r  r, that is, such that 0 M ⊆ r(M) for all R-modules M. (4) {r M : M ∈ R-Mod} is a closure system (resp., left semiexact, left exact) if and only if r is a radical (resp., idempotent, left exact). The following theorem is a straightforward consequence of the previous two lemmas.

5.5 Preradicals, Pretorsion Theories, and Systems of Closure Operations

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Theorem 5.5.22. Let R be a ring. (1) There is a monotone Galois injection from the ordered class of all preradicals on R-Mod to the ordered class of all functorial systems of operations on R-Mod given by the pair of maps r −→ {r M : M ∈ R-Mod} and  −→ r  . In other words, for any preradical r on R-Mod and any functorial system  of operations on R-Mod, one has {r M : M ∈ R-Mod}   if and only if r  r  , and the map  −→ r  is a left inverse to the map r −→ {r M : M ∈ R-Mod}. Moreover, for any functorial system of operations  on R-Mod, the system {(r  ) M : M ∈ R-Mod} is the largest functorial left exact preinterior system on R-Mod less than or equal to . (2) The map r −→ {r M : M ∈ R-Mod} induces an isomorphism from the ordered class of all preradicals (resp., idempotent preradicals, radicals, idempotent radicals, left exact preradicals, left exact radicals) on R-Mod to the ordered class of all functorial left exact preinterior systems (resp., functorial left exact interior systems, functorial left exact right semiexact preinterior systems, functorial left exact right semiexact interior systems, functorial left exact basic interior systems, and functorial left exact right semiexact basic interior systems) on RMod, with inverse  −→ r  . (3) There is a monotone Galois surjection from the ordered class of all functorial systems of operations on R-Mod to the ordered class of all preradicals on R-Mod given by the pair of maps  −→ r and r −→ {r M : M ∈ R-Mod}. In other words, for any functorial system  of operations on R-Mod and any preradical r on R-Mod, one has   {r M : M ∈ R-Mod} if and only if r  r, and the map r −→ {r M : M ∈ R-Mod} is a right inverse to the map  −→ r . Moreover, for any functorial system of operations  on R-Mod, the system {(r ) M : M ∈ R-Mod} is the smallest functorial right exact preclosure system on R-Mod greater than or equal to . (4) The map r −→ {r M : M ∈ R-Mod} induces an isomorphism from the ordered class of all preradicals (resp., radicals, idempotent preradicals, idempotent radicals, left exact preradicals, and left exact radicals) on R-Mod to the ordered class of all functorial right exact preclosure systems (resp., functorial right exact closure systems, functorial right exact left semiexact preclosure systems, functorial right exact left semiexact closure systems, exact preclosure systems, and exact closure systems) on R-Mod, with inverse  −→ r . Definition 5.5.23. Let R be a ring and r a preradical on R-Mod. (1) The pretorsion theory associated to r is the pair σr = (Tr , Fr ). (2) The ideals in the set Dr = {I ∈ Il (R) : R/I ∈ Tr } = {I ∈ Il (R) : I r = R} R

are called the r-dense ideals of R.

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5 Noncommutative Rings and Closure Operations on Submodules

(3) The ideals in the set Cr = {I ∈ Il (R) : R/I ∈ Fr } = {I ∈ Il (R) : I r = I } R

are called the r-closed ideals of R. Definition 5.5.24. Let R be a ring. (1) A pretorsion theory on R-Mod is a pair σ = σr = (Tr , Fr ) for some preradical r on R-Mod. For such a pair σ, we write Tσ = Tr and Fσ = Fr . (2) The class of all pretorsion theories on R-Mod is partially ordered by the relation  defined by (T , F)  (T  , F  ) if T ⊆ T  and F ⊇ F  . (3) A pretorsion theory (T , F) is radical (resp., idempotent) if T = Tr and F = Fr for some radical (resp., idempotent preradical) r on R-Mod. (4) Let σ be a pretorsion theory on R-Mod. The R-modules in Tσ are said to be σ-torsion, and the ideals in the set Dσ = {I ∈ Il (R) : R/I ∈ Tσ } are called the σ-dense ideals of R. (5) Let σ be a pretorsion theory on R-Mod. The R-modules in Fσ are said to be σ-torsion-free, and the ideals in the set Cσ = {I ∈ Il (R) : R/I ∈ Fσ } are called the σ-closed ideals of R. Note that every preradical r determines a unique pretorsion theory σr , but different preradicals may determine the same pretorsion theory. Definition 5.5.25. Let R be a ring and M a class of R-modules. (1) M is closed under isomorphisms if any R-module isomorphic to an element of M is in M. (2) M is closed under submodules if any submodule of an element of M is in M. (3) M is closed under quotient modules if any quotient module of an element of M is in M. (4) M is closed under direct sums if the direct sum of any indexed subset of M lies in M. (5) M is closed under direct products if the direct product of any indexed subset of M lies in M. (6) M is a pre-pretorsion class of R-modules if it is closed under isomorphisms and quotient modules. (7) M is a pretorsion class of R-modules if it is closed under isomorphisms, quotient modules, and direct sums. (8) M is a pre-pretorsion-free class of R-modules if it is closed under isomorphisms and submodules. (9) M is a pretorsion-free class of R-modules if it is closed under isomorphisms, submodules, and direct products.

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Proposition 5.5.26 ([177, Proposition VI.1.2]). Let R be a ring and r a preradical on R-Mod. Then Tr is a pretorsion class of R-modules, and Fr is a pretorsion-free class of R-modules.  A pretorsion class M is closed under  internal sums, since λ Mλ ∈ M (the sum being isomorphic to a quotient of λ Mλ ) for any collection {Mλ } ⊆ M of R-submodules of an R-module M. Dually,a pretorsion-free class M is closed under “co-intersections,” in the  sense that M/ λ Mλ ∈ M (the quotient being isomorphic to a submodule of λ M/Mλ ) for any collection {Mλ } of R-submodules of an R-module M such that M/Mλ ∈ M for all λ. Definition 5.5.27. Let R be a ring and M be a class of R-modules. For any R-module M, let r M : R-Mod −→ R-Mod and rM : R-Mod −→ R-Mod be defined, respectively, by  r M (M) = {L ∈ Mod R (M) : L ∈ M}, rM (M) =

 {L ∈ Mod R (M) : M/L ∈ M}

for all R-modules M. Corollary 5.5.28. Let R be a ring and M a class of R-modules. (1) Suppose that M is a pre-pretorsion class. Then r M is a preradical on R-Mod, and Tr M is the smallest pretorsion class containing M. Moreover, M is a pretorsion class if and only if M = Tr M , if and only if r M is an idempotent preradical. (2) Suppose that M is a pre-pretorsion-free class. Then rM is a preradical on R-Mod, and FrM is the smallest pretorsion-free class containing M. Moreover, M is a pretorsion-free class if and only if M = FrM , if and only if rM is a radical. The following two propositions expand on [177, Propositions VI.1.4–1.5]. Proposition 5.5.29. Let R be a ring. (1) The associations T −→ r T and r −→ Tr define a monotone Galois injection (resp., monotone Galois connection) from the ordered class of all pretorsion classes (resp., pre-pretorsion classes) of R-modules to the ordered class of all preradicals on R-Mod.  (2)  r = r Tr : M −→ {L ∈ Mod R (M) : r(L) = L} for any preradical r on R-Mod  is the largest idempotent preradical less than or equal to r, and Nr M = {L ∈ Mod R (M) : L ⊆ N , r(L) = L} for all R-modules M and all N ∈ Mod R (M). (3) The association T −→ r T defines an isomorphism from the ordered class of all pretorsion classes of R-modules to the ordered class of all idempotent preradicals on R-Mod, with inverse r −→ Tr .

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Proposition 5.5.30. Let R be a ring. (1) The associations F −→ rF and r −→ Fr define an antitone Galois injection (resp., antitone Galois connection) from the ordered class of all pretorsion-free classes (resp., pre-pretorsion-free classes) of R-modules to the ordered class of all preradicals on  R-Mod. (2) qr = rFr : M −→ {L ∈ Mod R (M) : r(M/L) = 0} for any preradical r on  M R-Mod is the smallest radical greater than or equal to r, and N qr = {L ∈ Mod R (M) : L ⊇ N , r(M/L) = 0} for all R-modules M and all N ∈ Mod R (M). (3) The association F −→ rF is an anti-isomorphism from the ordered class of all pretorsion-free classes of R-modules to the ordered class of all radicals on R-Mod, with inverse r −→ Fr . Remark 5.5.31. Let r be a preradical on R-Mod, and let T and F denote the full subcategories of R-Mod whose classes of objects are Tr and Fr , respectively. If r is idempotent, then r : M −→ r(M) defines a functor r : R-Mod −→ T that is a right adjoint to the inclusion T −→ R-Mod and therefore T a coreflective subcategory of R-Mod. Moreover, r is an idempotent comonad on R-Mod, and T is isomorphic to the category of (Eilenberg) coalgebras over the comonad r. Dually, if r is a radical, then r−1 : M −→ M/r(M) defines a functor r−1 : R-Mod −→ F that is a left adjoint to the inclusion F −→ R-Mod and therefore F is a reflective subcategory of R-Mod. Moreover, r−1 is an idempotent monad on R-Mod, and F is isomorphic to the category of (Eilenberg) algebras over the monad r−1 . Theorem 5.5.32. Let R be a ring. There are natural isomorphisms between the following ordered classes. (1) The ordered class of all radicals r on R-Mod. (2) The opposite of the ordered class of all pretorsion-free classes F of (left) R-modules. (3) The ordered class of all radical pretorsion theories σ on R-Mod. (4) The ordered class of all functorial right exact closure systems  on R-Mod. (5) The ordered class of all functorial, left exact, right semiexact preinterior systems  on R-Mod. Under these natural correspondences, one has the following. (a) F = Fσ = Fr . (b) σ = σr = (Tr , Fr ). M (c)  = {r M : M ∈ R-Mod}, where L r /L = r(M/L) for all R-modules M and all L ∈ Mod R (M). M (d) N r = {L ∈ Mod R (M) : L ⊇ N , M/L ∈ F} for all R-modules M and all N ∈ Mod R (M). M (e) r = rF , and r(M) = 0 M = 0r for all R-modules M. (f)  = {r M : M ∈ R-Mod}, where L r M = r(L) for all R-modules M and all L ∈ Mod R (M).

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Proof. The equivalence of (1)–(3) follows from Proposition 5.5.30 and is wellknown, and the equivalence of (1), (4), and (5) follows from Theorem 5.5.22.  Proposition 5.5.33. Let R be a ring. If r is a radical on R-Mod, then r R is a semiprime operation on R. Conversely, if s is a semiprime operation on R, then 0s − : M −→ 0s M is the smallest radical r on R-Mod with s = r R . Proof. Let r be a radical on R-Mod. By Theorem 5.5.32, the closure system  = {r M : M ∈ R-Mod} is functorial and right exact and therefore, by Theorem 5.5.14(3),  is compatible with r R . In particular, r R is compatible with r R , and therefore r R is a semiprime operation. Conversely, let s be a semiprime operation on R. Then the closure system {s M : M ∈ R-Mod} is functorial and right exact by Theorem 5.5.14(2), so 0s − is a radical on R-Mod by Theorem 5.5.32. Moreover, if r is any radical on R-Mod with s = r R , M then r M is compatible with s, hence r M  s M , so that r(M) = 0r ⊇ 0s M , for all R-modules M.  Definition 5.5.34. Let R be a ring, and let M be a class of R-modules. (1) M is hereditary if M is closed under submodules. (2) M is strongly hereditary if for every R-module M one has M ∈ M if and only if Rx ∈ M for all x ∈ M. Proposition 5.5.35. Let R be a ring and r a preradical on R-Mod. (1) Any strongly hereditary class of R-modules is hereditary. (2) A pretorsion class of R-modules is hereditary if and only if it is strongly hereditary. (3) The class Fr of all r-torsion-free R-modules is hereditary. (4) r is hereditary if and only if the class Tr of all r-torsion R-modules is (strongly) hereditary. (5) r is cohereditary if and only if the class Fr of all r-torsion-free R-modules is strongly hereditary. Remark 5.5.36. Let R be a ring, and let M be a class of R-modules. (1) Let us say that M is cohereditary if for every R-module M one has M ∈ M if Rx ∈ M for all x ∈ M. Then M is strongly hereditary if and only if M is hereditary and cohereditary. Moreover, a preradical r on R-Mod is cohereditary if and only if the class Fr of all r-torsion-free  R-modules is cohereditary. (2) Let us say that M is strongly cohereditary if N ∈N N ∈ M for any subset N of M. For any preradical r on R-Mod, the class Tr of all r-torsion R-modules is strongly cohereditary. By the following theorem, a radical is cohereditary if and only if it is of the form 0s − for a (unique) semiprime operation s on R; equivalently, the semiprime operations on a ring R are categorically equivalent to the cohereditary radicals on R-Mod.

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Theorem 5.5.37. Let R be a ring. The isomorphisms of Theorem 5.5.32 restrict to isomorphisms between the following posets. (1) The poset of all cohereditary radicals r on R-Mod. (2) The opposite of the poset of all strongly hereditary pretorsion-free classes F of (left) R-modules. (3) The poset of all rigid closure systems  on R-Mod. (4) The opposite of the poset of all left ideal systems C on R. (5) The poset of all left semiprime operations s on R. Under these natural correspondences, one has the following. (a) (b) (c) (d) (e)

F = Fr = {M ∈ R-Mod : ann R (x) ∈ C for all x ∈ M}. C = Cr = {I ∈ Il (R) : R/I ∈ F} = {I ∈ Il (R) : I s = I }. s = rR = R .  = {r M : M ∈ R-Mod} = {s M : M ∈ R-Mod}. r = rF = 0s − .

Proof. The isomorphism between the posets (1) and (2) follows from Theorem 5.5.32, the isomorphism between (3) and (5) from Theorem 5.5.14(5), and the isomorphism between (4) and (5) from Proposition 5.2.21. Thus, it remains only to prove (1) and (3) isomorphic. Let r be a radical on R-Mod with corresponding closure system  = {r M : M ∈ R-Mod} on R-Mod, and let s = r R . By Theorem 5.5.32 and Theorem 5.5.14(2), we need only show that r is cohereditary if and only if  is rigid. Suppose that  is rigid, so that r M = s M for all R-modules M. Then r(M) = 0 ⇔ 0s M = 0 ⇔ ∀x ∈ M ((0 : R x)s = (0 : R x)) for all R-modules M. Moreover, for all left ideals I of R, one has I s = I ⇔ I s /I = 0 ⇔ r(R/I ) = 0, and therefore, for all x ∈ M, one has (0 : R x)s = (0 : R x) ⇔ r(R/(0 : R x)) = 0 ⇔ r(Rx) = 0. Therefore, one has r(M) = 0 if and only if r(Rx) = 0 for all x ∈ M, that is, r is cohereditary. Conversely, suppose that r is cohereditary. Then one has r(M) = 0 ⇔ ∀x ∈ M (r(Rx) = 0) ⇔ 0s M = 0 for any R-module M. Therefore, for all L ⊆ M one has L r = L ⇔ r(M/L) = 0 ⇔ 0s(M/L) = 0 ⇔ L s M = L , M

and thus  = {s M : M ∈ R-Mod} is rigid.



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Lemma 5.5.38. Let R be a ring. Given any hereditary class M of R-modules, the class M = {M ∈ R-Mod : Rx ∈ M for all x ∈ M} is the smallest strongly hereditary class containing M, and M is a pretorsion-free class if M is a pretorsion-free class. Corollary 5.5.39. Let R be a ring. (1) There is a monotone Galois connection between the poset Clos(Il (R)) of all closure operations on Il (R) and the ordered class of all radicals on R-Mod given by the pair of maps s −→ 0s − and r −→ r R . Thus, for any closure operation s on Il (R) and any radical r on R-Mod, one has 0s −  r if and only if s  r R . (2) For any closure operation s on Il (R), one has (0s − ) R = s . R (3) For any radical r on R-Mod, the radical rcoher = 0r − is the largest cohereditary radical less than or equal to r, and Frcoher = {M ∈ R-Mod : Rx ∈ Fr for all x ∈ M} is the smallest strongly hereditary (pretorsion-free) class containing Fr . (4) The map s −→ 0s − , with inverse r −→ r R , is an isomorphism from the poset of all semiprime operations on R to the poset of all cohereditary radicals on R-Mod.

5.6 Torsion Theories and Left Exact Preradicals Definition 5.6.1. Let R be a ring, and let M be a class of R-modules. We let T(M) = {X ∈ R-Mod : Hom R (X, Y ) = 0 for all Y ∈ M}, F(M) = {Y ∈ R-Mod : Hom R (X, Y ) = 0 for all X ∈ M}. The class M is said to be closed under taking extensions if for all R-modules M, N with N ⊆ M one has M ∈ M provided that N ∈ M and M/N ∈ M. The pair of operations M −→ T(M) and M −→ F(M) comprise an antitone Galois connection from classes of R-modules to classes of R-modules in that they are operation-reversing and for any classes M and N of R-modules one has N ⊆ T(M) if and only if M ⊆ F(N ), if and only if Hom R (X, Y ) = 0 for all X ∈ M and all Y ∈ N . Consequently, one has M ⊆ T(F(M)) and F(T(F(M))) = F(M), and also M ⊆ F(T(M)) and T(F(T(M))) = T(M), for any class M of R-modules. The following proposition follows from [129, Proposition 0.1] and [177, Propositions VI.2.2–2.3]. Proposition 5.6.2. Let R be a ring, and let T and F be nonempty classes of R-modules. The following conditions are equivalent. (1) (T , F) = σr for a (unique) idempotent radical r on R-Mod. (2) T = T(F) and F = F(T ) (or equivalently T = T(F(T )) and F = F(T ), or equivalently F = F(T(F)) and T = T(F)).

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(3) (a) Hom R (X, Y ) = 0 for all X ∈ T and all Y ∈ F (or equivalently F ⊆ F(T ), or equivalently T ⊆ T(F)). (b) T and F are maximal with respect to condition (a). (4) T is a pretorsion class and is closed under taking extensions, and F = F(T ). (5) F is a pretorsion-free class and is closed under taking extensions, and T = T(F). (6) (a) T ∩ F = {0}. (b) T is a pre-pretorsion class. (c) F is a pre-pretorsion-free class. (d) For every R-module M, there exists a short exact sequence 0 → T → M → F → 0 of R-modules with T ∈ T and F ∈ F. (7) rF = r T . T | T = r (using the notation of Propositions 5.5.29 and 5.5.30). (8) r F = r and r F T (9) T = TrF and F = Fr . Proof. Statements (1)–(6) are equivalent by [129, Proposition 0.1] and [177, Propositions VI.2.2–2.3], statements (1) and (7)–(9) are equivalent by Propositions 5.5.29 and 5.5.30.  Definition 5.6.3. Let R be a ring. A torsion theory on R-Mod is a pair (T , F) of classes of R-modules satisfying the equivalent conditions of Proposition 5.6.2. A torsion class of R-modules is the class Tσ of R-modules for some torsion theory σ. A torsion-free class of R-modules is the class Fσ of R-modules for some torsion theory σ. Corollary 5.6.4. Let R be a ring. A torsion class of R-modules is equivalently a pretorsion class of R-modules that is closed under taking extensions. A torsion-free class of R-modules is equivalently a pretorsion-free class of R-modules that is closed under taking extensions. Remark 5.6.5. Some authors, like Lambek [129], call a torsion theory a “pretorsion theory” and a “hereditary” torsion theory a “torsion theory.” Theorem 5.5.32 and Proposition 5.6.2 yield the following. Theorem 5.6.6. Let R be a ring. The isomorphisms of Theorem 5.5.32 restrict to isomorphisms between the following ordered classes. (1) (2) (3) (4) (5)

The ordered class of all idempotent radicals on R-Mod. The ordered class of all torsion theories on R-Mod. The ordered class of all torsion classes of R-modules. The opposite of the ordered class of all torsion-free classes of R-modules. The ordered class of all functorial, right exact, left semiexact closure systems on R-Mod. (6) The ordered class of all functorial, left exact, right semiexact interior systems on R-Mod.

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The equivalence of the ordered classes (1)–(4) of the theorem above is well-known. Definition 5.6.7. A semiprime operation s on a ring R is torsion if the radical 0s −

on R-Mod corresponding to s is idempotent (or equivalently if s = r R for some cohereditary idempotent radical r on R-Mod). Theorem 5.6.8. Let R be a ring. The isomorphisms of Theorem 5.5.37 restrict to isomorphisms between the following ordered classes. (1) The poset of all cohereditary idempotent radicals on R-Mod. (2) The poset of all torsion theories on R-Mod whose torsion-free class is strongly hereditary. (3) The opposite of the poset of all strongly hereditary torsion-free classes of R-modules. (4) The poset of all left semiexact rigid closure systems  on R-Mod. (5) The poset of all torsion semiprime operations s on R. Corollary 5.6.9. Let R be a ring. For any idempotent radical r on R-Mod, the pair σr = (Tr , Fr ) is a torsion theory on R-Mod. Conversely, for any torsion theory σ on R-Mod, there is a unique idempotent radical r = rσ on R-Mod such that σ = σr . Moreover, r(M) for any R-module M is the largest R-submodule of M in Tσ , it is the smallest R-submodule N of M such that M/N ∈ Fσ , and it is the unique R-submodule N of M such that N ∈ Tσ and M/N ∈ Fσ . For any preradical r, one has r  r  qr, where r is the largest idempotent preradical less than or equal to r, and where qr is the smallest radical greater than or equal to r. These preradicals are constructed in Propositions 5.5.29 and 5.5.30, respectively, and an alternative construction is given in [177, Chapter VI Section 1]. By [177, Proposition VI.1.6], if r is a radical, then so is r; likewise, if r is idempotent, then so qr =  qr is an idempotent radical. is qr. It follows that  Definition 5.6.10. Let R be a ring and r a preradical on R-Mod. The idempotent qr =  qr, which we denote by rtor , is called the idempotent radical associated radical  to r. Definition 5.6.11 ([177, p. 139]). Let R be a ring and M a class of R-modules. Then (T(F(M)), F(M)) is the smallest torsion theory σ on R-Mod such that M ⊆ Tσ and is called the torsion theory generated by M. Dually, (T(M), F(T(M)) is the largest torsion theory σ on R-Mod such that M ⊆ Fσ and is called the torsion theory cogenerated by M. Propositions 5.5.29 and 5.5.30 yield the following. Proposition 5.6.12. Let R be a ring and r a preradical on R-Mod, and let σ = (T , F) be the pretorsion theory associated to r. One has the following. (1) σr = (T , F(T )). (2) σqr = (T(F), F).

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(3) σrtor = (T(F), F(T )). (4) σrtor = (Tqr , Fr ) is the torsion theory on R-Mod generated by Tr , and it is the torsion theory on R-Mod cogenerated by Fr . By Corollary 5.6.4, a torsion-free class is equivalently a pretorsion-free class that is closed under taking extensions. Moreover, by Theorem 5.5.37, there is a one-to-one correspondence between strongly hereditary pretorsion-free classes and semiprime operations, where the pretorsion-free class associated to a semiprime operation s is {M ∈ R-Mod : ann R (x)s = ann R (x) for all x ∈ M}. Therefore, we also have the following. Proposition 5.6.13. Let R be a ring and s a semiprime operation on R. The following conditions are equivalent. (1) s is torsion. (2) The class {M ∈ R-Mod : ann R (x)s = ann R (x) for all x ∈ M} of R-modules is closed under taking extensions. (3) If M and N are R-modules with N ⊆ M such that ann R (x) is s-closed for all x ∈ N and (N : R x) is s-closed for all x ∈ M, then ann R (x) is s-closed for all x ∈ M. Example 5.6.14. Let R be a commutative ring. √ (1) The semiprime operation rad : I −→ I is not torsion for R = Z. Indeed, note the exact sequence 0 −→ 2Z/4Z −→ Z/4Z −→ Z/2Z −→ 0 of Z-modules, where 2Z/4Z and Z/2Z are rad-torsion-free yet Z/4Z is not. (2) Trivially, the identity semiprime operation d is torsion. Thus, for example, if R is weakly F-regular, that is, if R is Noetherian of prime characteristic and every ideal of R is tightly closed, then tight closure on R is a torsion semiprime operation. All regular rings, for example, are weakly F-regular. It is unlikely that there are any other Noetherian rings of prime characteristic for which tight closure is torsion. Definition 5.6.15. Let L be a uniform ideal filter on a ring R. We let r L (M) = {x ∈ M : ann R (x) ∈ L} for any R-module M. The association M −→ r L (M) defines a preradical r L on R. An R-module M is L-torsion, or L-discrete, if r L (M) = M, and M is L-torsion-free if r L (M) = 0. Proposition 5.6.16 ([177, p. 146]). Let R be a ring. A Gabriel filter on R is equivalently a uniform ideal filter L on R such that the class of all L-torsion R-modules is closed under taking extensions. Definition 5.6.17. A pretorsion theory σ is hereditary if the pretorsion class Tσ of σ is hereditary (that is, closed under submodules). In the following theorem, the isomorphisms between (1) and (5) are well-known [177, Chapter VI Section 4].

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Theorem 5.6.18. Let R be a ring. There are natural isomorphisms between the following posets. (1) (2) (3) (4) (5) (6) (7) (8)

The poset of all left exact preradicals r on R-Mod. The poset of all hereditary pretorsion classes T of R-modules. The poset of all hereditary pretorsion theories σ on R-Mod. The poset of all uniform ideal filters L on R. The poset all linear topologies O on R. The poset of all stable intersection-preserving presemiprime operations s on R. The poset of all exact preclosure systems  on R-Mod. The poset of all functorial, left exact, basic interior systems  on R-Mod.

Under these natural correspondences, one has the following. T = Tσ = Tr = {M ∈ R-Mod : r(M) = M} = {M ∈ R-Mod : M is L-torsion}. σ = (Tr , Fr ). L = Dσ = {I ∈ Il (R) : R/I ∈ T }. L = Il (R) ∩ O = {I ∈ Il (R) : I s = R}. r = r L , that is, r(M) = {x ∈ M : ann R (x) ∈ L} = {x ∈ M : Rx ∈ T } for all R-modules M.  (f) I s = {a ∈ R : (I : R a) ∈ L} = {(J \ R I ) : J ∈ L} for all I ∈ Il (R). (g)  = {r M : M ∈ R-Mod}. (h)  = {r M : M ∈ R-Mod}.

(a) (b) (c) (d) (e)

Proof. The isomorphisms (1)–(5) are proved in [177, Chapter VI Section 4], the isomorphism between (4) and (6) follows from Theorem 5.3.23, and the isomorphism between (1), (7), and (8) follows from Theorem 5.5.22.  Remark 5.6.19. A hereditary pretorsion class of R-modules is equivalently a Serre class of R-modules that is strongly complete, in the sense of [94]. Remark 5.6.20. If r is a left exact preradical on R-Mod, then the left exact endofunctor r of the category of R-modules has a right derived functor R i r given by R i r(M) = H i (r(I · )) for any injective resolution M −→ I · of M. One has R 0 r(M) = M H 0 (r(I · )) = r(M) for all R-modules M, and therefore L r /L = r(M/L) = R 0 r(M/L) for all R-modules M, L with L ⊆ M. The case where r is the left exact radical given by r(M) = (0 : M J ∞ ), where J is a finitely generated ideal of R, yields the derived functor H Ji = R i r of local cohomology, as discussed, for example, in [182, Section 4.6]. Thus, left exact preradicals, whose corresponding cohomology theories are discussed in [185], generalize the local cohomology theories HIi . Definition 5.6.21. Let R be a ring, and let M be a class of R-modules. We let Mher = {N ∈ R-Mod : N ⊆ M for some M ∈ M}. Lemma 5.6.22. Let R be a ring, and let M be a class of R-modules. Then Mher is the smallest hereditary class of R-modules containing M. Moreover, if M is a pretorsion class, then Mher is the smallest hereditary pretorsion class containing M.

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Corollary 5.6.23. Let R be a ring and r a preradical on R-Mod. (1) Tr is a pretorsion class. (2) Ther r = {N ∈ R-Mod : N ⊆ M for some M ∈ Tr } is the smallest hereditary pretorsion class containing Tr . (3) The left exact preradical rher : M −→ {x ∈ M : Rx ∈ Ther r } = {x ∈ M : x ∈ N for some N ∈ Tr } associated to the hereditary pretorsion class Ther r is the smallest left exact preradical greater than or equal to r. Definition 5.6.24. Let R be a ring. We denote the injective envelope of an R-module M by E(M). If M is an R-module, then an R-module N is an essential extension of M if M is an R-submodule of N and N ∩ L = 0 for every R-submodule L = 0 of M. Note for example that a left ideal I of a ring R is essential in the sense of Definition 5.6.38 if and only the R-module R is an essential extension of I . Proposition 5.6.25 ([177, Chapter VI Section 3]). Let R be a ring. Let σ be a torsion theory on R-Mod and r = rσ its associated idempotent radical. The following conditions are equivalent. σ is hereditary. Tσ is hereditary. Fσ is closed under injective hulls, or equivalently under essential extensions. r is left exact. σ is generated by some class of R-modules that is closed under submodules and quotient modules. (6) σ is cogenerated by some class of injective R-modules.  (7) σ is cogenerated by the injective R-module {E(R/I ) : I ∈ Cσ }. (1) (2) (3) (4) (5)

If σ is hereditary, then rσ = rDσ , that is, rσ (M) = {x ∈ M : ann R (x) ∈ Dσ } = {x ∈ M : Rx ∈ Tσ } for all R-modules M. Definition 5.6.26. Let R be a ring. A system of order-preserving operations  on R-Mod is stable if  M is  R -stable for all R-modules M. Lemma 5.6.27. Let R be a ring. (1) A closure system  on R-Mod is stable if and only if  is rigid and  R is stable. (2) The association s −→ {s M : M ∈ R-Mod} defines an isomorphism, with inverse  −→  R , from the poset of all stable left semiprime operations on R to the poset of all stable closure systems on R-Mod.

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Proof. By Theorem 5.5.14(5), the association s −→ {s M } : M ∈ R-Mod} defines an isomorphism from the poset of all left semiprime operations on R to the poset of all rigid closure systems on R-Mod. Moreover, by Corollary 5.3.7, a closure system  on R-Mod is stable if and only if  is rigid and  R is stable, and therefore a semiprime operation s is stable if and only if the rigid closure system {s M } : M ∈ R-Mod} is stable. The proposition follows.  In the following theorem, the isomorphisms between (1) and (6) are well-known [177, Chapter VI Section 5] [129, Chapter 0]. Theorem 5.6.28. Let R be a ring. The isomorphisms of Theorems 5.5.37 and 5.6.18 restrict to isomorphisms between the following posets. (1) (2) (3) (4) (5) (6) (7) (8) (9)

The poset of all left exact radicals r on R-Mod. The poset of all hereditary torsion theories σ on R-Mod. The poset of all hereditary torsion classes T of R-modules. The opposite of the poset of all torsion-free classes F of R-modules that are closed under injective hulls, or equivalently under essential extensions. The poset of all Gabriel filters L on R. The poset of all Gabriel topologies on R. The poset of all stable left semiprime operations s on R. The poset of all stable closure systems  on R-Mod, equal to the poset of all exact closure systems on R-Mod. The poset of all functorial, left exact, right semiexact basic interior systems  on R-Mod.

Under these natural correspondences, one also has the following. (a) (b) (c) (d) (e) (f) (g) (h) (i)

T = T(F) = {M ∈ R-Mod : r(M) = M} = {M ∈ R-Mod : M is L-torsion}. F = F(T ) = {M ∈ R-Mod : r(M) = 0} = {M∈ R-Mod : M is L-torsion-free}. L = Dσ = {I ∈ Il (R) : I s = R}. r = rF = r T = r L .  r(M) = 0 M = {x ∈ M : ann R (x) ∈ L} = {L ∈ Mod R (M) : L ∈ T } for all R-modules M.  L  M = {x ∈ M : (L : R x) ∈ L} = {(J \ M L) : J ∈ L} for all R-modules M and all L ∈ Mod R (M).  s =  R , and I s = {a ∈ R : (I : R a) ∈ L} = {(J \ R I ) : J ∈ L} for all I ∈ Il (R).  M = s M = r M for all R-modules M.  = {r M : M ∈ R-Mod}.

Proof. The isomorphisms between the posets (1) and (6) are well-known [177, Chapter VI Section 5] [129, Chapter 0], the isomorphism between (5) and (7) follows from Theorem 5.3.23, and the isomorphism between (1) and (9) follows from Theorem 5.5.22(2). By Lemma 5.6.27, the association s −→ {s M } : M ∈ R-Mod} defines an isomorphism from the poset (7) to the poset of all stable closure systems on R-Mod. Moreover, by Theorem 5.5.22(4), the association r −→ {r M : M ∈ R-Mod}

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defines an isomorphism from the poset (1) to the poset of all exact closure systems on R-Mod. But if r = 0s − is the left exact radical corresponding to a given stable semiprime operation s, then r M = s M for all R-modules M. It follows that a closure system  on R-Mod is stable if and only if it is exact, and the association s −→ {s M } : M ∈ R-Mod} defines an isomorphism from the poset (7) to the poset (8).  Corollary 5.6.29. Let R be a ring. A closure system  on R-Mod is stable if and only if  is rigid and  R is stable, if and only if  is exact. Corollary 5.6.30. A nonempty set L of left ideals of a ring R is a Gabriel filter on R if and only if (1) if J is a left ideal of R containing I for some I ∈ L, then J ∈ L, (2) if I ∈ L and a ∈ R, then (I : R a) ∈ L, and (3) if J is a subset  of L and X = {a J : J ∈ J } is a subset of R indexed by J such that R X ∈ L, then {J a J : J ∈ J } ∈ L. Proof. Sufficiency of the three conditions is clear. Suppose that L is a Gabriel filter on R, and let s be the corresponding semiprime operation on R. Let J and X be as in condition (3). Then



so

s  s  s {J s a J : J ∈ J } = {Ra J : J ∈ J } = (R X )s = R, {J a J : J ∈ J } =

 {J a J : J ∈ J } ∈ L.



A left exact preradical on R-Mod is of finite type if its corresponding uniform ideal filter on R is of finite type, or equivalently if its corresponding linear topology on R is of finite type. Corollary 5.6.31. A stable semiprime operation is of finite type if and only if its corresponding Gabriel filter is of finite type, if and only if its corresponding left exact radical is of finite type. Note also that, by [112, Proposition 5.22], a left exact radical is of finite type if and only if it commutes with direct limits. Remark 5.6.32. The isomorphism between the posets (5) and (7) of the theorem is also proved in [34, Theorem 3.3], while the isomorphism between the posets (1) and (6) follows from [55, Proposition 7.1.3]. Remark 5.6.33. The paper [185] proves an equivalence, for commutative Noetherian rings R, between left exact radicals on R-Mod and what the authors call abstract local cohomology functors on R-Mod. Let R be any ring, and let r be a left exact radical on R-Mod with corresponding Gabriel filter L. Recall from Remark 5.6.20 that the left exact endofunctor r of the category of R-modules has a right derived functor R i r given by R i r(M) = H i (r(I · )) for any injective resolution M −→ I · of M. Since Hom R (R/J, M) R 0 r(M) = r(M) = lim −→ J ∈L

5.6 Torsion Theories and Left Exact Preradicals

421

and direct limit is exact, we also have ExtiR (R/J, M), R i r(M) = lim −→ J ∈L

for all R-modules M. In the case where R is commutative and Noetherian, the functor R i r is characterized as an abstract local cohomology functor on R-Mod, and these are equivalent to left exact radicals on R-Mod [185]. It remains to be seen if a more general characterization holds for all rings R. Recall that a map f : S −→ T of posets is reduced if inf S does not exist or else f (inf S) = inf T . A closure operation  on Mod R (M) is reduced if and only if 0 = 0. Example 5.6.34 ([177, Chapter VI]). Let R be a commutative ring. (1) Let S be a commutative R-algebra. The map I −→ I S ∩ R on I(R) is a semiprime operation on R. The given semiprime operation is stable if S is a flat R-algebra, and the corresponding Gabriel filter is the set {I ∈ I(R) : I S = S}. (2) The ordered monoid Ireg (R) of all regular ideals of R is a Gabriel filter on R. The corresponding hereditary torsion theory is that of ordinary R-torsion R-modules and R-torsion-free  R-modules. The corresponding semiprime operation r 1 acts by r 1 : I −→ J ∈Ireg (R) (I : R J ) = I K ∩ R, where K is the total quotient ring of R. (3) If r is a radical on R-Mod, then R is r-torsion-free if and only if the semiprime operation r R on R associated to r is reduced. The largest reduced semiprime operation on R is the semiprime operation r : I −→ ann R (ann R (I )), and the largest reduced stable semiprime  operation on R is the semiprime operation  r : I −→ J ∈I(R):J r =R (I : R J ) = J ∈I(R):ann R (J )=0 (I : R J ). (4) The torsion theory cogenerated by the injective envelope E(R) of R is the largest hereditary torsion theory on R-Mod with respect to which R is torsion-free. The corresponding Gabriel filter is the set D(R) of all dense ideals of R, where an ideal I of R is said to be dense if ann R (I ) = 0, or equivalently, if I r = R. The corresponding Gabriel topology on R is called the dense topology on R. The corresponding (stable) semiprime operation on R is the largest reduced  stable semiprime operation r on R described in (3), acting also by I −→ J ∈D(R) (I : R J ). (5) Let M be an R-module. The torsion theory cogenerated by the injective envelope E(M) of M is the largest hereditary torsion theory on R-Mod with respect to which M is torsion-free. Let D(M) denote the corresponding Gabriel filter. One has D(M) = {I ∈ I(R) : (0 : MI ) = 0}. The corresponding (stable) semiprime operation on R acts by I −→ J ∈D(M) (I : R J ) for all I ∈ I(R). (6) An ideal of R is essential if its intersection with any nonzero ideal of R is nonzero. For example, any dense ideal is essential. The set G(R) of all essential ideals of R is a Gabriel filter on R, and the corresponding Gabriel topology on R is called the Goldie topology on R. If g denotes the corresponding stable

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semiprime operation on R, then g  r, and one has g = r if and only if g is reduced, if and only if every essential ideal of R is dense, if and only if the ring R is reduced. We now generalize examples (4) and (6) to noncommutative rings. Definition 5.6.35. A left ideal I of a ring R is dense if annr (I : R a) = 0 for all a ∈ R. Proposition 5.6.36 (cf. [177, Propositions VI.5.6 and VI.6.4]). Let R be a ring. The torsion theory cogenerated by the injective envelope E(R) of R is the largest hereditary torsion theory on R-Mod with respect to which R is torsion-free. The corresponding Gabriel filter is the set Dl (R) of all dense left ideals of R. The corresponding stable semiprime operation on R is the largest reduced stable semiprime operation r on R and acts by r : I −→ {a ∈ R : (I : R ra)r = R for all r ∈ R} = {a ∈ R : (I : R a) ∈ Dl (R)} for all I ∈ Il (R). Definition 5.6.37. Let R be a ring. The Gabriel topology on R corresponding to the Gabriel filter Dl (R) of all dense left ideals of R is called the (left) dense topology on R. Definition 5.6.38. Let R be a ring. A left ideal I of R is essential if its intersection with any nonzero left ideal of R is nonzero. A left ideal I of R is Goldie dense if I ⊆ J for some essential left ideal J of R such that (I : R a) is essential for all a ∈ J . Every dense left ideal is essential and every essential left ideal is Goldie dense. The set Gl (R) ⊇ Dl (R) of all Goldie dense left ideals of R is the smallest Gabriel filter containing the essential left ideals, and the corresponding Gabriel topology on R is called the Goldie topology on R [177, Chapter VI Section 6]. If g denotes the corresponding stable semiprime operation on R, then g  r, and one has g = r if and only if g is reduced, if and only if every essential left ideal of R is dense; such a ring R is said to be (left) nonsingular. A commutative ring is nonsingular if and only if it is reduced. Definition 5.6.39. Let R be a ring and M be a class of R-modules. We let Ther (M) = {X ∈ R-Mod : Hom R (X, E(Y )) = 0 for all Y ∈ M}, Fher (M) = {Y ∈ R-Mod : Hom R (X, E(Y )) = 0 for all X ∈ M}. Proposition 5.6.40 ([129, p. 5]). Let R be a ring and M a class of R-modules. (1) For all R-modules X and Y , one has X ∈ Ther (M) ⇔ X  ∈ T(M) for all X  ∈ Mod R (X ),

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423

Y ∈ Fher (M) ⇔ E(Y ) ∈ F(M). (2) The pair (Ther (Fher (M)), Fher (M)) is the smallest hereditary torsion theory (T , F) on R-Mod such that M ⊆ T . (3) The pair (Ther (M), Fher (Ther (M))) is the largest hereditary torsion theory (T , F) on R-Mod such that M ⊆ F. Definition 5.6.41. Let R be a ring and M a class of R-modules. The smallest hereditary torsion theory (T , F) on R-Mod such that M ⊆ T and is called the hereditary torsion theory generated by M. Dually, the largest hereditary torsion theory (T , F) on R-Mod such that M ⊆ F and is called the hereditary torsion theory cogenerated by M. Corollary 5.6.42. Let R be a ring, and let r be a preradical on R-Mod and σ = (Tr , Fr ) its associated pretorsion theory. (1) The hereditary torsion theory σ her generated by Tr is the smallest hereditary torsion theory (T , F) such that Tr ⊆ T . Its corresponding left exact radical r. rhertor is the smallest left exact radical greater than or equal to (2) The hereditary torsion theory σher cogenerated by Fr is the largest hereditary torsion theory (T , F) such that Fr ⊆ F. Its corresponding left exact radical rhertor is the largest left exact radical less than or equal to qr. (3) If σ is a torsion theory, then σ her is the smallest hereditary torsion theory greater than or equal to σ, and σher is the largest hereditary torsion theory less than or equal to σ. Corollary 5.6.43. Let R be a ring and s a semiprime operation on R with corresponding cohereditary radical r. Let s denote the stable semiprime operation on R with corresponding left exact radical rhertor . Then s is the largest stable semiprime operation on R less than or equal to s. Remark 5.6.44. Antistable semiprime operations like radical, integral closure, and tight closure cannot be studied using hereditary torsion theories or Gabriel filters, because the latter are useful only if s is a suitable approximation for s. It is conceivable that these three semiprime operations s are even “antitorsion” in the sense that stor = d. (We leave this as an open problem.) (Since stor  s, antitorsion implies antistable.) Thus, even torsion theories may have limited use in this regard. Remark 5.6.45. The method we used in Section 5.3 for constructing sw can be applied to construct the largest torsion semiprime operation stor less than or equal to a given semiprime operation s, as follows. The operations r −→  r and r −→ rcoher are interior operations on the complete lattice of all radicals on R-Mod. Therefore their infimum exists; denote it by r −→ ridcoher . One has r = ridcoher if and only if r is idempotent and cohereditary. It follows that stor = ((0s − )idcoher ) R is the largest torsion semiprime operation less than or equal to a given semiprime operation s. Moreover, one has (inf )tor = inf{stor : s ∈ } for any set  of semiprime operations on R. Open Problem 5.6.46. Must a cohereditary idempotent radical be left exact? Equivalently, must a torsion semiprime operation be stable?

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5.7 Semistar Operations on Algebras Let R be a ring, and let T be an R-algebra, by which we mean a ring T together with a ring homomorphism R −→ T . The ring T is endowed with the structure of an R-module, and the ring homomorphism R −→ T acts by a −→ a1T , where 1T is the identity element of T . For the sake of notational simplicity, we blur the distinction between a left ideal I in R and its image I 1T in T , which is an R-submodule of T . Also, if I is an R-submodule of T , then we write I ∩ R = {a ∈ R : a1T ∈ I } for the preimage of I in R. Recall from Section 3.6 that, if R is a commutative ring and T is a commutative R-algebra, then a T -semistar operation on R is a nucleus on the ordered monoid Mod R (T ). This definition generalizes to noncommutative algebras over noncommutative rings as follows. Definition 5.7.1. Let R be a ring and T an R-algebra. A left T -semistar operation on R is a closure operation  on the complete lattice Mod R (T ) of all left R-submodules of T such that I  x ⊆ (I x) for all I ∈ Mod R (T ) and all x ∈ T . A right T -semistar operation on R is a closure operation  on the complete lattice Mod(T ) R of all right R-submodules of T such that x I  ⊆ (x I ) for all I ∈ Mod(T ) R and all x ∈ T . A left or right T -semistar operation  is unital if the image of R in T is -closed. By a T -semistar operation we mean, by convention, a left T -semistar operation. Any semiprime operation on the left R-module T is clearly a T -semistar operation on R. We will see that in various special situations that the converse is also true, as, for example, in Example 5.7.2(2) and Proposition 5.7.16. Advantages of T -semistar operations over and above semiprime operations on the R-module T are that the former are more general and working with them does not require knowledge of the R-endomorphisms of T . Example 5.7.2. Let R be a commutative ring with total quotient ring K = T (R). (1) A semiprime operation on a ring R is equivalently an R-semistar operation on R. (2) A semistar operation on R is equivalently a K -semistar operation on R, or equivalently a semiprime operation on the R-module K . (3) If R and T are commutative, then a T -semistar operation is equivalently a nucleus on the ordered monoid Mod R (T ). Examples are provided by Propositions 3.6.18 and 3.6.21 and the operations bTR , a TR , and cTR studied in Section 4.4. Proposition 5.7.3. Let R be a ring, let T be an R-algebra, and let  be a closure operation on Mod R (T ). The following conditions are equivalent. (1)  is a T -semistar operation. (2) f (I  ) ⊆ f (I ) for all I ∈ Mod R (T ) and all T -module endomorphisms f of the left T -module T . (3) I  X ⊆ (I X ) for all I ∈ Mod R (T ) and all subsets X of T . (4) I  ⊆ ((I a) :T a) for all I ∈ Mod R (T ) and all a ∈ T .

5.7 Semistar Operations on Algebras

(5) (6) (7) (8) (9)

425

I  ⊆ ((I X ) :T X ) for all I ∈ Mod R (T ) and all subsets X of T . (I :T a) ⊆ (I  :T a) for all I ∈ Mod R (T ) and all a ∈ T . (I :T X ) ⊆ (I  :T X ) for all I ∈ Mod R (T ) and all subsets X of T . I  = I implies (I :T a) = (I :T a) for all I ∈ Mod R (T ) and all a ∈ T . I  = I implies (I :T X ) = (I :T X ) for all I ∈ Mod R (T ) and all subsets X of T .

Proposition 5.7.4. Let R be a ring. (1) The poset of all closure operations on Mod R (T ) is a complete lattice. If  is a set of T -semistar operations, then inf  and sup  are T -semistar operations. (2) For any closure operation  on Mod R (T ), there exists a smallest T -semistar operation  greater than or equal to , and there exists a largest T -semistar operation  less than or equal to . (3) sup  = sup{  :  ∈ } and inf  = inf{ :  ∈ } for any set  of T -semistar operations. (4) For any T -semistar operation , the operation t : I −→

 {J  : J ∈ Mod R (T ) is f.g. and J ⊆ I }

on Mod R (T ) is the largest finite type T -semistar operation less than or equal to . If  is a closure operation on Mod R (T ), then the notations  and  have a different meaning if T is considered an R-module rather than an R-algebra. If T is an R-algebra, then in this section we use the notation as in Proposition 5.7.4. Corollary 5.2.35 generalizes as follows. Proposition 5.7.5. Let R be a ring, let T be an R-algebra, and let  be a closure operation on Mod R (T ). One has Mod R (T )  = {J ∈ Mod R (T ) : (J :T a) = (J :T a) for all a ∈ T }. Proof. Let C  = {J ∈ Mod R (T ) : (J :T a) = (J :T a) for all a ∈ T}. Let J ⊆ J , it follows J :T a = C  . For all a ∈ T , since (J :T a) is -closed for all J ∈   J ∈ C  . Hence C  is {(J :T a) : J ∈ J } is also -closed, and therefore a submodule closure system on the R-module T . By Proposition 5.2.23, therefore, we may let 0 be the closure operation on Mod R (T ) corresponding to C  , so that C  = Mod R (T )0 and   0 . Let J ∈ Mod R (T ) with J 0 = J . Then ((J :T a) :T b) = (J :T ba) = (J :T ba) = ((J :T a) :T b) for all a, b ∈ T , and therefore (J :T a)0 = (J :T a). Therefore 0 is a T -semistar operation. Let  be any T -semistar operation greater than or equal to . Let J ∈ Mod R (T ) be  -closed. Then,  for all a ∈ T , one has (J :T a) = (J :T a), hence (J :T a) = (J :T a). Hence J  is 0 -closed. Thus 0   . It follows, then, that 0 =  . Proposition 5.7.6. Let R be a ring and f : S −→ T an R-algebra homomorphism, and let  be a T -semistar operation on R. Then the operation f −1 () : I −→

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f −1 ( f (I ) ) on Mod R (S) is an S-semistar operation on R. Moreover, one has f −1 ()t  f −1 (t ), and if f is injective then equality holds. −1

Proof. For all I ∈ Mod R (S) and all a ∈ S one has f (I f () a) = f ( f −1 ( f (I ) )a) = f ( f −1 ( f (I ) )) f (a) ⊆ f (I ) f (a) ⊆ ( f (I ) f (a)) = f (I a) and therefore If

−1

()

a ⊆ f −1 ( f (I a) ) = (I a) f

−1

()

.

Therefore f −1 () is an S-semistar operation on R. The rest of the proposition follows from the fact that if J ⊆ I is finitely generated then f (J ) ⊆ f (I ) is finitely generated and, if f is injective, then conversely every finitely generated J  ⊆ f (I ) is of the form f (J ) for some finitely generated J ⊆ I .  Note that if f : M −→ N is a homomorphism of R-modules, then one does not expect that f −1 () : L −→ f −1 ( f (L) ) will be a semiprime operation on M if  is a semiprime operation on N . Proposition 5.7.6 implies the following. Proposition 5.7.7. Let R be a ring and T an R-algebra, and let  be a T -semistar operation. Then R acts by I −→ I  ∩ R. Moreover, if the map R −→ T is injective, then t R = R t , and in particular, if  is of finite type, then R is of finite type. Corollary 5.7.8. Let R be a ring and T an R-algebra. A closure operation  on Mod R (T ) is stable if and only if (I  : R a) = (I : R a) ∩ R for all I ∈ Mod R (T ) and all a ∈ T . Proposition 5.7.9. Let R be a ring and T an R-algebra such that the map R −→ T is injective, and let  be a T -semistar operation. The following conditions are equivalent. (1)  is stable. (2)  = R T and R is stable. (3) I  = {x ∈ T : (I : R x) = R  } for all I ∈ Mod R (T ). Proof. Conditions (1) implies condition (3) by Corollary 5.3.8 since J  = R  if and only if J R = R for all J ∈ Il (R). Suppose that (3) holds. Then I R = {x ∈ R : (I : R x)R = R} for all I ∈ Il (R), whence R is stable by Corollary 5.3.7. Moreover,  = R T by Corollary 5.3.7. Therefore (3) implies (2). Finally, (2) implies (1), again by Corollary 5.3.7.  Corollary 5.7.10. Let R be a ring and T an R-algebra such that the map R −→ T is injective, and let  be a T -semistar operation. Then  = R T is the largest stable T -semistar operation less than or equal to  and acts by  : I −→ I  = {a ∈ T : (I : R ra) = R  for all r ∈ R}. Moreover, one has R = R as semiprime operations on R.

5.7 Semistar Operations on Algebras

427

Corollary 5.7.11. Let R be a ring and T an R-algebra such that the map R −→ T is injective. The map s −→ s T , with inverse  −→ R , is an isomorphism from the poset of all stable semiprime operations on R to the poset of all stable T -semistar operations on R. Proof. Let s be a stable semiprime operation on R with corresponding Gabriel filter L. Then s T is a stable T -semistar operation, and since I s T = {a ∈ T : (I : R a) ∈ L} for all I ∈ Mod R (T ), one has I s T R = I s T ∩ R = {a ∈ R : (I : R a) ∈ L} = I s for all I ∈ Il (R), and therefore s T R = s. Conversely, by Proposition 5.7.9, R is a stable semiprime operation on R for any stable T -semistar operation , and by Corollary 5.7.10 one has R T = .  Example 5.7.12. Applying Theorem 5.6.28 and Corollary 5.7.11 to the case where R is an integral domain and T is the quotient field of R, we recover [62, Theorem 2.10 and Corollary 2.11]: there is an isomorphism between the poset of all Gabriel filters on R and the poset of all stable semistar operations on R. Theorem 5.2.48 generalizes as follows. Theorem 5.7.13. Let R be a ring and T an R-algebra such that the map R −→ T is injective. (1) For any closure operation  on Mod R (T ), the following conditions are equivalent. (a)  is an R-surprime T -semistar operation on R. (b)  is of the form s T for some semiprime operation s on R. (c) I  = I R T for all I ∈ Mod R (T ). (d) I  = I if (and only if) (I : R x) = (I : R x) ∩ R for all x ∈ T , for all I ∈ Mod R (T ). (e) I  = I if (and only if) for all x ∈ T one has (I : R x) = J  ∩ R for some J ∈ Mod R (T ), for all I ∈ Mod R (T ). (2) For any closure operation s on Il (R), the following conditions are equivalent. (a) s is a T -subprime semiprime operation on R. (b) s = R for some semiprime operation  on the R-module T . (c) s = R for some T -semistar operation  on R. (d) I s = I s T ∩ R for all I ∈ Il (R). (e) I s = I (if and) only if I = J ∩ R for some J ∈ Mod R (T ) such that (J : R x)s = (J : R x) for all x ∈ T , for all I ∈ Il (R). (3) The association s −→ s T , with inverse  −→ R , defines an isomorphism from the poset of all T -subprime semiprime operations on R to the poset of all R-surprime T -semistar operations on R. (4) The association s −→ s T , with inverse  −→ R , defines an isomorphism from the poset of all finite type T -subprime semiprime operations on R to the poset of all finite type R-surprime T -semistar operations on R. Corollary 5.4.3 generalizes to T -semistar operations in a straightforward manner, as follows.

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Proposition 5.7.14. Let R be a ring and T an R-algebra. (1) Let J ∈ Mod R (T ). There is a largest T -semistar operation  = v(J ) such that J = J. (2) For all I ∈ Mod R (T ), one has I v(J ) = (J :T (I \T J ))  = {H ∈ Mod R (T ) : ∀a ∈ T (I a ⊆ J ⇒ H a ⊆ J )}  = {H ∈ Mod R (T ) : (I \T J ) ⊆ (H \T J )},

(3)

(4) (5) (6)

and I v(J ) is the largest H ∈ Mod R (T ) such that I a ⊆ J implies H a ⊆ J for all a ∈ T . Moreover, I ∈ Mod R (T ) is v(J )-closed if and only if I = (J :T X ) for some subset X of T . Let J ⊆ Mod R (T ). The operation v(J ) = inf{v(J ) : J ∈ J } is the largest T -semistar operation  such that J  = J for all J ∈ J . For any T -semistar operation , one has   v(J ) if and only if J  = J for all J ∈ J . One has  = v(Mod R (T ) ) for any T -semistar operation . Thus, every T -semistar operation  is of the form v(J ) for some J ⊆ Mod R (T ). v = v(R) is the largest unital T -semistar operation. r = v((0)) is the largest reduced T -semistar operation.

Using Lemma 5.2.24 (3) as needed, we can define on the complete lattice of all T -semistar operations the following interior operations. d :  −→ , s :  −→ , t :  −→ t , r :  −→ red = inf{, r}, v :  −→ v = inf{, v}, w = inf{s, t} :  −→ w , rs = inf{s, r} = s ◦ r :  −→ red , vs = inf{s, v} = s ◦ v :  −→ v , rt = inf{t, r} = t ◦ r :  −→ (red )t , vt = inf{t, v} = t ◦ v :  −→ (v )t ,

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429

rv = inf{v, r} = v ◦ r :  −→ (red )v , rvw = inf{s, t, v, r} = w ◦ v ◦ r :  −→ ((red )v )w . rvs = inf{s, v, r} = s ◦ v ◦ r :  −→ (red )v , rvt = inf{t, v, r} = t ◦ v ◦ r :  −→ ((red )v )t , rvw = inf{s, t, v, r} = w ◦ v ◦ r :  −→ ((red )v )w . These operations are ordered according to the following lattice. d s

t r

rs

w

vs rw

v rt

rv

rvs

vt

vw rvt

rvw Proposition 5.7.15. Let R be a ring and T an R-algebra, and let  be a T -semistar operation on R. (1) w = w is the largest finite type stable T -semistar operation less than or equal to . (2) red = rs is the largest reduced stable T -semistar operation less than or equal to . (3) v = vs is the largest unital stable T -semistar operation less than or equal to . (4) (red )t = rt is the largest reduced finite type T -semistar operation less than or equal to . (5) (v )t = vt is the largest finite type unital T -semistar operation less than or equal to .

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5 Noncommutative Rings and Closure Operations on Submodules

(6) (red )v = rv is the largest reduced unital T -semistar operation less than or equal to . (7) (red )w = rw is the largest reduced finite type stable T -semistar operation less than or equal to . (8) (v )w = vw is the largest unital finite type stable T -semistar operation less than or equal to . (9) (red )v = rvs is the largest reduced unital stable-type stable T -semistar operation less than or equal to . (10) ((red )v )t = rvt is the largest reduced unital finite type T -semistar operation less than or equal to . (11) ((red )v )w = rvw is the largest reduced unital finite type stable T -semistar operation less than or equal to . Let L be any Gabriel filter on R with corresponding left exact radical r and stable semiprime operation s. For any R-module M, we let E L (M) = M s E(M) = {x ∈ E(M) : (M : R x) ∈ L} denote the L-injective envelope of M and ML = E L (M/r(M)) denote the module of quotients of M with respect to L, also known as the localization of M with respect to L [177, Chapter IX]. The R-module RL has the structure of an R-algebra, and the R-module ML has the structure of an RL -module. One may Hom R (I, M/r(M)), and one may construct ML alternatively as the direct limit lim −→ I ∈L

characterize both E L (M) and ML via universal properties [177, Chapter IX]. The association M −→ ML is functorial in M and defines a left exact endofunctor of R-Mod, also denoted Q L : M −→ Q L (M), or alternatively Q σ : M −→ Q σ (M), where σ is the hereditary torsion theory corresponding to L [112, 177]. Proposition 5.7.16. Let R be a ring and L a left Gabriel filter on R. A left RL -semistar operation on R is equivalently a semiprime operation on the left R-module RL . Proof. If M and N are R-modules, then by [177, Proposition IX.1.8] and the proof of [177, Proposition IX.1.11] the canonical map Hom R (ML , NL ) −→ Hom R (M, NL ) is an isomorphism. In particular, the canonical maps End R (RL ) −→ Hom R (R, RL ) −→ RL −→ End RL (RL ) are isomorphisms. The proposition therefore follows from the equivalence of (1) and (2) of Proposition 5.7.3.  In the special case where L = Dl (R) is the Gabriel filter of all dense left ideals of R, the localization RL is equal to the maximal (or complete) left ring of quotients Q(R) = Q lmax (R) of R [177, p. 200]. Moreover, by Proposition 5.6.36, the following conditions are equivalent for any Gabriel filter L on R with corresponding stable

5.7 Semistar Operations on Algebras

431

semiprime operations: (1) R is L-torsion-free; (2) L is contained in Dl (R); (3) s  r; (4) R is contained in RL ; (5) RL is contained in Q(R); and (6) RL = R s Q(R) . In particular, Dl (R) is the largest Gabriel filter L on R such that R ⊆ RL . Definition 5.7.17 ([177, p. 205]). Let R be a ring. The canonical topology on R is the largest Gabriel topology on R with respect to which R is its own ring of quotients, that is, the corresponding Gabriel filter is the largest Gabriel filter L on R such that R = RL . By [177, p. 205], the hereditary torsion theory corresponding to the canonical topology on R is the torsion theory on R-Mod cogenerated by the injective R-module E(R) ⊕ E(E(R)/R). We now consider T -semistar operations in the situation where T = Q(R). Definition 5.7.18. Let R be a ring. A left Q-semistar operation on R is a left Q lmax (R)-semistar operation (or equivalently a semiprime operation on the left R-module Q lmax (R)). A right Q-semistar operation on R is a right Q rmax (R)semistar operation (or equivalently a semiprime operation on the right R-module Q rmax (R)). By a Q-semistar operation we mean, by convention, a left Q-semistar operation. Proposition 5.7.19. Let R be a ring. Then v = v(R) : I −→ (R : Q(R) (I \ Q(R) R)) is the largest unital Q-semistar operation on R. Moreover, v = vR : I −→ (R : R (I \ Q(R) R)) is the largest semiprime operation s on R such that the Q-semistar operation s Q(R) is unital, and v Q(R) is the largest unital R-surprime Q-semistar operation on R. Definition 5.7.20. Let R be a ring. (1) (2) (3) (4) (5) (6) (7)

v = v(R) is the largest unital Q-semistar operation. t = vt . w = vw . v = vR . t = vt = tR . w = vw = wR . u = vred .

The semiprime operations v, t, and w are the semiprime analogues of the semistar operations v, t, and w. Note that v = vR , and u = vred = inf{v, r} = inf{v, r}, uw = (vred )w = inf{v, r}w = inf{w, r w }. Corollary 5.7.21. Let R be a ring. The semiprime operation v (resp., t, v, w, vred , (vred )t , u, uw ) is the largest (resp., largest finite type, largest stable, largest finite

432

5 Noncommutative Rings and Closure Operations on Submodules

type stable, largest reduced, largest reduced finite type, largest reduced stable, and largest reduced finite type stable) semiprime operation s on R such that the Q-semistar operation s Q(R) is unital. The following lattice diagram shows the relationships between twelve important semiprime operations. v

r

v

t

r

rt

w

rw

vred (vred )t

u

uw Note that I v = (R : R (I \ Q(R) R)) for all left ideals I of R, so in particular 0v = (R : R Q(R)). Therefore v = vred if and only if v  r, if and only if v is reduced, if and only if (R : R Q(R)) = 0. When these conditions hold, the lattice above simplifies as follows. r

r

v = vred

rt

v=u

rw

t = (vred )t

w = uw

Definition 5.7.22. Let R be a ring. A left ideal I of R is a (left) Glaz–Vasconcelos ideal if the restriction map Hom R (R, R) −→ Hom R (I, R) is an isomorphism. We let GV(R) denote the set of all Glaz–Vasconcelos ideals of R.

5.7 Semistar Operations on Algebras

433

Proposition 5.7.23. Let R be a ring and I a left ideal of R. The following conditions are equivalent. (1) (2) (3) (4) (5) (6)

I is open in (or equivalently, lies in the Gabriel filter of) the canonical topology. I u = R. I ∈ Dl (R) and I v = R. I ∈ Dl (R) and (I : R a)v = R for all a ∈ R. I ∈ Dl (R) and ((I : R a) \ Q(R) R) = R for all a ∈ R. (I : R a) ∈ GV(R) for all a ∈ R.

Proof. Conditions (2) and (3) are equivalent because u = inf{r, v}. Conditions (3) and (4) are equivalent by Theorem 5.3.25, and conditions (4) and (5) are equivalent by Proposition 5.7.19. Note that, if I ∈ GV(R), then annr (I ) = 0; indeed, if I a = 0 for some a ∈ R, then the map R −→ R acting by x −→ xa is the zero map when restricted to I , hence is the zero map, whence a = 0. Consequently, if condition (6) holds, then I is dense. Moreover, if I is dense, then the map (I \ Q(R) R) −→ Hom R (I, R) acting by a −→ (x −→ xa) is an isomorphism. These two facts yield the equivalence of conditions (5) and (6). Finally, note that R = RL if and only if R is L-torsion-free and R = E L (R), if and only if L ⊆ Dl (R) and (I \ Q(R) R) = R for all I ∈ L, if and only if I u = R for all I ∈ L. It follows that the stable semiprime operation corresponding to the canonical topology is the semiprime operation u. Consequently, conditions (1) and (2) are equivalent.  If R is commutative, then condition (6) of the proposition may be replaced by I ∈ GV(R). Corollary 5.7.24. Let R be a ring. The Gabriel topology on R corresponding to the stable semiprime operation u is the canonical topology on R. The Gabriel topology on R corresponding to the stable semiprime operation uw is the largest finite type topology contained in the canonical topology on R, which is the largest finite type Gabriel topology on R with respect to which R is its own ring of quotients. Corollary 5.7.25. Let R be a ring, and let s be a stable semiprime operation on R and L its corresponding Gabriel filter. The following conditions are equivalent. (1) (2) (3) (4)

R = RL . s  u. Every element of L is open in the canonical topology. L ⊆ GV(R).

We end this section with a brief application to w-envelopes. Let R be a ring and M an R-module. Let L denote the Gabriel filter corresponding to the semiprime operation uw , or equivalently the Gabriel filter corresponding to the largest finite type Gabriel topology contained in the canonical topology. If R is commutative, then the w-envelope Mw of M [184] is defined to be the R-submodule

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5 Noncommutative Rings and Closure Operations on Submodules

Mw = {x ∈ E(M) : I x ⊆ M for some f.g. I ∈ GV(R)} of E(M). In that case, one has Mw = E L (M) = M uw E(M) , that is, the w-envelope of M is equal to the L-injective envelope of M. Thus, for any ring R (possibly noncommutative) and any R-module M, we may define the w-envelope Mw of M by Mw = E L (M) = M uw E(M) . One typically studies w-envelopes for R-modules M that are GV-torsion-free [184], which is equivalent to L-torsion-free. Thus, for a general R-module M, one may instead study the R-module ML = E L (M/r(M)), where r is the left exact radical corresponding to the Gabriel filter L and stable semiprime operation uw . Thus, w-envelopes can be seen in the more general context of semiprime operations, particularly the semiprime operation uw . This opens a path to generalizing results on w-envelopes to noncommutative rings.

Exercises Section 5.2 1. Prove Lemma 5.2.6 and Proposition 5.2.7. 2. a) Prove Proposition 5.2.8. b) Let R be a ring. Verify that the R-module homomorphisms R −→ M for any R-module M are precisely those of the form a −→ ax for some fixed x ∈ M. c) Verify Corollary 5.2.10. 3. Verify the claims made in Remark 5.2.13. 4. Verify all of the unproved claims in Remark 5.2.14. 5. Verify Proposition 5.2.17. 6. Prove Proposition 5.2.19. 7. Prove Proposition 5.2.23. 8. Prove Lemma 5.2.24. 9. Prove statements (1)–(3) of Proposition 5.2.27 and statements (1), (3), and (4) of Proposition 5.2.28. 10. Complete the verification of statements (2) and (3) of Theorem 5.2.30, and prove Corollaries 5.2.32, 5.2.33, 5.2.34, 5.2.35, 5.2.36, and 5.2.37. 11. Let R be a commutative ring with total quotient ring K , and let M be an Rmodule. A closure operation  on Mod R (M) is said to be reduced if (0) = (0). Let s be a semiprime operation on R. Prove the following.

Exercises

12. 13. 14. 15.

16.

17.

435

a) I ∈ Mod R (K ) is s K -closed if and only if a I ∩ R is s-closed for all regular a ∈ R. b) I /R ∈ Mod R (K /R) is s K /R -closed if and only if I is s K -closed. c) The following conditions are equivalent. 1) s K is unital. 2) s K /R is reduced. 3) Every regular principal ideal of R is s-closed 4) I −1 /R is s K /R -closed for every ideal I of R. d) Let b : I −→ I denote the integral closure semiprime operation on R. Then b K /R is reduced if and only if R is integrally closed. Prove Theorem 5.2.40 and Corollaries 5.2.41, 5.2.42, 5.2.43, 5.2.44, and 5.2.45. Verify Example 5.2.46. Prove Theorem 5.2.48. In this exercise we provide an alternative description of the semiprime operations on a module over a ring as the nuclei on a certain right quantale module. Let S be a semigroup. An right S-module is a set Q together with a right action Q × S −→ Q of S on Q such that, for all a, b ∈ S and all x ∈ Q, one has (xa)b = x(ab). An ordered right S-module is a right S-module Q together with a partial ordering on Q such that, for all a ∈ S and all x, y ∈ Q, if x  y, then xa  ya. A right quantale S-module is an ordered right S-module Q that is a complete lattice and satisfies sup(X A) = (sup X )A for all subsets X of Q and all subsets A of S, where X A = {xa : x ∈ X, a ∈ A}. Let S be a semigroup and Q an ordered right S-module. A nucleus on Q over S is a closure operation  on Q such that x  a  (xa) for all x ∈ Q and all a ∈ S. a) Let R be a ring. Prove the following. i) Let R • denote the monoid R under multiplication. The ordered semigroup Il (R) of all left ideals of R is a right quantale R • -module under the action (I, a) −→ I a for I ∈ Il (R) and a ∈ R. ii) A left semiprime operation on R is equivalently a nucleus on the right quantale R • -module Il (R). b) Let R be a ring and M an R-module. Prove the following. i) End R (M) is a ring under the operation of pointwise addition and the operation ( f, g) −→ f g = g ◦ f of multiplication. ii) The poset Mod R (M) is a right quantale End R (M)• -module under the action (N , f ) −→ f (N ) for N ∈ Mod R (M) and f ∈ End R (M)• . iii) A semiprime operation on M is equivalently a nucleus on the right quantale End R (M)• -module Mod R (M). Let R be a ring and M an R-module. Let s be an order-preserving operation on Il (R) and  a closure operation on Mod R (M). Show that, if  is compatible with s, then I s N ⊆ (I N ) for all I ∈ Il (R) and all N ∈ Mod R (M). Moreover, show that the converse holds if R is commutative, but not necessarily if R is noncommutative. (Provide a counterexample in the noncommutative case.) (∗∗) Prove Corollary 5.2.48 directly from Theorem 5.2.48, without using Theorem 4.3.32.

436

5 Noncommutative Rings and Closure Operations on Submodules

Section 5.3 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

Fill in the missing details in the proof of Proposition 5.3.2. Prove Proposition 5.3.4 and Corollary 5.3.6. Prove Corollaries 5.3.7 and 5.3.8. Verify that conditions (1)–(4) of Corollary 5.3.9 are equivalent. Prove Proposition 5.3.11 and Corollary 5.3.12. Verify the equivalence of the five conditions in Example 5.3.13. Verify the claims made in Remark 5.3.16. Prove Proposition 5.3.20. Prove Lemma 5.3.22. Prove Theorem 5.3.23. Prove Theorem 5.3.25. Let R be a ring and s a filtered presemiprime operation on R. Show that s∞ = s∞ . Prove Proposition 5.3.27. Prove Proposition 5.3.31. Verify the examples in Example 5.3.32. Prove Proposition 5.3.34. Prove Proposition 5.3.38. Let s be a semiprime operation on a ring R and  a semiprime operation on an R-module M. Prove or give a counterexample. a) If  is s-stable, then s is M-subprime. b) If  is s-stable, then  is R-surprime. 19. Let s be a semiprime operation on a ring R and  a semiprime operation on an R-module M. Show that the following conditions are equivalent. 1)  is stable. 2) For every R-submodule L of M, one has L  = L if (and only if) for all x ∈ M one has x ∈ L whenever y ∈ ((L : R x)y) for all y ∈ M. 3)  is R-semiprime and, for every R-submodule L of M, if for all x ∈ M one has x ∈ L whenever y ∈ ((L : R x)y) for all y ∈ M, then for all x ∈ M and all a ∈ R one has ax ∈ L whenever ay ∈ ((L : R x)y) for all y ∈ M. 20. Let R be a commutative ring and M an R-module, and let  be a closure operation on Mod R (M). a) Show that if  and R are stable, then (I \ M N ) = (I \ M N  ) for every N ∈ Mod R (M) and every finitely generated I ∈ I(R). b) (∗∗) If  is stable, then must (I \ M N ) = (I \ M N  ) for every N ∈ Mod R (M) and every finitely generated I ∈ I(R)? Conversely, if (I \ M N ) = (I \ M N  ) for every N ∈ Mod R (M) and every finitely generated I ∈ I(R), then must  be stable? 21. (∗∗) Let R be a Noetherian commutative ring of prime characteristic, let ∗ denote the tight closure semiprime operation on R, and for any finitely generated Rmodule M let ∗ M denote the tight closure operation on Mod R (M), as defined in [102]. For which rings R and R-modules M does one have ∗ M = ∗ M , that is, for which rings R and R-modules M is tight closure on M the smallest semiprime operation on M that is compatible with tight closure on R? Answer

Exercises

437

the analogous problem for the condition (∗ M )t = ∗ M , as well as for various other well-known closure operations on submodules. 22. (∗∗) Let R be a ring and s a semiprime operation on R. If s is of finite type, then must s also be of finite type? Equivalently, must one have sw = st ? If not, provide an explicit description of the semiprime operation sw . Section 5.4 1. Verify the unsubstantiated claims in Definition 5.4.1. 2. Let M, N , and P be sets, let X ⊆ N , Y ⊆ M, and Z ⊆ P, and let H ⊆ M N and K ⊆ P M . Let K ◦ H = { f ◦ g : f ∈ K, g ∈ H} ⊆ P N . Show the following. a) (X · H) · K = X · (K ◦ H). b) ((Z : K) : H) = (Z : K ◦ H). c) (Y \K Z ) ◦ (Y \H X ) ⊆ (X \K◦H Z ). 3. a) Verify the claims made in Remark 5.4.6. b) Verify statements (1)–(17) used in the proof of Proposition 5.4.11 and then complete the proof of the proposition. 4. Let D be a DVR. Show that every semiprime s operation on D satisfies I s = I + (0)s for all I ∈ I(R) or else (0)s = (0) and there is a nonzero H ∈ I(R) such that I s = I + H for all I ∈ I(R). 5. Show that there are uncountably many reduced semiprime operations (in fact, 2ℵ0 of them) on Z but only countably many non-reduced semiprime operations on Z. 6. Verify all of the unsubstantiated claims made in Example 5.4.10. 7. Use [177, Propositions VI.5.6 and VI.6.4] to verify Proposition 5.6.36. 8. Let R be a ring and M an R-module. Let us say that an operation  on Mod R (M) is a Kirby closure on M if  is a closure operation on Mod R (M) such that I N  ⊆ (I N ) for all N ∈ Mod R (M) and all I ∈ Il (R). Such closure operations, for R commutative, were first considered by D. Kirby in 1969 [119]. Prove the following. a) Let  be a self-map of Mod R (M). The following conditions are equivalent. 1)  is a Kirby closure on M. 2)  is a closure operation on Mod R (M) such that a N  ⊆ (a N ) for all N ∈ Mod R (M) and all a ∈ R. 3) I N ⊆ P  if and only if I N  ⊆ P  for all N , P ∈ Mod R (M) and all I ∈ Il (R). b) The poset of all Kirby closures on M is complete. c) Let N be an R-submodule of M. Define L v M (N ) = ((N : R L) \ M N ) for all L ∈ Mod R (M). Then v M (N ) : L −→ L v M (N ) is the largest Kirby closure  on M such that N  = N . Moreover, L ∈ Mod R (M) is v M (N )closed if and only if L = (I \ M N ) for some (two-sided) ideal I of R. d) Let N be a subset of Mod R (M). Then v M (N ) = inf{v M (N ) : N ∈ N } is the largest Kirby closure  on M such that N ⊆ Mod R (M) .

438

9.

10. 11. 12.

5 Noncommutative Rings and Closure Operations on Submodules

e) One has  = v M (Mod R (M) ) for any Kirby closure  on M. Thus, every Kirby closure on M is of the form v M (N ) for some N ⊆ Mod R (M). f) A semiprime operation on M need not be a Kirby closure on M. g) Suppose that R is commutative. Then any semiprime operation on M is a Kirby closure on M, but a Kirby closure on M need not be a semiprime operation on M. Let R be a ring. Prove the following. a) The set Gl (R) of all Goldie dense left ideals of R is the smallest Gabriel filter of R containing the essential left ideals of R. b) If g denotes the stable semiprime operation on R corresponding to the Gabriel filter Gl (R), then g  r, and one has g = r if and only if g is reduced, if and only if every essential left ideal of R is dense; such a ring R is said to be (left) nonsingular. c) A commutative ring is nonsingular if and only if it is reduced. Prove Proposition 5.4.13. Prove Theorem 5.4.14. Let R be a ring, and let P ⊆ H ⊆ M be R-modules. Prove the following. a) There exists a smallest closure operation  on Mod R (M) such that P  = H , and  acts by   : N −→

N + H if P ⊆ N N otherwise.

b) There exists a smallest semiprime operation  = u M (P, H ) on M such that P  ⊇ H . For all N ∈ Mod R (M), one has N u M (P,H ) =



{L ∈ Mod R (M) : L ⊇ N and ∀ f ∈ End R (M) ( f (P) ⊆ L (5.1)

⇒ f (H ) ⊆ L)}. Equivalently, N u M (P,H ) is the smallest L ∈ Mod R (M) containing N such M that H ⊆ P v M (L) . c) The following conditions are equivalent. 1) There exists a semiprime operation  on M such that P  = H . 2) For all f ∈ End R (M), if f (P) ⊆ H , then f (H ) ⊆ H . 3) P u M (P,H ) = H . 4) u M (P, H ) is the smallest semiprime operation  on M such that P = H . M 5) P v M (H ) = H . M 6) v M (H ) is the largest semiprime operation  on M such that P  = H .

Exercises

439

Section 5.5 1. Let R be a ring, let r be a preradical on R-Mod, and let M and N be R-modules with N ⊆ M. Prove the following. a) If M is r-torsion-free, then N is r-torsion-free. b) If M is r-torsion, then M/N is r-torsion. c) For any set M of R-modules, one has  r



 ⊆

M

M∈M



and r

 M∈M



r(M).

M∈M

 M

=



r(M).

M∈M

 2. Prove Proposition 5.5.5. (Hint: Consider the natural surjection x∈M Rx −→ M, and use the previous exercise.) 3. Fill in the missing details in the proof of Proposition 5.5.6. 4. Verify Lemma 5.5.11. 5. Verify the unproved assertions in statements (4), (5), and (6) of Remark 5.5.12. 6. Fill in the missing details in the proof of Theorem 5.5.14. 7. Prove Lemmas 5.5.19 and 5.5.21. 8. Prove Theorem 5.5.22. 9. Prove Corollary 5.5.28. 10. Fill in the missing details in the proof of Proposition 5.5.6. 11. Prove Propositions 5.5.29 and 5.5.30. 12. Verify Lemma 5.5.38 and Corollary 5.5.39. 13. (∗∗) Must a left exact radical or preradical be strongly cohereditary? Section 5.6 1. Prove Proposition 5.6.2 and Corollary 5.6.4 with assistance from [129, Proposition 0.1] and [177, Propositions VI.2.2–2.3]. 2. Prove Theorem 5.6.6. 3. Prove Theorem 5.6.8 and Corollary 5.6.9. 4. Prove Proposition 5.6.12. 5. Fill in the missing details in the proof of Theorem 5.6.18 with assistance from [177, Chapter VI Section 4]. 6. Prove Proposition 5.6.16. 7. Verify Lemma 5.6.22 and Corollary 5.6.23. 8. Prove Proposition 5.6.25 with assistance from [177, Chapter VI Section 3]. 9. Fill in the missing details in the proof of Theorem 5.6.28 with assistance from [177, Chapter VI Section 5]. 10. Verify the claims made in Example 5.6.34 with assistance from [177, Chapter VI]. 11. Prove Proposition 5.6.40 and Corollaries 5.6.42 and 5.6.43.

440

5 Noncommutative Rings and Closure Operations on Submodules

Section 5.7 1. Let R be a ring and T an R-algebra. Show that the complete lattice Mod R (T ) is a right quantale module over the semigroup T • under the action (I, x) −→ I x, and a T -semistar operation is equivalently a nucleus on the right quantale module Mod R (T ) over the semigroup T • . (See Exercise 5.2.15 for the definitions of these terms.) 2. Let R be a ring and T an R-algebra. Show that any semiprime operation on the R-module T is a T -semistar operation. 3. Prove Proposition 5.7.3. 4. Prove Proposition 5.7.4. 5. Prove Proposition 5.7.7 and Corollary 5.7.8. 6. Prove Corollaries 5.7.10 and 5.7.11. 7. Prove Theorem 5.7.13. 8. Prove Proposition 5.7.15. 9. Prove Proposition 5.7.19 and Corollary 5.7.21. 10. Verify Corollaries 5.7.24 and 5.7.25.  (Z/2Z), and let R denote the subring of T generated by m = 11.  Let T = ∞ n=1 ∞ (Z/2Z). Prove the following. n=1 a) T and R are von Neumann regular and therefore are total quotient rings. b) m is a maximal ideal of R and an ideal of T . c) m is a dense ideal of R. d) R = T (R)  Q(R) = T . e) m = 0v = 0t = (R : R Q(R)) = (m : R Q(R)). f) I v = I t = I + m for every ideal I of R. g) Int Q (R) = { f ∈ Q(R)[X ] : f (R) ⊆ R} is a ring properly containing the ring Int(R) = { f ∈ T (R)[X ] : f (R) ⊆ R}. 12. (∗∗) Let the rings R and Int Q (R) be as defined in the previous exercise. a) Determine all of the elements of the ring Int Q (R). b) Describe all ideals of R and all semiprime (or equivalently, semistar) operations on R. c) Describe all R-submodules of Q(R) and all Q-semistar operations on R. 13. (∗∗) Find an example of a T -semistar operation that is not a semiprime operation on the R-module T , where R is a commutative ring and T is a commutative R-algebra. 14. (∗∗) Let R be a ring. Must one have uw = (vred )t ? If not, give an explicit description of the semiprime operation uw .

Chapter 6

Closure Operations and Nuclei

This chapter generalizes many of the requisite facts on semistar, star, and semiprime operations on commutative rings employed in Chapters 1–4 to nuclei on ordered magmas satisfying various algebro-order-theoretic hypotheses. This serves to situate the theories of semistar, star, and semiprime operations on commutative rings in the broader contexts of closure operations, order theory, and the theories of ordered monoids, quantales, and multiplicative lattices. In order that this chapter may be self-contained, some of the definitions from Chapter 0 are repeated here.

6.1 Ordered Magmas and Nuclei Definition 6.1.1. Let S be a poset. For any subset X of S, we denote the supremum of X in S, if it exists, by sup S X , or sup X if S is understood, and similarly for the infimum inf S X = inf X . Note that sup S ∅ = inf S S and inf S ∅ = sup S S if either exists. Definition 6.1.2. (1) A poset S is complete (resp., near sup-complete, bounded complete) if sup S X exists for every subset (resp., every nonempty subset, every nonempty bounded above subset) X of S. (2) A join semilattice (resp., meet semilattice) is a poset in which every pair of elements of has a supremum (resp., infimum). (3) A lattice, or lattice-ordered poset, is a poset that is both a join and meet semilattice. A near sup-complete poset is equivalently a bounded complete poset with a largest element, and a complete poset is equivalently a near sup-complete poset with a smallest element. Every complete poset is a lattice, and every near sup-complete poset is a join semilattice. A poset S is complete (resp., near sup-complete, bounded complete) if and only if inf S X exists for every subset (resp., every bounded below subset, every nonempty bounded below subset) X of S. © Springer Nature Switzerland AG 2019 J. Elliott, Rings, Modules, and Closure Operations, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-24401-9_6

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Definition 6.1.3. For any self-map  of a set X , we write x  = (x) for all x ∈ X and Y  = (Y ) for all Y ⊆ X , and we write Closed() = {x ∈ X : x  = x}. Elements of Closed() are said to be -closed. A self-map  of a set X is idempotent if  ◦  = , that is, if (x  ) = x  for all x ∈ S. If  is a self-map of a set X , then Closed() ⊆ X  , and equality holds if and only if  is idempotent. Definition 6.1.4. A closure operation on a poset S is a self-map  of S satisfying the following conditions. (1)  is order-preserving: x  y implies x   y  for all x, y ∈ S. (2)  is expansive: x  x  for all x ∈ S. (3)  is idempotent: (x  ) = x  for all x ∈ S. A preclosure on a poset S is a self-map of S that is order-preserving and expansive. A closure operation is equivalently an idempotent preclosure. A closure operation on S is equivalently a self-map  of S such that x  y  if and only if x   y  for all x, y ∈ S. Definition 6.1.5. Let f : S −→ T be a map of posets. (1) f is order-preserving if x  y implies f (x)  f (y) for all x, y ∈ S. (2) f is sup-preserving (resp., near sup-preserving) if f (sup S X ) = supT f (X ) for all subsets (resp., all nonempty subsets) X of S such that sup S X exists. Any near sup-preserving map of posets is order-preserving. A map f : S −→ T of posets is sup-preserving if and only if f is near sup-preserving and f (inf S S) = inf T f (S) provided that inf S S exists. Lemma 6.1.6. Let  be a closure operation on a poset S and let X ⊆ S. (1) (2) (3) (4)

sup S  X  = (sup S X  ) = (sup S X ) if sup S X exists. inf S  X  = (inf S X  ) = inf S X  if inf S X  exists. The map  : S −→ S  is sup-preserving. If S is complete (resp., near sup-complete, bounded complete, a join semilattice, a meet semilattice, a lattice), then so is S  .

Definition 6.1.7. A magma is a set M equipped with a binary operation on M, which by convention we write multiplicatively. A magma is unital (resp., left unital, right unital) if it has an identity (resp., left identity, right identity) element. A semigroup is an associative magma, and a monoid is a unital semigroup. For any subsets X, Y of a magma M, we write X Y = {x y : x ∈ X, y ∈ Y }. Definition 6.1.8. If  is a self-map of a magma M, then the binary operation (x, y) −→ x  y = (x y) on M is called -multiplication. We then consider the set M  as a magma under -multiplication restricted to M  .

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Definition 6.1.9. An ordered magma is a magma M equipped with a partial ordering  on M such that x  x  and y  y  implies x y  x  y  for all x, x  , y, y  ∈ M. Definition 6.1.10. A nucleus on an ordered magma M is a closure operation  on M such that x  y   (x y) for all x, y ∈ M. A left nucleus on M is a closure operation  on M such that x y   (x y) for all a, b ∈ M. A right nucleus on M is a closure operation  on M such that x  y  (x y) for all x, y ∈ M. Nuclei were first studied in the contexts of ideal lattices and locales and later in the context of quantales as quantic nuclei [156]. If  is a nucleus on an ordered magma M, then the set M  is an ordered magma under -multiplication, by Lemma 6.1.6(1) the corestriction M  | : M −→ M  of  to M  is a sup-preserving morphism of ordered magmas, and if 1 is an identity (resp., left identity, right identity) element of M then 1 is an identity (resp., left identity, right identity) element of M  . The following elementary result provides several equivalent characterizations of nuclei. Proposition 6.1.11. The following conditions are equivalent for any closure operation  on an ordered magma M. (1)  is a nucleus on M. (2)  is a left nucleus on M and a right nucleus on M. (3) The map  : M −→ M  is a magma homomorphism, or equivalently, (x  y  ) = (x y) for all x, y ∈ M. Moreover, if M is an ordered monoid, then the conditions above hold if and only if -multiplication on M is associative. A right nucleus on an ordered magma M is equivalently a left nucleus on the ordered magma M op with underlying poset M whose binary operation ◦ is given by x ◦ y = yx for all x, y ∈ M. Thus, any proposition about left nuclei has a dual statement for right nuclei, and vice versa, and, due to Proposition 6.1.11, there is often a corresponding statement for nuclei obtained by forming the conjunction of the two. Occasionally, we state a result for left or right nuclei and leave its dual and its conjunction with its dual for the reader to formulate. In this chapter, we require definitions of several important classes of ordered magmas. To assist the reader in assimilating the definitions, we characterize the most important of them in the following table.

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6 Closure Operations and Nuclei ordered magma

sup X exists

complete near sup-complete bounded complete

for all X for all X = ∅ for all X = ∅ bounded above for all directed X for all directed X bounded above for X = ∅ for all X for all X = ∅ for all finite or bounded X = ∅ if ∃x, y : X = {z : zy  x} or X = {z : yz  x} if X = ∅ and ∃x, y : X = {z : zy  x} or X = {z : yz  x}

dcpo bdcpo with annihilator prequantale near prequantale semiprequantale residuated

near residuated

Scott-topological

sup(X Y ) = sup(X ) sup(Y )

for X = ∅ or Y = ∅ for all X, Y for all X, Y = ∅ for all (finite or) bounded X, Y = ∅ for all X, Y such that sup X , sup Y exist for all X, Y = ∅ such that sup X , sup Y exist for all directed X, Y such that sup X , sup Y exist

As we will see, one has the following implications.

residuated

near residuated

Scott-topological

prequantale

near prequantale

semiprequantale

complete

near sup-complete

bounded complete

dcpo

bdcpo

Definition 6.1.12. Let x and a be elements of an ordered magma M. We let x/a, when it exists, denote the largest element z of M such that za  x. Likewise, we let a\x, when it exists, denote the largest element z of M such that az  x. An ordered magma M is right residuated (resp., left residuated) if x/a exists (resp., a\x exists) for all x, a ∈ M. An ordered magma is residuated if it is both left residuated and right residuated.

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Note that, if x/a exists, then (x/a)a  x, and, for all z, one has za  x ⇔ z  x/a. Similarly, if a\x exists, then a(a\x)  x, and, for all z, one has az  x ⇔ z  a\x. If M is a residuated ordered magma, then an easy argument shows that for all a ∈ M the left and right multiplication by a maps on M (a· : x −→ ax and ·a : x −→ xa, respectively) are sup-preserving [76, Theorem 3.10]. More generally, if M is a left residuated (resp., right residuated) ordered magma, then for all a ∈ M the left multiplication (resp., right multiplication) by a maps on M are sup-preserving. Example 6.1.13. A Heyting algebra is a bounded lattice that, as an ordered monoid under the operation (x, y) −→ inf{x, y} of infimum, is residuated [76, Section 1.1.4]. Equivalently, a Heyting algebra is a bounded lattice equipped with a binary operation → such that inf{c, a}  b is equivalent to c  a → b. Lemma 6.1.14. Let M be an ordered magma and  a left nucleus on M, and let x, y ∈ M. (1) If x  /y exists, then x  /y  = x  /y. (2) If y\x  exists, then (y\x  ) = y\x  . Proposition 6.1.15. The following conditions are equivalent for any self-map  of a left unital ordered magma M. (1)  is a left nucleus on M. (2) x y  z  ⇔ x y   z  for all x, y, z ∈ M. (3) x  x  , and x y  z  ⇒ x y   z  , for all x, y, z ∈ M. If M is left unital and right residuated, then the conditions above are equivalent to the following. (4) x  /y = x  /y  for all x, y ∈ M. Definition 6.1.16. (1) A prequantale is an ordered magma Q such that the supremum sup X of X exists and sup(X Y ) = sup(X ) sup(Y ) for all subsets X, Y of Q [173, Definition 2.4.2]. (2) A quantale is an associative prequantale. (3) A multiplicative lattice is a commutative unital quantale. (4) A locale, or frame, is a quantale in which multiplication is the operation (x, y) −→ inf{x, y} of infimum. Remark 6.1.17. Contrary to tradition we do not require the identity element of a multiplicative lattice to be its largest element. If M is a multiplicative lattice in the sense defined here, then {x ∈ M : x  1} is a multiplicative lattice in the traditional sense. Remark 6.1.18. The term “quantale” was coined by C. J. Mulvey in 1986 [151], and the term “prequantale” by K. Rosenthal in 1990 [173]. However, prequantales were defined and studied as early as 1972, in the work of K. Keimel, who called them “completely m-distributive complete multiplicative lattices” [116, 1.4].

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Prequantales provide a natural context in which to study nuclei, as the nuclei on a prequantale classify its quotient objects in the category of prequantales. (See Exercise 6.1.19.) As the following examples indicate, prequantales are ubiquitous in mathematics. Example 6.1.19. (1) The power set 2 M of a magma M is a prequantale under the operation (X, Y ) −→ X Y = {x y : x ∈ X, y ∈ Y }. It is the free prequantale on the magma M. The prequantale 2 M is a quantale if and only if M is a semigroup, in which case 2 M is the free quantale on M. (2) The poset of all sup-preserving self-maps of a complete lattice is a unital quantale under the operation of composition. (3) A locale is equivalently a complete lattice in which finite infima distribute over arbitrary suprema, or equivalently a unital prequantale (or quantale) in which every element is idempotent and whose largest element is the identity element, or, equivalently still, a complete Heyting algebra. (4) The complete lattice of all open subsets of a topological space is a locale. A locale isomorphic to a locale of this form is said to be spatial. It is well-known that the category of spatial locales is equivalent to the category of sober topological spaces, where a topological space X is said to be sober if every irreducible closed subset of X is the closure of exactly one point of X . (5) Any totally ordered complete lattice is a locale, as is any complete Boolean algebra. (6) For any algebra T over a ring R (by which we mean a ring T together with a ring homomorphism R −→ T ), the complete lattice Mod R (T ) of all R-submodules ofthe left R-module T is a quantale under the operation (I, J ) −→ I J = { x∈X,y∈Y x y : X ⊆ I, Y ⊆ J finite}. (7) The poset of all closed linear subspaces of a unital C∗ -algebra A is a unital quantale with involution [126, Definition 2.4] under the operation (M, N ) −→ M N and by [126, Theorem 5.4] is a complete invariant of A. In the definitions above of prequantales, quantales, and multiplicative lattices, restricting subsets X to be among certain subclasses of subsets of Q yields even more general structures with respect to which it is natural to study nuclei. This is required, for example, to accommodate the theory of star operations, since the ordered monoids on which star operations are defined are bounded complete but generally not complete. Thus we make the following definitions. Definition 6.1.20. (1) A near prequantale (resp., semiprequantale) is an ordered magma Q such that sup X exists and sup(X Y ) = sup(X ) sup(Y ) for all nonempty subsets X, Y (resp., all nonempty finite or bounded subsets X, Y ) of Q. (2) A near quantale is an associative near prequantale. (3) A semiquantale is an associative semiprequantale. (4) A near multiplicative lattice is a commutative unital near quantale.

6.1 Ordered Magmas and Nuclei

447

(5) A semimultiplicative lattice is a commutative unital semiquantale. A near prequantale is equivalently a semiprequantale Q with a largest element, and a prequantale is equivalently a near prequantale Q with a smallest element inf Q that annihilates every element of Q. Many known results about quantales and multiplicative lattices generalize to these more general structures. In this chapter we generalize many of the results of Sections 2.4 and 4.2 on semistar operations, as well as their analogues in Section 4.1 for star and semiprime operations, to the context of right nuclei (resp., left nuclei) on coherent unital residuated lattice-ordered semiquantales. (Coherence is defined in Section 6.5.) Example 6.1.21. (1) If a is an idempotent element of a prequantale Q, then the set Q a = {x ∈ Q : x  a} is a sub prequantale of Q, while the set Q a = {x ∈ Q : x  a} is a sub near prequantale of Q. (2) If Q is a prequantale and x y = 0 implies x = 0 or y = 0 for all x, y ∈ Q, where 0 = inf Q, then Q − {0} is a sub near prequantale of Q. In particular, the set 2 M − {∅} of all nonempty subsets of a magma M is a sub near prequantale of the prequantale 2 M , and it is the free near prequantale on the magma M. (3) If R is a commutative ring, then the ordered monoid K(R) of all Kaplansky fractional ideals of R is a multiplicative lattice, and the ordered monoid Freg (R) of all regular fractional ideals of R is a semimultiplicative lattice. We say that an ordered magma M is with annihilator if 0 = inf M exists and 0x = 0 = x0 for all x ∈ M. Proposition 6.1.22. The following conditions are equivalent for any ordered magma M. (1) (2) (3) (4)

M is a prequantale. M is complete and residuated. M is a near prequantale with annihilator. M is complete with annihilator and the multiplication map M × M −→ M is sup-preserving. (5) M is complete and the left and right multiplication by a maps on M are suppreserving for all a ∈ M, that is, a(sup X ) = sup(a X ) and (sup X )a = sup(Xa) for all a ∈ M and all subsets X of M. (6) The map 2 M −→ M acting by X −→ sup X is a well-defined magma homomorphism. Remark 6.1.23. We say that an ordered magma M is near residuated if for all x, a ∈ M such that za  x (resp., az  x) for some z ∈ M there exists a largest such element z = x/a (resp., z = a\x) of M. Many of the results in this chapter that assume an ordered monoid M is residuated may be generalized to the situation where M is near residuated [51]. The following conditions are equivalent for any ordered magma M.

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(1) M is a near prequantale. (2) M is near sup-complete and near residuated. (3) The ordered magma M0 = M {0}, where 0x = 0 = x0 and 0  x for all x ∈ M0 , is a prequantale. (4) M is near sup-complete and the multiplication map M × M −→ M is near suppreserving. (5) M is near sup-complete and the left and right multiplication by a maps on M are near sup-preserving for all a ∈ M, that is, a(sup X ) = sup(a X ) and (sup X )a = sup(Xa) for all a ∈ M and all nonempty subsets X of M. (6) The map 2 M −{∅} −→ M acting by X −→ sup X is a well-defined magma homomorphism. Example 6.1.24. Let R be a ring. (1) The ordererd semigroup Il (R) of all left ideals of R under multiplication is a left unital quantale, hence a complete and residuated ordered semigroup, with residuation given byI /J = (I : R J ) = {H ∈ Il (R) : H J ⊆ I } = {x ∈ R : x J ⊆ I } and J \I = {H ∈ Il (R) : J H ⊆ I } = {x ∈ R : Jr x ⊆ I for all r ∈ R} for all I, J ∈ Il (R). (2) Let T be an R-algebra (by which we mean a unital ring T together with a unital ring homomorphism R −→ T ). The ordererd semigroup Mod R (T ) of all R-submodules of the left R-module T under multiplication is a left unital quantale,hence a complete and residuated ordered semigroup, with I /J = (I :T J ) = {H ∈ Mod R (T ) : H J ⊆ I } = {x ∈ T : x J ⊆ I } and J \I =  {H ∈ Mod R (T ) : J H ⊆ I } = {x ∈ T : Jr x ⊆ I for all r ∈ R} for all I, J ∈ Mod R (T ). Proposition 6.1.25. Let  be a nucleus on an ordered magma M. If M is a prequantale (resp., near prequantale, semiprequantale, quantale, near quantale, semiquantale, multiplicative lattice, near multiplicative lattice, semimultiplicative lattice, residuated), then so is M  . Moreover, if M is residuated, then (x/y) = x/y  = x/y and (y\x) = y  \x = y\x for all x ∈ M  and all y ∈ M. Proof. Suppose that M is a prequantale. By Lemma 6.1.6(4), the partial ordering on M  is complete. For any a ∈ M  and X ⊆ M  we have a  sup M  X = a  (sup M X ) = (a sup M X ) = (sup M a X ) = (sup M (a X ) ) = (sup M (a  X )) = sup M  (a  X ), and therefore a  sup M  X = sup M  (a  X ). By symmetry the corresponding equation holds for right multiplication. Thus M  is a prequantale. The proof of the rest of the first statement of the proposition is similar. The second statement of the proposition follows from Lemma 6.1.14.  The following result characterizes nuclei on residuated ordered magmas. Proposition 6.1.26. Let C be a subset of a poset S. (1) There exists a closure operation  = C on S with C = S  if and only if inf{a ∈ C : a  x} exists in S for all x ∈ S. For any such closure operation  one has x  = inf{a ∈ C : a  x} for all x ∈ S, and therefore C is uniquely determined by C.

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(2) If S = M is a left residuated ordered magma and C exists, then C is a left nucleus on M if and only if for all x ∈ C and y ∈ M one has y\x ∈ C. (3) If S = M is a right residuated ordered magma and C exists, then C is a right nucleus on M if and only if for all x ∈ C and y ∈ M one has x/y ∈ C. (4) If S = M is a residuated ordered magma and C exists, then C is a nucleus on M if and only if for all x ∈ C and y ∈ M one has x/y ∈ C and y\x ∈ C. Proof. Statement (1) is well-known and easy to check. We prove (3), from which (2) and (4) immediately follow. If  is a right nucleus, then it follows from Proposition 6.1.25 that x/y ∈ C for all x, y as in statement (2). Conversely, suppose that this condition on C holds. Let x, y ∈ M. Then we claim that x  y = inf{a ∈ C : a  x}y is less than or equal to (x y) = inf{b ∈ C : b  x y}. For let b ∈ C with b  x y. Set a = b/y, whence a  x. By hypothesis one has a ∈ C. Therefore x  y  ay = (b/y)y  b. Taking the infimum over all such b, we see that  x  y  inf{b ∈ C : b  x y} = (x y) . It follows that  is a right nucleus. Corollary 6.1.27. Let C be a nonempty subset of a bounded complete poset S. (1) There exists a closure operation  on S with C = S  if and only if inf X ∈ C for all nonempty X ⊆ C bounded below. (2) If S = M is a bounded complete and left residuated ordered magma, then there exists a left nucleus  on M with C = M  if and only if inf X ∈ C for all nonempty X ⊆ C bounded below and for all x ∈ C and y ∈ M one has y\x ∈ C. (3) If S = M is a bounded complete and right residuated ordered magma, then there exists a right nucleus  on M with C = M  if and only if inf X ∈ C for all nonempty X ⊆ C bounded below and for all x ∈ C and y ∈ M one has x/y ∈ C. (4) If S = M is a bounded complete and residuated ordered magma, then there exists a nucleus  on M with C = M  if and only if inf X ∈ C for all nonempty X ⊆ C bounded below and for all x ∈ C and y ∈ M one has x/y ∈ C and y\x ∈ C. Definition 6.1.28. A subset S of an ordered magma M is a left sup-spanning subset (resp., right sup-spanning subset) of M if x y = sup{ay : a ∈ S and a  x} (resp., x y = sup{xa : a ∈ S and a  y}) for all x, y ∈ M. A subset S of M is a supspanning subset of M of S is both a left sup-spanning subset and a right sup-spanning subset of M. If M is right unital and right residuated, or more generally if M is right unital and for all a ∈ M the right multiplication by a maps on M are sup-preserving, then S is a left sup-spanning subset of M if and only if every element of M is the supremum of some subset of S. For example, the set of all principal left ideals of a possibly noncommutative ring R is a sup-spanning subset of the left unital quantale Il (R) of all left ideals of R. Proposition 6.1.29. Let  be a closure operation on an ordered magma M and let S be any left sup-spanning subset of M. Then  is a left nucleus on M if and only if ax   (ax) for all a ∈ S and all x ∈ M, and in that case S  is a left sup-spanning subset of M  .

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Proof. Necessity of the given condition is clear. Suppose that ax   (ax) for all a ∈ S and all x ∈ M. Then   x y =

sup

ay  

a∈S: ax

sup

ay

= (x y) ,

a∈S: ax

so that  is a left nucleus. That S  is a left sup-spanning subset of M  follows easily from Lemma 6.1.6(1).  Example 6.1.30. Let R be a commutative ring with total quotient ring K . (1) The set of all principal ideals of R is a sup-spanning subset of the ordered monoid I(R) of all ideals of R. Therefore, by the proposition, if  is a closure operation on I(R), then  is a semiprime operation on R (that is,  is a a nucleus on I(R)) if and only if a I  ⊆ (a I ) for all a ∈ R and all I ∈ I(R). (2) The set of all principal fractional ideals of R is a sup-spanning subset of the ordered monoid K(R) of all Kaplansky fractional ideals of R. Therefore, by the proposition, if  is a closure operation on K(R), then  is semistar operation on R (that is,  is a nucleus on K(R)) if and only if a I  ⊆ (a I ) for all a ∈ K and all I ∈ K(R).

6.2 The Poset of All Nuclei In this section, we study properties of the poset Nucl(M) of all nuclei on an ordered magma M, defined below. There are analogues for all the results of this section for the posets Nucll (M) and Nuclr (M) of all left nuclei and right nuclei, respectively, on M, which we leave to the reader to formulate. Definition 6.2.1. Let S be a poset and X a set. The set S X of all functions from X to S is partially ordered, where f  g if f (x)  g(x) for all x ∈ X . The set Clos(S) ⊆ S S of all closure operations on S inherits a partial ordering from the poset S S . For any 1 , 2 ∈ Clos(S) one has 1  2 if and only if S 1 ⊇ S 2 . The identity operation, or smallest closure operation, on S is denoted d. If sup S exists, then there is a largest closure operation e on S, given by x e = sup S for all x ∈ S. If M is an ordered magma, then we let Nucl(M) denote the subposet of Clos(M) consisting of all nuclei on M. Proposition 6.2.2. One has the following. (1) Let S be a bounded complete poset. Then Clos(S) is bounded complete. More over, one has S sup  = ∈ S  for all bounded subsets  of Clos(S). (2) Let M be a bounded complete and residuated ordered  magma. Then Nucl(M) is bounded complete. Moreover, one has M sup  = ∈ M  for all bounded subsets  of Nucl(M).

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451

Proof. One easily checks that the intersection of nonempty subsets C of S that satisfy the condition of statement (1) of Corollary 6.1.27 itself satisfies the same condition  and is nonempty given the boundedness condition on . Therefore, since S  ⊆ S    if and only if    for all ,  ∈ Clos(S), statement (1) follows. Statement (2) is proved in a similar fashion.  Recall that for any self-map  of a set X , we let Closed() = {x ∈ X : x  = x} denote the set of all -closed elements of X . Our next proposition allows one in many circumstances to find, and in some sense compute, the smallest closure operation that is larger than a given preclosure operation. Definition 6.2.3. A prenucleus on an ordered magma M is a preclosure + on M such that x + y  (x y)+ and x y +  (x y)+ for all x, y ∈ M. A nucleus is equivalently an idempotent prenucleus. Proposition 6.2.4. Let S be a bounded complete poset, and let + be a preclosure on S that is bounded above by some closure operation on S. (1) There exists a smallest closure operation +∞ on S that is larger than +, and one ∞ ∞ has S + = Closed(+) and x + = inf{y ∈ Closed(+) : y  x} for all x ∈ S. 0 α+1 α (2) Define x + = x, and define x + = (x + )+ for all successor ordinals α + 1 α β and x + = sup{x + : β < α} for all limit ordinals α, for all x ∈ S. One has +∞ = +α for all sufficiently large ordinals α. (3) If S = M is a bounded complete and residuated ordered magma and + is a prenucleus on M, then the closure operation +∞ as defined in (1) is a nucleus on M. Proof. (1) The set {y ∈ Closed(+) : y  x} is nonempty and bounded by the hypothesis on +. Define x  = inf{y ∈ Closed(+) : y  x} for all x ∈ S. Clearly,  is a preclosure on S with Closed(+) ⊆ Closed(). For the reverse inclusion note that (x + ) = inf{y ∈ Closed(+) : y  x + } = inf{y ∈ Closed(+) : y  x} = x  and therefore x  x +  (x + ) = x  , whence Closed() ⊆ Closed(+). Therefore (x  ) = inf{y ∈ Closed(+) : y  x  } = inf{y ∈ Closed(+) : y  x} = x  , whence  is a closure operation on S. It is then clear that S  = Closed(+) and  is the smallest closure operation on S that is larger than +. (2) This follows readily from statement (1). (3) Let a, x ∈ M. Let z be any element of M such that z  ax and z + = z. Set y = a\z. Then y  x and y + = (a\z)+  a\z + = a\z = y, hence y + = y. Moreover, we have ay  z. Therefore we have

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ax   inf{ay : y  x, y + = y}  inf{z ∈ M : z  ax, z + = z} = (ax) . By symmetry we also have x  a  (xa) , whence  is a nucleus.  Definition 6.2.5. Let S be a poset and  ⊆ Clos(S). Define partial self-maps  of S by and x   = inf{x  :  ∈ },





x   = inf{y ∈ S : y  x and ∀ ∈  (y  = y)}, respectively, for all x ∈ S such that the respective infima exist. Lemma 6.2.6. Let S be a near sup-complete poset (resp., bounded complete poset). Then Clos(S) is a complete lattice (resp., bounded complete poset), and one has the following.  (1) inf Clos(S)  =  and S   ⊇ ∈ S  for all subsets (resp., all nonempty subsets)  of Clos(S).   and S   = ∈ S  for all subsets (resp., bounded subsets) (2) supClos(S)  =  of Clos(S).

 

Proof. (1)

on S, and for all x ∈ S one has (x  ) = inf (x  )    is a preclosure  x = x . Therefore   is a closure operation on S, from which it inf . The given inclusion follows immediately from follows that   = inf the definition of  . 

∈







∈

 

Clos(S)

(2) Define x + = sup{x  :  ∈ } for all x ∈ S, which exists since the x  are bounded above by x supClos(S)  . Clearly, + is a preclosure on S bounded above by supClos(S) , and one has x + = x if and only if x  = x for all  ∈ . There is a closure operation on S by Proposition fore  6.2.4, and it follows easily  = supClos(S) . Finally, that S   = ∈ S  follows from Proposithat tion 6.2.2(1).

 

 The following result generalizes [173, Proposition 3.1.3]. Proposition 6.2.7. Let M be a near sup-complete ordered magma (resp., bounded complete ordered magma). Then Nucl(M) is a complete lattice (resp., bounded complete poset), and one has the following.  (1) inf Nucl(M)  =  and M   ⊇ ∈ M  for all subsets (resp., all nonempty subsets)  of Nucl(M).   and M   = ∈ M  for all subsets (resp., all bounded (2) supNucl(M)  = subsets)  of Nucl(M) if M is a near prequantale (resp., is bounded complete and residuated).

 

6.2 The Poset of All Nuclei

453

Proof. (1) By Lemma 6.2.6, it suffices to show that

  is a nucleus. We have

x y   = x inf y   inf x y   inf (x y) = (x y)  , ∈

∈

∈



and similarly x   y  (x y)  , for all x, y ∈ M, whence  is a nucleus. (2) By Lemma 6.2.6, it suffices to show that  is a nucleus. Since d = ∅ is a nucleus we may assume that  is nonempty. Let + be the preclosure on M defined in the proof of Lemma 6.2.6(2), that is, define x + = sup∈ x  for all x ∈ M. For all x, y ∈ M we have





x y + = x sup y  = sup x y   sup(x y) = (x y)+ , ∈

∈

∈

and similarly x + y  (x y)+ . Therefore  is a nucleus by Proposition 6.2.4.  Remark 6.2.8. If an ordered magma M is a meet semilattice, then Nucl(M) is also a meet semilattice, and statement (1) of Proposition 6.2.7 holds for all nonempty finite subsets  of M. The proof is similar to that of Proposition 6.2.7. If N is a submagma of a near sup-complete ordered magma M, then, since Nucl(M) is complete, for any nucleus  on N we may define ind M () = inf Nucl(M) { ∈ Nucl(M) :  | N = }, ind M () = supNucl(M) { ∈ Nucl(M) :  | N = }, both of which are nuclei on M. (However, the set { ∈ Nucl(M) :  | N = } may be empty.) Proposition 6.2.9. Let M be a near sup-complete ordered magma and  a nucleus on a submagma N of M. (1) Suppose that { ∈ Nucl(M) :  | N = } is nonempty. Then ind M () is the smallest nucleus on M whose restriction to N is equal to . Moreover, if M is a near prequantale, then ind M () is the largest nucleus on M whose restriction to N is equal to , and one has  | N =  if and only if ind M ()    ind M (). (2) If M is a near prequantale and N is a sup-spanning subset of M, then x ind M () = inf{y ∈ M : y  x and ∀z ∈ N (z  y ⇒ z   y)} for all x ∈ M, and ind M ()| N = . Proof. Statement (1) follows easily from Proposition 6.2.7. To prove (2), first we define x + = sup{y  : y ∈ N and y  x} for all x ∈ M. As N is a sup-spanning

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subset of M one has x  x + for all x ∈ M. Since + is order-preserving it follows that + is a preclosure on M. One has y + = y if and only if ∀z ∈ N (z  y ⇒ z   y), whence by Lemma 6.2.4(1), it follows that the operation  M on M defined by x  M = inf{y ∈ M : y  x and ∀z ∈ N (z  y ⇒ z   y)} is a closure operation on M. Now, for all a ∈ N and x ∈ M one has ay + =

sup

z∈N : zy

az  

sup (az) 

z∈N : zy

sup

w∈N : way

w  = (ay)+ .

Since N is a sup-spanning subset of M, it follows that x y +  (x y)+ , and by symmetry x + y  (x y)+ , for all x, y ∈ M. Thus  M is a nucleus by Proposition 6.2.4(3). Now, + is the smallest preclosure on M whose restriction to N is , and by Proposition 6.2.4(1) the operation  M is the smallest closure operation on M that is larger than +, and one has ( M )| N = . It follows that  M is the smallest closure operation on M whose restriction to N is . Therefore, since  M is a nucleus, we must have  M = ind M () by statement (1).  Remark 6.2.10. The set of all prenuclei (resp., all left prenuclei, all right prenuclei) on an ordered magma M is itself an ordered monoid under composition. However, the poset Nucl(M) of all nuclei on M need not be closed under composition. Let 1 and 2 be nuclei on an ordered magma M. Then sup{1 , 2 } exists in Nucl(M) and equals the n-fold composition  = · · · ◦ 2 ◦ 1 ◦ 2 for some positive integer n if and only if  is larger than the n-fold composition · · · ◦ 1 ◦ 2 ◦ 1 . In particular, sup{1 , 2 } = 1 ◦ 2 for two nuclei 1 and 2 if and only if 1 ◦ 2  2 ◦ 1 . The operation ◦ on closure operations, though not necessarily closed, is studied in [164, Section 2] and [181].

6.3 Divisorial Nuclei In this section, we generalize the generalized divisorial closure star and semistar operations to the context of residuated ordered semigroups. Definition 6.3.1. Let a be an element of an ordered magma M. (1) If there exists a largest nucleus  on M such that a  = a, then we denote it by v(a) = v M (a) and call it the divisorial nucleus on M with respect to a. (2) If there exists a largest left nucleus  on M such that a  = a, then we denote it by vl (a) = vl,M (a) and call it the divisorial left nucleus on M with respect to a. (3) If there exists a largest right nucleus  on M such that a  = a, then we denote it by vr (a) = vr,M (a) and call it the divisorial right nucleus on M with respect to a.

6.3 Divisorial Nuclei

455

Example 6.3.2. Let M be an ordered magma. (1) If sup M exists, then v(sup M) exists and equals sup Nucl(M). (2) If also 0 = inf M exists and one has x y = 0 if and only if x = 0 or y = 0 for all x, y ∈ M, then v(0) exists, and x v(0) = 0 if x = 0, and x v(0) = sup M otherwise. The proof of the following result is straightforward. Proposition 6.3.3. Let M be an ordered magma. (1) Let a ∈ M. If v(a) exists, then for all  ∈ Nucl(M), one has a  = a if and only if   v(a). (2) Let a ∈ M. Then v(a) exists if and only if va = sup{ ∈ Nucl(M) : a  = a} exists and a va = a, in which case v(a) = va . (3) If  is a nucleus on M such that v(a) exists for all a ∈ M  , then  = inf{v(a) : a ∈ M  }. The next proposition follows from Propositions 6.3.3 and 6.2.7(2). Proposition 6.3.4. Let Q be a near prequantale. Then v(a) exists and equals sup{ ∈ Nucl(Q) : a  = a} for all a ∈ Q, and one has  = inf{v(a) : a ∈ Q  } for all  ∈ Nucl(Q). The preceding two propositions have obvious analogues for left nuclei and right nuclei. The divisorial nuclei and left and right nuclei defined previously generalize as follows. Definition 6.3.5. Let S be a subset of an ordered magma M. (1) If there exists a largest nucleus  on M such that S ⊆ M  (that is, such that x  = x for all x ∈ S), then we denote it by v(S) = v M (S) and call it the divisorial nucleus on M with respect to S. (2) If there exists a largest left nucleus  on M such that S ⊆ M  , then we denote it by vl (S) = vl,M (S) and call it the divisorial left nucleus on M with respect to S. (3) If there exists a largest right nucleus  on M such that S ⊆ M  , then we denote it by vr (S) = vr,M (S) and call it the divisorial right nucleus on M with respect to S. Note that v(a) = v({a}) if either exists. Proposition 6.3.3 generalizes as follows. Proposition 6.3.6. Let S be a subset of an ordered magma M. (1)  = v(M  ) for any nucleus  on M. (2) Suppose that v(S) exists. Then, for all  ∈ Nucl(M), one has S ⊆ M  if and only if   v(S). (3) The following conditions are equivalent. (a) v(S) exists.

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(b) v S = sup{ ∈ Nucl(M) : S ⊆ M  } exists and S ⊆ M vS . (c) There is a smallest subset C of M containing S such that C = M  for some  ∈ Nucl(M). Moreover, if the conditions above hold, then v S =  = v(S).  (4) Suppose that S = λ Tλ and v(Tλ ) exists for all λ. Then v(S) exists if and only if inf λ v(Tλ ) exists, in which case they are equal. Corollary 6.3.7. Let Q be a near prequantale. Then v(S) exists for any subset S of Q and equals inf{v(a) : a ∈ S}. The proof of the following proposition is also straightforward. Proposition 6.3.8. Let Q be a bounded complete and residuated ordered magma. If v(S) exists for some subset S of Q, then v(T ) exists for any subset T of Q containing S. Again, the preceding results have analogues for left nuclei and right nuclei. Example 6.3.9. Let R be a commutative ring. The ordered monoid Freg (R) of all regular fractional ideals of R is a residuated semiquantale, and in particular is bounded complete and residuated. Thus by Proposition 6.2.7 the poset Nucl(Freg (R)) of all nuclei on Freg (R) is bounded complete. Moreover, v(R) = v = ((−)−1 )−1 exists. It follows that the set Star(R) = Nucl(Freg (R))v of all star operations on R is complete. Next we show that, if M is a residuated ordered semigroup, then vl (a) and vr (a) exist for all a ∈ M, and we derive a formula for both in that case. Lemma 6.3.10. Let M be an ordered semigroup, and let a ∈ M. If for any x ∈ M there is a largest element x  of M such that (1) x  a implies x   a and (2) r x  a implies r x   a for all r ∈ M, then vl (a) exists and equals  : x −→ x  . Proof. First we show that the map  defined in the statement of the lemma is a left nucleus on M. Let x, y, r ∈ M. If x  y, then (1) y  a ⇒ x  a ⇒ x   a and (2) r y  a ⇒ r x  a ⇒ r x   a; therefore x   y  . Clearly x  x  . Moreover, one has (1) x  a ⇒ x   a ⇒ (x  )  a and (2) r x  a ⇒ r x   a ⇒ r (x  )  a, whence (x  )  x  , whence equality holds. Thus  is a closure operation on M. Next, note that (1) x y  a ⇒ x y   a and (2) r x y  a ⇒ r x y   a, whence x y   (x y) . Thus  is a left nucleus on M. Next we observe that a  a ⇒ a   a, whence a  = a. Finally, let  be any left  nucleus on M with a  = a. Then for any x ∈ M we have (1) x  a ⇒ x   a  = a   and (2) r x  a ⇒ r x   a  = a for all r ∈ M, whence x   x  . Thus   . It follows that  is the largest left nucleus on M such that a  = a, so vl (a) exists and equals .  Theorem 6.3.11. Let M be a residuated ordered semigroup, and let a ∈ M.

6.3 Divisorial Nuclei

457

(1) vl (a) exists and is given by x vl (a) = (a/x)\a = sup{y ∈ M : a/x  a/y} for all x ∈ M. Consequently, if M is also bounded complete, then vl (S) exists for any subset S of M and equals inf{vl (a) : a ∈ S}. (2) vr (a) exists and is given by x vr (a) = a/(x\a) = sup{y ∈ M : x\a  y\a} for all x ∈ M. Consequently, if M is also bounded complete, then vr (S) exists for any subset S of M and equals inf{vr (a) : a ∈ S}. (3) If M is complete, or near sup-complete, then v(a) exists and equals the supremum in Nucl(M) of the set { ∈ Nucl(M) :   vl (a),   vr (a)}, and v(S) exists for any subset S of M and equals inf{v(a) : a ∈ S}. Proof. We prove (1), from which (2) follows by duality, from which then (3) follows in a straightforward manner. For any x ∈ M, let x  = (a/x)\a. The identity (a/x)\a = sup{y ∈ M : a/x  a/y} for x ∈ M is easy to check. We claim that x  is the largest element y of M such that r x  a implies r y  a for all r ∈ M. Let x ∈ M. Then r x  a ⇒ r  a/x ⇒ r x   (a/x)((a/x)\a)  a. Suppose that y  is any element of M such that r x  a ⇒ r y   a for all r ∈ M. Then since (a/x)x  a we have (a/x)y   a, whence y   (a/x)\a = x  . This proves our claim. Finally, we note that (a/x)x  a, whence x  (a/x)\a = x  . Therefore the theorem follows from Lemma 6.3.10.  Corollary 6.3.12. Let M be a residuated ordered commutative semigroup. Then v(a) exists for any element a of M and is given by x v(a) = a/(a/x) = sup{y ∈ M : a/x  a/y} for all x ∈ M. Consequently, if M is also bounded complete, then v(S) exists for any subset S of M and equals inf{v(a) : a ∈ S}. Example 6.3.13. Let R be a ring. Since K(R), Freg (R), and I(R) are residuated and bounded complete ordered commutative monoids, we may deduce that the divisorial closure semistar, star, and semiprime operations exist and are described as in Sections 2.4, 4.1, 4.2, and 4.3. For applications to noncommutative rings and algebras, see Corollary 5.4.3 and Proposition 5.7.14. Lemma 6.3.14. Let M be an ordered monoid and a ∈ M. If for any x ∈ M there is a largest element x  of M such that r xs  a implies r x  s  a for all r, s ∈ M, then v(a) exists and equals  : x −→ x  . Proof. First we show that the map  defined in the statement of the lemma is a nucleus on M. Let x, y ∈ M. If x  y, then r ys  a ⇒ r xs  a ⇒ r x  s  a, whence x  

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y  . Likewise r xs  a ⇒ r xs  a, whence x  x  . Moreover, one has r xs  a ⇒ r x  s  a ⇒ r (x  ) s  a, whence (x  )  x  , whence equality holds. Thus  is a closure operation on M. Next, note that r x ys  a ⇒ r x  ys  a ⇒ r x  y  s  a, whence x  y   (x y) . Thus  is a nucleus on M. Next we observe that 1a1  a, which implies 1a  1  a, whence a  = a. Finally,  let  be any nucleus on M with a  = a. Then for any x ∈ M we have r xs  a ⇒    r x s  a = a for all r, s ∈ M, whence x   x  . Thus   . It follows that  is   the largest nucleus on M such that a = a, so v(a) exists and equals . If Q is a unital near quantale, then we can likewise determine an explicit formula for the divisorial nuclei v(a). Proposition 6.3.15. Let Q be a unital near quantale and a ∈ Q. Then one has x v(a) = sup{y ∈ Q : ∀r, s ∈ Q (r xs  a ⇒ r ys  a)} for all x ∈ Q; alternatively, x v(a) is the largest element y of Q such that r xs  a implies r ys  a for all r, s ∈ Q. Proof. Let x  = sup{y ∈ Q : ∀r, s ∈ Q (r xs  a ⇒ r ys  a)} for all x ∈ Q. Since Q is a near quantale, one has y  x  if and only if r xs  a implies r ys  a for all r, s ∈ Q. Therefore by Lemma 6.3.10 one has  = v(a).  The above proposition has a generalization to (possibly nonassociative, noncommutative, nonunital) near prequantales: see Exercise 6.3.8.

6.4 Finitary Nuclei In this section, we study the poset of all nuclei on an ordered magma that satisfy an important finiteness, or continuity, condition called Scott continuity. There are appropriate analogues for left nuclei and right nuclei that we leave for the reader to formulate. Definition 6.4.1. (1) A nonempty subset  of a poset S is directed if every finite subset of  has an upper bound in . (2) A poset S is directed complete, or a dcpo, if each of its directed subsets has a supremum in S. (3) A poset S is a bdcpo if every directed subset of S that is bounded above has a supremum in S. Definition 6.4.2. Let X be a subset of a poset S. (1) X is downward closed (resp., upward closed) if y ∈ X whenever y  x (resp., y  x) for some x ∈ X .

6.4 Finitary Nuclei

459

(2) X is Scott closed if X is a downward closed subset of S and for any directed subset  of X one has sup  ∈ X if sup  exists. (3) X is Scott open if its complement is Scott closed, or equivalently if X is an upward closed subset of S and X ∩  = ∅ for any directed subset  of S with sup  ∈ X . Definition 6.4.3. The Scott open subsets of a poset S form a topology on S called the Scott topology on S. Definition 6.4.4. A map f : S −→ T of posets is Scott continuous if f is continuous when S and T are endowed with their respective Scott topologies. Equivalently, f is Scott continuous if and only if f (sup ) = sup f () for every directed subset  of S for which sup  exists. Clearly for any map f : S −→ T of posets, we have the implications sup-preserving ⇒ near sup-preserving ⇒ Scott-continuous ⇒ order-preserving.

Definition 6.4.5. We say that an ordered magma M is Scott-topological if the multiplication map M × M −→ M is Scott continuous, where M × M is given the obvious pairwise ordering. If the multiplication map M × M −→ M of an ordered magma M is continuous when M × M is endowed with the product topology of the Scott topologies on M, then M is a Scott-topological magma, but the converse seems to be false. Lemma 6.4.6. The following conditions are equivalent for any ordered magma M. (1) The map M × M −→ M of multiplication in M is sup-preserving and inf M is an annihilator of M if inf M exists (resp., multiplication in M is near suppreserving, multiplication in M is Scott continuous). (2) For all a ∈ M, the left and right multiplication by a maps on M are suppreserving (resp., near sup-preserving, Scott continuous). (3) a(sup X ) = sup(a X ) and (sup X )a = sup(Xa) for any a ∈ M and any subset (resp., any nonempty subset, any directed subset) X of M such that sup X exists. (4) sup(X Y ) = sup(X ) sup(Y ) for any subset (resp., any nonempty subset, any directed subset) X and Y of M such that sup X and sup Y exist. (5) M is a prequantale (resp., near prequantale, Scott-topological ordered magma). Corollary 6.4.7. Every semiprequantale is Scott-topological. A closure operation  : S −→ S on a poset S may not be Scott continuous, even though the corestriction S  | : S −→ S  of  to S  is Scott continuous and in fact is sup-preserving, by Lemma 6.1.6(1). (However, the inclusion S  −→ S need not be Scott continuous.) Thus, we make the following definition.

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Definition 6.4.8. If a closure operation  : S −→ S on a poset S is Scott continuous then we say that  is finitary. (Such closure operations are also said to be algebraic.) Equivalently this means that (sup S ) = sup S ( ) (or equivalently sup S ( ) ∈ S  ) for any directed subset  of S such that sup S  exists. Example 6.4.9. A semistar, star, or semiprime operation on a commutative ring is of finite type if and only if it is finitary as a map of posets. Likewise, for any ring R, a semiprime operation on an R-module is of finite type if and only if it is finitary as a map of posets. Definition 6.4.10. For any poset S, let Clos f (S) denote the poset of all finitary closure operations on S, and for any ordered magma M, let Nucl f (M) = Nucl(M) ∩ Clos f (M) denote the poset of all finitary nuclei on M. Next we study properties of the poset Nucl f (M). A summary of these results and the results for Nucl(M) from the previous section is provided in the following table. If an ordered magma M is . . . then

Nucl(M) is . . .

near sup-complete bounded complete meet semilattice Scott-topological dcpo Scott-topological bdcpo Scott-topological, near sup-complete Scott-topological, bounded complete algebraic meet semilattice

complete bounded complete meet semilattice

complete bounded complete meet semilattice

Nucl f (M) is . . .

complete bounded complete complete bounded complete meet semilattice

In particular, if M is a near prequantale (resp., semiprequantale), then the previous table implies that both Nucl(M) and Nucl f (M) are complete (resp., bounded complete). This generalizes the corresponding results for semistar operations (resp., nonunital star operations). Definition 6.4.11. An element x of a poset S is compact if, whenever x  sup  for some directed subset  of S such that sup  exists, one has x  y for some y ∈ . We let K(S) denote the set of all compact elements of S. Example 6.4.12. (1) Any minimal element of a poset is compact. (2) For any set X , the set K(2 X ) of compact elements of the complete lattice 2 X of subsets of X is equal to the set of all finite subsets of X . (3) For any algebra T over a ring R, both possibly noncommutative, the compact elements of the quantale Mod R (T ) of all R-submodules of the left R-module T are precisely the finitely generated R-submodules of T . (4) For any commutative ring R, the compact elements of the ordered monoid K(R) of Kaplansky fractional ideals of R are precisely the finitely generated fractional ideals of R.

6.4 Finitary Nuclei

461

In algebra, unlike in topology, posets tend to be algebraic, in the following sense. Definition 6.4.13. An element x of a poset S is algebraic if x is the supremum of a set of compact elements of S, or equivalently, if one has x = sup{y ∈ K(S) : y  x}. A poset is algebraic if all of its elements are algebraic. An algebraic lattice is a complete lattice that is algebraic. Example 6.4.14. For any set X , the poset 2 X is an algebraic lattice, and the poset 2 X −{∅} is an algebraic near sup-complete poset with K(2 X −{∅}) = K(2 X )−{∅}. For any set X and any set  of self-maps of X , let  denote the submonoid generated by  of the monoid of all self-maps of X under composition. Lemma 6.4.15. Let S be a poset. (1) If S is a dcpo (resp., bdcpo), then Clos f (S) is complete (resp., bounded complete) and one has supClos f (S)  = supClos(S)  and x supClos(S)  = sup{x γ : γ ∈ } for all x ∈ S and for any subset (resp., any bounded subset)  of Clos f (S). (2) If S is an algebraic meet semilattice, then Clos f (S) is a sub meet semilattice of the meet semilattice Clos(S). Proof. (1) Define a preclosure σ of S by x σ = sup{x γ : γ ∈ } for all x ∈ S. The map σ is defined because the set {x γ : γ ∈ } is directed (and bounded above if  is bounded above) for any x ∈ S. Let x ∈ S, and let X = {x γ : γ ∈ }. Then (sup X ) = sup{(x γ ) : γ ∈ }  sup X for all  ∈ , whence (sup X )γ = sup X , and therefore (x σ )γ = x σ , for all γ ∈ . Therefore (x σ )σ = x σ , so σ is a closure operation on S, and clearly σ = supClos(S) . It remains only to show that σ is finitary, since then it will follow that σ = supClos f (S) . Let  be a directed subset of S. Then we have (sup )σ = sup {(sup )γ : γ ∈ } = sup {sup(γ ) : γ ∈ } = sup{x γ : x ∈ , γ ∈ } = sup(σ ). Thus σ is finitary. (2) Let ,  ∈ Clos f (S). By Lemma 6.2.6, it suffices to show that  ∧  ∈ Clos(S) is finitary. Let  be a directed subset of S such that x = sup  exists, and let   z = x ∧ . Note first that z is an upper bound of ∧ . Let u be any upper bound  of ∧ . We claim that u  z. To show this, let t be any compact element of S    with t  z = x  ∧ x  . Then t  x  = sup( ) and t  x  = sup( ), whence  t  y  and t  (y  ) for some y, y  ∈ . Since  is directed, we may choose   w ∈  with y, y   w. Then t  w  ∧ w  = w ∧  u. Therefore, taking the  supremum over all compact t  z, we see that z  u. Therefore (sup )∧ =  z = sup(∧ ). Thus  ∧  is finitary.  Theorem 6.4.16. Let M be an ordered magma. (1) If M is a Scott-topological dcpo magma (resp., Scott-topological bdcpo magma) then Nucl f (M) is complete (resp., bounded complete) and one has supNucl f (M)  = supNucl(M)  and x supNucl(M)  = sup{x γ : γ ∈ } for all x ∈ M and for any subset (resp., any bounded subset)  of Nucl f (M).

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(2) If M is an algebraic meet semilattice, then Nucl f (M) is a sub meet semilattice of the meet semilattice Nucl(M). Proof. By the lemma it suffices to observe that the closure operation x −→ sup{x γ : γ ∈ }, when defined for  ⊆ Clos f (M), is a nucleus if M is Scotttopological.  Example 6.4.17. Let R be a commutative ring. Since Freg (R) is a semimultiplicative lattice, it is Scott-topological and bounded complete, and therefore Nucl f (Freg (R)) is bounded complete and the poset Star f (R) = Nucl f (Freg (R))v of all finite type star operations on R is complete. Its largest element is the t-operation.

6.5 Associated Finitary Nuclei In this section, we generalize to nuclei (and to left nuclei and right nuclei) the association  −→ t of the largest finite type semistar operation t less than or equal to a given semistar operation , and in particular we generalize the t -operation t = v t . We also study the related notion of a precoherent ordered magma. The proof of the following lemma is clear. Lemma 6.5.1. Let S be a join semilattice. Then x ∈ S is compact if and only if, whenever x  sup X for some subset X of S such that sup X exists, one has x  sup Y for some finite subset Y of X . Moreover, K(S) is closed under finite suprema and is therefore directed. Lemma 6.5.2. Let  be a closure operation on a poset S. (1) If x  = x ∨ sup{y  : y ∈ K(S), y  x} for all x ∈ S, then  is finitary. (2) If  is finitary and S is an algebraic join semilattice, then x  = sup{y  : y ∈ K(S), y  x} for all x ∈ S, and one has K(S  ) ⊆ K(S) . (3) If  is finitary and S is a bdcpo, then K(S) ⊆ K(S  ). (4) Suppose that  is finitary. If S is an algebraic bounded complete join semilattice (resp., algebraic near sup-complete poset, algebraic lattice), then so is S  , and K(S  ) = K(S) . Proof. (1) Let  be a directed subset of S such that x = sup  exists. We show that sup( ) exists and equals x  . Clearly, x  is an upper bound of  . If w is any upper bound of  , then we claim that x   w. Let y ∈ K(S) with y  x. Then y  z for some z ∈ . Therefore y   z  , where z  ∈  . Thus we have y   w. Taking the supremum over all such y, we see that sup{y  : y ∈ K(S), y  x}  w. Moreover, we have x = sup   w, and therefore x  = x ∨ sup{y  : y ∈ K(S), y  x}  w, as claimed.

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(2) Let x ∈ S. One has x = sup , where  = {y ∈ K(S) : y  x} is directed by Lemma 6.5.1. Therefore x  = sup( ). Suppose that x ∈ K(S  ). Then x = sup( ) and  ⊆ S  is directed, so x  y  for some y ∈ , whence x = y  . Therefore K(S  ) ⊆ K(S) . (3) Let x ∈ K(S). Suppose that x   sup S   for some directed subset  of S  such that sup S   exists. Then  is directed and bounded above, whence sup S  exists. It follows that sup S   = (sup S ) = sup S ( ) = sup S . Therefore x  x   sup S , so x  y for some y ∈ , whence x   y  = y. Thus we have x  ∈ K(S  ). (4) Since K(S) ⊆ K(S  ) by (3), it follows from (2) that S  is algebraic and K(S) = K(S  ). Since S  is bounded complete (resp., near sup-complete, complete) by Lemma 6.1.6(4), it follows that S  is an algebraic bounded complete join semilattice (resp., algebraic near sup-complete poset, algebraic lattice).  Definition 6.5.3. For any closure operation  on a bounded complete poset S, let t denote the operation on S defined by x t = sup{y  : y ∈ K(S) and y  x} for all x ∈ S. Proposition 6.5.4. Let  be a closure operation on an algebraic bounded complete join semilattice S. Then t is the largest finitary closure operation on S that is smaller than , one has x t = x  for all x ∈ K(S), and K(S t ) = K(S)t . Proof. Clearly, t is a preclosure on S. Let x ∈ S. The set {y  : y ∈ K(S), y  x} is directed by Lemma 6.5.1. Therefore if z ∈ K(S) and z  x t then z  y  for some y ∈ K(S) such that y  x, in which case z   y  . Therefore (x t )t = sup{z  : z ∈ K(S), z  x t }  sup{y  : y ∈ K(S), y  x} = x t . Thus  is a closure operation on S, and the rest of the proposition follows from Lemma 6.5.2.  The following definition generalizes [173, Definition 4.1.1]. Definition 6.5.5. An ordered magma M is precoherent if M is algebraic and K(M) is closed under multiplication. An ordered magma M is coherent if M is precoherent and unital with 1 compact. Example 6.5.6. (1) For any magma M, the prequantale 2 M is precoherent, and if M is unital then 2 M is coherent. (2) A K-lattice is a precoherent multiplicative lattice. For example, if M is a commutative monoid, then 2 M is a K-lattice. (3) Let T be an algebra over a ring R, both possibly noncommutative. The unital quantale Mod R (T ) of all R-submodules of the left R-module T is coherent. In particular, if R and T are commutative, then Mod R (T ) is a K-lattice. (4) For any commutative ring R, the ordered monoid Freg (R) of all regular fractional ideals of R is a coherent unital semimultiplicative lattice.

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Theorem 6.5.7. If  is a nucleus on a precoherent semiprequantale (resp., precoherent near prequantale, precoherent prequantale) Q, then the operation t defined by x t = sup{y  : y ∈ K(Q) and y  x} for all x ∈ Q is the largest finitary nucleus on Q that is smaller than , the ordered magma Q t is also a precoherent semiprequantale (resp., precoherent near prequantale, precoherent prequantale), and K(Q t ) = K(Q)t . Proof. By Proposition 6.5.4, to prove the first claim we need only show that t is a nucleus. Let x ∈ Q and a ∈ K(Q). We have (ax)t = sup{y  : y ∈ K(Q), y  ax} and ax t = sup{az  : z ∈ K(Q), z  x}. Let z ∈ K(Q) with z  x, and let y = az. By our hypotheses on Q we have y ∈ K(Q), and y = az  ax. Therefore az   y   (ax)t . Taking the supremum over all such z ∈ K(Q) we see that ax t  (ax)t . Then, since Q is precoherent, so that one has y = sup{a ∈ K(Q) : a  y} for all y ∈ Q, a straightforward argument shows that yx t  (yx)t for all y. By symmetry, then, one also has y t x  (yx)t for all x, y ∈ Q. It follows, then, that t is a nucleus on Q. To prove the second claim we may assume  = t is finitary. Then Q  is a an algebraic semiprequantale (resp., algebraic near prequantale, algebraic prequantale) by Lemma 6.5.2(3) and Proposition 6.1.25. Let x, y ∈ K(Q  ), and suppose that x  y  sup Q   for some directed subset  of Q  such that sup Q   exists. Then sup Q  exists, and x y  sup Q   = (sup Q ) = sup Q . Therefore, since x y ∈ K(Q), we have x y  z, whence x  y  z, for some z ∈ . Thus x  y ∈ K(Q  ) and  Q  is precoherent. Finally, the third claim follows from Proposition 6.5.4. We also note the following. Theorem 6.5.8. If  is a left nucleus (resp., right nucleus) on a precoherent semiprequantale Q, then the operation t defined by x t = sup{y  : y ∈ K(Q) and y  x} for all x ∈ Q is the largest finitary left nucleus (resp., right nucleus) on Q that is smaller than . Theorem 6.5.9. Let  be a left nucleus on a precoherent residuated semiquantale Q. The nucleus tl (a) = vl (a)t for any a ∈ Q is the largest finitary left nucleus  on Q such that a  = a, and one has t = inf{tl (a) : a ∈ Q t }. More generally, the nucleus tl (S) = vl (S)t for any subset S of Q is the largest finitary left nucleus  on Q such that S ⊆ Q  , and one has t = tl (Q t ). Theorem 6.5.10. Let  be a nucleus on a precoherent residuated semimultiplicative lattice or precoherent near prequantale Q. Then the nucleus t (a) = v(a)t for any a ∈ Q is the largest finitary nucleus  on Q such that a  = a, and one has t = inf{t (a) : a ∈ Q t }. More generally, the nucleus t (S) = v(S)t for any subset S of Q is the largest finitary nucleus  on Q such that S ⊆ Q  , and one has t = t (Q t ). In particular, since K(R), Freg (R), and I(R) for any commutative ring R are coherent residuated semimultiplicative lattices, the theorem above generalizes the corresponding results for semistar, star, and semiprime operations. In the next section, we also generalize the stable semistar (resp., star) operation associated to a given

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semistar (resp., star) operation, assuming only one additional hypothesis, namely, that the given semimultiplicative lattice is also a meet semilattice and therefore a lattice.

6.6 Stable Nuclei Let R be a commutative ring with total quotient ring K . Recall from Section 4.2 that a semistar operation  on R is stable if the following two conditions hold. (1) (I ∩ J ) = I  ∩ J  for all I, J ∈ K(R). (2) (I : K J ) = (I  : K J ) for all I, J ∈ K(R) with J finitely generated. Also recall that, for any semistar operation  on a commutative ring R, there exists a largest stable semistar operation  on R that is less than or equal to , given by I =



{(I : K J ) : J  = R}

for all I ∈ K(R). In this section we generalize this result to any coherent residuated lattice-ordered semiquantale. In particular, the results also apply to star and semiprime operations. Definition 6.6.1. An order-preserving self-map  of an ordered magma M is left stable (resp., right stable) if the following two conditions hold. (1) (inf X ) = inf(X  ) for all finite subsets X of M such that inf X exists. (2) (t\x) = t\x  for all x, t ∈ M with t compact such that t\x exists (resp., (x/t) = x  /t for all x, t ∈ M with t compact such that x/t exists). Moreover,  is stable if  is both left stable and right stable. In general, condition (1) of Definition 6.6.1 does not necessarily imply condition (2). However, one has the following proposition below. For any magma M, let U(M) denote the set of all u ∈ M for which there exists u −1 ∈ M such that the left and right multiplication by u −1 maps are inverses, respectively, to the left and right multiplication by u maps. Proposition 6.6.2. Let M be an ordered magma such that sup(X Y ) = sup(X ) sup(Y ) for all nonempty finite subsets X and Y of M and every compact element of M is the supremum of a finite subset of U(M). A left nucleus  on M is left stable (resp., a right nucleus  on M is right stable) if and only if (inf X ) = inf(X  ) for any finite subset X of M such that inf X exists. Moreover, if M is also a meet semilattice, then x/t and t\x exist in M for all x, t ∈ M with t compact. Proof. Let  be a right nucleus on M such that (inf X ) = inf(X  ) for any finite subset X of M such that inf X exists. Let x, t ∈ M with t compact, and suppose that x/t exists. We may write t = u 1 ∨ u 2 ∨ · · · ∨ u n with each u i ∈ U(M). Since

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x/u exists and equals xu −1 , it straightforward to check that x/u 1 ∧ x/u 2 ∧ · · · ∧ x/u n exists and equals x/t. Moreover, since (x/u) = (xu −1 ) = x  u −1 = x  /u for all u ∈ U(M), one has (x/t) = (x/u 1 ) ∧ · · · ∧ (x/u n ) = x  /u 1 ∧ · · · ∧ x  /u n = x  /t. Therefore  is right stable. Also, if M is a meet semilattice and x, t ∈ M with t compact, then, writing t = u 1 ∨ u 2 ∨ · · · ∨ u n with each u i ∈ U(M), one easily  verifies that x/t = x/u 1 ∧ x/u 2 ∧ · · · ∧ x/u n exists, and likewise for t\x. Example 6.6.3. Let R be ring. Then U(Freg (R)) = Inv(R), and every compact element of Freg (R) is the supremum of a finite subset of U(Freg (R)) if and only if R is quasi-Marot. Consequently, as an application of Proposition 6.6.2 one obtains Proposition 2.7.14. Definition 6.6.4. Let M be a unital ordered magma with identity 1, and let  ∈ Nucl(M). We say that z ∈ M is a -dense ideal if z  1 and z  = 1 . We let -DI(M) denote the set of all -dense ideals of M, which is a submagma of M. If X ⊆ -DI(M) is nonempty, then sup X ∈ -DI(M) if sup X exists, and inf X ∈ -DI(M) if inf X exists and X is finite. Definition 6.6.5. Let Q be a unital semiquantale. If  is a left nucleus on Q, then we let x  = sup{x/z : z ∈ -DI(Q)} for all x ∈ Q, which is well-defined since x/z  x  exists for all z ∈ -DI(Q). Indeed, x z  x, and yz  x implies y  y1 = yz   x  , so {y ∈ M : yz  x} is nonempty and bounded above, whence x/z = sup{y ∈ M : yz  x} exists. Lemma 6.6.6. Let  be a left nucleus on a unital semiquantale Q. (1)  is a preclosure on Q that is less than or equal to , and one has x y   (x y) for all x, y ∈ Q. (2) For any x, t ∈ Q with t compact one has t  x  if and only if t z  x for some z ∈ -DI(Q). (3) If Q is algebraic, then (inf X ) = inf(X  ) for any finite subset X of Q such that inf X exists. (4) If Q is precoherent, then  is a left stable left nucleus on Q. Proof. (1) One has x   sup{x  /z : z ∈ -DI(Q)} = sup{x  /z  : z ∈ -DI(Q)} = x  for all x ∈ Q, whence   , and the rest of statement (1) is equally trivial to verify. (2) Let x, t ∈ Q with t compact. If t z  x for some z ∈ -DI(Q), then t  x/z  x  . Conversely, if t  x  , then we have t  x/z 1 ∨ x/z 2 ∨ · · · ∨ x/z n  x/(z 1 z 2 · · · z n ) for some z 1 , z 2 , . . . , z n ∈ -DI(Q), whence t z  x, where z = z 1 z 2 · · · z n ∈ -DI(Q). (3) Let X = {x1 , x2 , . . . , xn } be a finite subset of Q such that a = inf X exists. We must show that a  = inf(X  ). Clearly a   x  for all x ∈ X . Let b be any lower

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467

bound of X  . If t is any compact element of Q such that t  b, then, by statement (2), for each i there exists z i ∈ -DI(Q) such that t z i  xi , whence t z  inf X = a, where z = z 1 z 2 · · · z n ∈ -DI(Q). Thus t  a  . Taking the supremum over all such t we see that b = sup{t ∈ K(Q) : t  b}  a  . Therefore inf(X  ) exists and equals a  . (4) We first show that  is idempotent and therefore a left nucleus on Q. Let x ∈ Q. Let t be any compact element such that t  (x  ) . Then t z  x  for some z ∈ -DI(Q). If u is a compact element of Q such that u  z, then tu  x  and tu is compact, whence tuz u  x for some z u ∈ -DI(Q). Let z  = sup{uz u : u ∈ K(Q) and u  z}. Then z   1 and (z  ) = (sup{uz u : u ∈ K(Q) and u  z}) = z  = 1 , whence z  ∈ -DI(Q). Moreover, one has t z  = sup{tuz u : u ∈ K(Q) and u  z}  x. It follows that t  x  . Taking the supremum over all t, we see that (x  )  x  . Thus  is a nucleus on Q. To show that  is left stable, let x, t be elements of Q with t compact such that t\x exists. Clearly t\x  also exists and (t\x)  t\x  . Let u ∈ K(Q) with u  t\x  . Then tu  x  and tu is compact, whence tuz  x for some z ∈ -DI(Q). Therefore uz  t\x, whence u  (t\x) . Taking the supremum over all u we see that t\x   (t\x) , whence equality holds. Combining this with statement (3), we see that  is left stable.  Theorem 6.6.7. Let Q be a precoherent unital residuated semiquantale such that x ∧ 1 exists for all x ∈ Q (which holds, for example, if Q is a precoherent unital residuated lattice-ordered semiquantale, hence also if Q is a precoherent unital quantale). Let  be a left nucleus on Q. For any x ∈ Q let x  = sup{x/z : z ∈ -DI(Q)}. Then  is the largest left stable left nucleus on Q that is less than or equal to . Moreover, the following conditions on  are equivalent. (1) (2) (3) (4)

 is left stable. (x ∧ 1) = x  ∧ 1 and (t\x) = t\x  for all x, t ∈ Q with t compact. (t\x ∧ 1) = t\x  ∧ 1 for all x, t ∈ Q with t compact.  = .

Proof. We first show that the four statements of the proposition are equivalent. Clearly, we have (1) ⇒ (2) ⇒ (3), and by Lemma 6.6.6(4) we have (4) ⇒ (1). To show that (3) implies (4), we suppose that (3) holds, and then we need only show that x   x  for any x ∈ Q. Let t be any compact element of Q with t  x  . By the hypothesis on Q the element z = t\x ∧ 1 exists in Q. Condition (3) then implies that z  = t\x  ∧ 1 . Note then that, since t1  x  1  x  one has 1  t\x  , whence z  = t\x  ∧ 1 = 1 . Since also z  1 we have z ∈ -DI(Q). Therefore, since t z  t (t\x)  x, we have t  x  . Since this holds for all compact t  x  , we have x   x  , as desired. It remains only to show that  is the largest left stable left nucleus on Q that is less than or equal to . But by Lemma 6.6.6,  is itself a left stable left nucleus that is less than or equal to , and if  is any other such left nucleus,  then one has  =   , whence  is larger than  . Definition 6.6.8. Let w = t for any left nucleus  on a precoherent unital semiquantale Q.

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Theorem 6.6.9. Let  be a left nucleus on a coherent unital residuated semiquantale Q. (1) One has x w = sup{x/z : z ∈ -DI(Q) ∩ K(Q)} for all x ∈ Q, and w is finitary. In particular, if  is finitary, then so is  = w . (2) If x ∧ 1 exists for all x ∈ Q (for example, if Q is lattice-ordered or a quantale), then w is the largest finitary left stable left nucleus on Q that is less than or equal to . Proof. Statement (2) follows from statement (1) and Theorems 6.6.7 and 6.5.7. To prove (1), let x ∈ Q. One has x w = sup{x/y : y  1 and y t = 1t }. Let t be any compact element of Q with t  x w . Then t  x/y for some y ∈ Q with y  1 and y t = 1t . Since 1 is compact, the condition y t = 1t implies z  = 1 for some compact z  y. Note then that t  x/z since t z  t y  x. Therefore, since z ∈ -DI(Q) ∩ K(Q), one has t  sup{x/z : z ∈ -DI(Q) ∩ K(Q)}. Since this holds for all compact t  x w , it follows that x w  sup{x/z : z ∈ -DI(Q) ∩ K(Q)}, and therefore equality holds since the reverse inequality is obvious. Suppose now that  = t is finitary. To show that  is also finitary, we let x ∈ Q and verify that x  = sup{y  : y ∈ K(Q) and y  x}. Let t be any compact element of Q with t  x  . Since w = , one has x  = sup{x/z : z ∈ -DI(Q) ∩ K(Q)}. Since t  x  is compact and the set {x/z : z ∈ -DI(Q) ∩ K(Q)} is directed, it follows that t  x/z for some z ∈ -DI(Q) ∩ K(Q). Therefore t z  x = sup{y ∈ K(Q) : y  x}, whence t z  y for some compact y  x since t z is compact. Thus we have t  y/z, where z ∈ -DI(Q), whence t  y  . It follows that x   sup{y  : y ∈ K(Q) and y  x}, and the desired equality follows.  Example 6.6.10. If R is a commutative ring, then the ordered monoids K(R), Freg (R), and I(R) are coherent unital residuated lattice-ordered semiquantales, and therefore Theorems 6.6.7 and 6.6.9 yield several of our results on semistar, star, and semiprime operations from Chapters 2 and 4, namely, Propositions 2.4.23, 2.4.24, 2.4.40, 4.1.11, 4.2.8, 4.2.9 and 4.3.12. The following result follows from Theorems 6.5.10, 6.6.7, and 6.6.9. Theorem 6.6.11. Let Q be a coherent unital residuated lattice-ordered semiquantale, and let S ⊆ Q. (1) vl (S) exists and is the largest left nucleus  on Q such that S ⊆ Q  . (2) vl (S) = vl (S) = inf{vl (a) : a ∈ S} exists and is the largest left stable left nucleus  on Q such that S ⊆ Q  . (3) wl (S) = vl (S)w = tl (S) exists and is the largest finitary left stable left nucleus  on Q such that S ⊆ Q  . Finally, we note the following. Proposition 6.6.12. Let Q be a precoherent unital residuated lattice-ordered semiquantale. Then inf  is a left stable left nucleus on Q for any set  of left stable left nuclei on Q. Consequently, a left nucleus  on Q is left stable if and only if  = inf{vl (a) : a ∈ S} for some subset S of Q.

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Exercises Section 6.1 1. Prove that a poset S is complete (resp., near sup-complete, bounded complete) if and only if inf S X exists for every subset (resp., every bounded below subset, every nonempty bounded below subset) X of S. 2. Prove that a closure operation on a poset S is equivalently a self-map  of S such that x  y  if and only if x   y  for all x, y ∈ S. 3. Prove Lemma 6.1.6. 4. Prove Proposition 6.1.11. 5. Verify Example 6.1.21. 6. Prove Proposition 6.1.15. 7. Verify (1), (3), and (5) of Example 6.1.19. 8. Verify (6) of Example 6.1.19. 9. Verify (2) of Example 6.1.19. 10. Prove Proposition 6.1.22. 11. Verify Remark 6.1.23. 12. Prove statement (1) of Proposition 6.1.26. 13. Prove Corollary 6.1.27. sup X exists

sup(X Y ) = sup(X ) sup(Y )

ps

for all X for all X = ∅ for all directed X for all X = ∅ bounded above for X = M for X = ∅ for all X for all X = ∅ for all finite or bounded X = ∅ for all finite X

for X = ∅ or Y = ∅ for all X, Y for all X, Y = ∅ for all (finite or) bounded X, Y = ∅ for all finite X, Y

ms

for all finite X = ∅

for all finite X, Y = ∅

ordered magma M

abbr.

ordered magma sup-magma near sup-magma dcpo magma bounded complete

om s ns d bc

bounded above with annihilator prequantale near prequantale semiprequantale

b a p np sp

prequantic semilattice multiplicative semilattice Scott-topological

t

residuated

r

near residuated

nr

if ∃x, y : X = {z : zy  x} or X = {z : yz  x} if X = ∅ and ∃x, y : X = {z : zy  x} or X = {z : yz  x}

for all directed X, Y such that sup X , sup Y exist for all X, Y such that sup X , sup Y exist for all X, Y = ∅ such that sup X , sup Y exist

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14. Equivalent characterizations of various classes of ordered magmas are given in the table above. Show that the three lattices in Figure 6.1 are each full lattices of implications. 15. Let M be an ordered magma. Prove the following. a) If M is unital and residuated, or more generally if M is left or right unital and for all a ∈ M the left and right multiplication by a maps on M are sup-preserving, then is a sup-spanning subset of M if and only if every element of M is the supremum of some subset of . b) Let  be a closure operation on an ordered magma M and let be any supspanning subset of M. Then  is a nucleus on M if and only if ax   (ax) and x  a  (xa) for all a ∈ and all x ∈ M, and in that case  is a sup-spanning subset of M  . c) The set Prin(D) of all nonzero principal fractional ideals of an integral domain D is a sup-spanning subset of the residuated near multiplicative lattice Freg (D). d) A nucleus on Freg (D) is equivalently a closure operation ∗ on Freg (D) such that (a I )∗ = a I ∗ for all I ∈ Freg (D) and all nonzero a ∈ T (D). 16. For any self-map  of a magma M, we say that a ∈ M is transportable through  if (ax) = ax  and (xa) = x  a for all x ∈ M. We let T (M) denote the set of all elements of M that are transportable through . a) Show that any closure operation  on an ordered magma M such that T (M) is a sup-spanning subset of M is a nucleus on M. b) For any magma M we let U(M) denote the set of all u ∈ M for which there exists u −1 ∈ M such that the left and right multiplication by u −1 maps are inverses, respectively, to the left and right multiplication by u maps. Show that U(M) ⊆ T (M) for any nucleus  on an ordered magma M. 17. A weak ideal system on a commutative monoid M with annihilator 0 is a closure operation r on the multiplicative lattice 2 M such that 0 ∈ ∅r , cM ⊆ {c}r , and cX r ⊆ (cX )r for all c ∈ M and all X ∈ 2 M [95]. A weak ideal system on M is said to be an ideal system on M if (cX )r = cX r for all such c and X . Let M be a commutative monoid with annihilator 0. Prove the following. a) A weak ideal system on M is equivalently a nucleus r on the multiplicative lattice 2 M such that {0}r = ∅r and {1}r = M. b) A weak ideal system r on M is an ideal system on M if and only if every singleton in 2 M is transportable through r , in the sense of the previous exercise. c) Let R be a commutative ring. A semiprime operation may be seen equivalently as a weak ideal system on the commutative monoid R • of R under multiplication. d) (∗∗) Generalize the notion of a weak ideal system to accommodate the left semiprime operations on a (possibly noncommutative) ring. 18. For any magma M, let M0 denote the magma M {0}, where 0x = 0 = x0 for all x ∈ M0 is an annihilator of M0 . Let G be an abelian group. A module system on G is a closure operation r on the multiplicative lattice 2G 0 such that ∅r = {0} and (cX )r = cX r for all c ∈ G 0 and all X ∈ 2G 0 [96]. Prove the following.

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471

Fig. 6.1 Three full lattices of implications

a) A module system on G is equivalently a nucleus r on the multiplicative lattice 2G 0 such that ∅r = {0}. b) The following are equivalent for any self-map r of 2G 0 such that ∅r = {0}. 1) r is a module system on G. 2) r is a closure operation on the poset 2G 0 and r -multiplication on 2G 0 is associative. 3) r is a closure operation on the poset 2G 0 and (X r Y r )r = (X Y )r for all X, Y ∈ 2G 0 . 4) X Y ⊆ Z r if and only if X Y r ⊆ Z r for all X, Y, Z ⊆ G 0 . c) 2G 0 −2G is a sub multiplicative lattice of 2G 0 , and a module system on G may be seen equivalently as a nucleus on the multiplicative lattice 2G 0 −2G .

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d) Let D be an integral domain, K × the group of nonzero elements of the quotient field K of D, and  a self-map of K(D). Let ∅r = {0}r = {0} and X r = (D X ) for all subsets X of K containing a nonzero element. Then  is a semistar operation on D if and only if r is a module system on the abelian group K × . e) (∗∗) Generalize the notion of a module system in order to accommodate semistar operations on commutative rings. 19. A morphism f : N −→ M of complete ordered magmas is a sup-preserving map from N to M such that f (ab) = f (a) f (b) for all a, b ∈ N . Let f : Q → M be a morphism of complete ordered magmas, where Q is a prequantale. Prove the following. a) If  is any nucleus on Q, then the corestriction Q  | : Q −→ Q  of  to Q  is a surjective morphism of prequantales. b) There exists a unique nucleus  on Q such that f = ( f | Q  ) ◦ ( Q  |) and f | Q  is injective; moreover, one has x  = sup{y ∈ Q : f (y) = f (x)} for all x ∈ Q. c) f | Q  : Q  −→ M is an embedding of complete ordered magmas. d) f | Q  : Q  −→ im f is an isomorphism of ordered magmas. e) A quantale Q is said to be simple if every nontrivial sup-preserving semigroup homomorphism from Q is injective. A quantale Q is simple if and only if d : x −→ x and e : x −→ sup Q are the only nuclei on Q. f) A multiplicative lattice Q is simple if and only if Q = {0, 1}, where 1 = sup Q and 0 is the identity element of Q and 0 = inf Q is an annihilator of Q (and possibly 0 = 1). 20. State and prove a generalization of the previous exercise for a (near suppreserving) morphism from a near prequantale to a near sup-complete ordered magma. Section 6.2 1. 2. 3. 4. 5. 6. 7.

Fill in the missing details in the proof of Proposition 6.2.2. Prove statement (2) of Proposition 6.2.4. Verify Remark 6.2.8. Prove statement (1) of Proposition 6.2.9. Use Proposition 6.2.9 to prove statements (2) and (3) of Proposition 4.1.20. Verify Remark 6.2.10. State and prove analogues of all the results of this section for the poset Nucll (M) of all left nuclei on an ordered magma M. 8. a) Let  be a closure operation on a residuated semiquantale Q. For all x ∈ Q, let x  = inf{x, inf{y ∈ Q : y  x and ∀z, w ∈ Q (zw  y ⇒ z  w   y)}}. Show that  : x −→ x  is the smallest nucleus on Q larger than . Also show that, for all x ∈ Q, one has x  = x if and only if zw  x implies z  w   x for all z, w ∈ Q. b) State and prove an analogous result for left nuclei.

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Section 6.3 1. 2. 3. 4. 5. 6. 7.

8.

Prove Proposition 6.3.3. Prove Proposition 6.3.4. Prove Proposition 6.3.6 and Corollary 6.3.7. Let  be a closure operation on a near prequantale Q. Show that v(Q  ) is the largest nucleus on Q smaller than . Prove Proposition 6.3.8. Prove statement (3) of Theorem 6.3.11. For any ordered magma M, let U(M) denote the set of all u ∈ M such that the left and right multiplication by u maps are poset automorphisms of M. We say that a U-lattice (resp., near U-lattice, semi-U-lattice) is an ordered magma M that is complete (resp., near sup-complete, a bounded complete join semilattice) and such that U(M) is a sup-spanning subset of M. Prove the following. a) For any commutative ring R, one has U(K(R)) = U(Freg (R)) = Inv(R). b) For any commutative ring R, the ordered monoid Freg (R) is a semi-U-lattice if and only if R is quasi-Marot. c) For any integral domain D, the multiplicative lattice K(D) is a U-lattice. d) Any U-lattice (resp., near U-lattice, semi-U-lattice) is a prequantale (resp., near prequantale, semiprequantale). e) Let  be a nucleus on an ordered magma M. If M is a U-lattice (resp., near U-lattice, semi-U-lattice), then so is M  . f) Let Q be an associative unital semi-U-lattice, and let a ∈ Q. Then v(a) exists Q. and x v(a) = inf{uav : u, v ∈ U(Q) and x  uav} for all x ∈  g) Let D be an integral domain, and let J ∈ K(D). Then I v (J ) = {H J : H ∈ Inv(D) and I ⊆ H J } for all I ∈ K(D).  h) Let R be a quasi-Marot ring, and let J ∈ Freg (R). Then I v (J ) = {H J : H ∈ Inv(R) and I ⊆ H J } for all I ∈ Freg (R). Let M be any magma. For any r ∈ M, define self-maps L r and Rr of M by L r (x) = r x and Rr (x) = xr for all x ∈ M, which we call translations. Let Lin(M) denote the submonoid of the monoid of all self-maps of M generated by the translations. Prove the following. a) If M is a monoid, then any element of Lin(M) can be written in the form L r ◦ Rs = Rs ◦ L r for some r, s ∈ M. b) Let Q be a near prequantale and a ∈ Q. One has x v(a) = sup{y ∈ Q : ∀ f ∈ Lin(Q) ( f (x)  a ⇒ f (y)  a)} for all x ∈ Q. Alternatively, x v(a) for all x ∈ Q is the largest element of Q such that f (x)  a implies f (x v(a) )  a for all f ∈ Lin(Q).

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Section 6.4 1. Verify the claims made in Definitions 6.4.3 and 6.4.4. 2. a) Prove Lemma 6.4.6 and Corollary 6.4.7. b) Show that a semistar, star, or semiprime operation is of finite type if and only if it is finitary as a map of posets. 3. Given an example of an infinite poset all of whose elements are compact and an example of an infinite poset having no compact elements. 4. Verify Example 6.4.12. 5. Consider the following properties of maps between posets. 1) sup-preserving. 2) near sup-preserving. 3) Scott continuous. 4) order-preserving. a) Show that (1) ⇒ (2) ⇒ (3) ⇒ (4). b) Which of the four properties are preserved under composition of functions? Prove your answers correct. 6. A subset I of a poset S is said to be an ideal of S if I is a directed downward closed subset of S. For any x ∈ S, the set ↓x = {y ∈ S : y  x} is an ideal of S called the principal ideal generated by x. The ideal completion Idl(S) of S is the set of all ideals of S partially ordered by the subset relation. If S is a join semilattice, then, for any nonempty subset X of S, we let ↓X = {y ∈ S : y  sup T for some finite nonempty T ⊆ X }. Prove the following. a) Let S be a join semilattice. i) For any nonempty subset X of S, the set ↓X is the smallest ideal of S containing X . ii) The operation ↓ is a finitary closure operation on the algebraic near suplattice 2 S −{∅} with im↓ = Idl(S). iii) If S has a least element, then the operation ↓ extends uniquely to a finitary closure operation on the algebraic sup-lattice 2 S such that ↓∅ = inf S. b) Let L be an algebraic near sup-complete poset and S a join semilattice. i) Idl(S) is an algebraic near sup-complete poset. ii) K(L) is a sub join semilattice of L. iii) The map S −→ K(Idl(S)) acting by x −→↓x is a poset isomorphism. iv) The map Idl(K(L)) −→ L acting by I −→ sup I is a poset isomorphism with inverse acting by x −→ (↓x) ∩ K(L). 7. A multiplicative semilattice is an ordered magma M such that M is a join semilattice and a sup{x, y} = sup{ax, ay} and sup{x, y}a = sup{xa, ya} for all

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a, x, y ∈ M [21, Section XIV.4]. Show that the following conditions are equivalent for any ordered magma M. 1) M is a multiplicative semilattice with annihilator (resp., a multiplicative semilattice). 2) a(sup X ) = sup(a X ) and (sup X )a = sup(Xa) for any a ∈ M and any finite subset (resp., any finite nonempty subset) X of M. 3) The map K(2 M ) −→ M (resp., K(2 M )−{∅} −→ M) acting by X −→ sup X is a well-defined magma homomorphism. 4) sup(X Y ) = sup(X ) sup(Y ) for all finite subsets (resp., all finite nonempty subsets) X and Y of M. 8. Using Exercises 6 and 7, prove the following. a) The multiplicative semilattices with annihilator (resp., multiplicative semilattices) form a category, where a morphism is a magma homomorphism f : M −→ M  , with M and M  multiplicative semilattices with annihilator (resp., multiplicative semilattices), such that f (sup X ) = sup f (X ) for all finite subsets (resp., all finite nonempty subsets) X of M. b) The precoherent prequantales (resp., precoherent near prequantales) form a category, where a morphism is magma homomorphism f : Q −→ Q  , with Q and Q  precoherent prequantales (resp., precoherent near prequantales), such that f (K(Q)) ⊆ K(Q  ). c) Let M be a multiplicative semilattice. i) The operation ↓: X −→↓X is a finitary nucleus on the precoherent near prequantale 2 M −{∅}, one has Idl(M) =↓(2 M −{∅}), and Idl(M) is a precoherent near prequantale under ↓-multiplication. ii) Suppose that M is with annihilator. The operation ↓: X −→↓X is a finitary nucleus on the precoherent prequantale 2 M , one has Idl(M) =↓ (2 M ), and Idl(M) is a precoherent prequantale under ↓-multiplication. d) Let Q be a precoherent prequantale and let M be a multiplicative semilattice with annihilator. i) Idl(M) is a precoherent prequantale under ↓-multiplication. ii) K(Q) is a multiplicative semilattice with annihilator. iii) The map M −→ K(Idl(M)) acting by x −→↓x is an isomorphism of ordered magmas. iv) The map Idl(K(Q)) −→ Q acting by I −→ sup I is an isomorphism of ordered magmas. v) If f : Q −→ Q  is a morphism of precoherent prequantales, then the map K( f ) : K(Q) −→ K(Q  ) given by K( f )(x) = f (x) for all x ∈ K(Q) is a morphism of multiplicative semilattices with annihilator. vi) If g : M −→ M  is a morphism of multiplicative semilattices with annihilator, then the map Idl(g) : Idl(M) −→ Idl(M  ) given by Idl(g)(I ) =↓ (g(I )) for all I ∈ Idl(M) is a morphism of precoherent prequantales. vii) The associations K and Idl are functorial and provide an equivalence of categories between the category of precoherent prequantales and the category of multiplicative semilattices with annihilator.

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e) Generalize part (d) to show that there is an equivalence of categories between the category of precoherent near prequantales and the category of multiplicative semilattices. Section 6.5 1. Prove Lemma 6.5.1. 2. Verify Example 6.5.6. 3. a) Give an example of a multiplicative lattice that is not precoherent. b) Give an example of a precoherent multiplicative lattice that is not coherent. 4. Prove Theorem 6.5.8. 5. Prove Theorem 6.5.9. 6. Prove Theorem 6.5.10. 7. a) Let  and  be closure operations on an algebraic bounded complete lattice S. Show that inf{t , t } = (inf{,  })t . b) Let  and  be nuclei on a precoherent semiprequantale Q. Show that inf{t , t } = (inf{,  })t . Section 6.6 1. Fill in the missing details in the proof of Proposition 6.6.2. 2. Verify statement (1) of Lemma 6.6.6. 3. Let M be a unital ordered magma. Prove the following. a) -DI(M) is a submagma of M. b) If X ⊆ -DI(M) is nonempty, then sup X ∈ -DI(M) if sup X exists, and inf X ∈ -DI(M) if inf X exists and X is finite. 4. Verify Example 6.6.10. 5. Prove Theorem 6.6.11. 6. Prove Proposition 6.6.12. 7. Let Q be a coherent unital residuated lattice-ordered semiquantale. a) Show that, for any nucleus  on Q, there is a largest stable nucleus  on Q that is less than or equal to . b) Show that, for any nucleus  on Q, there is a largest stable finitary nucleus w on Q that is less than or equal to . c) (∗∗) If possible, find an explicit formula for  and for w . 8. (∗∗) Let Q be a precoherent prequantale. Characterize the subsets X of Q such that inf{t (a) : a ∈ X } is finitary.

Correction to: Rings, Modules, and Closure Operations

Correction to: J. Elliott, Rings, Modules, and Closure Operations, Undergraduate Topics in Computer Science, https://doi.org/10.1007/978-3-030-24401-9 In the original version of the book, the author missed to provide corrections in most of the chapters. The book now has been updated with all the corrections. Replacement in the Proposition: Statements (6) and (7) of Proposition 2.8.7 has been corrected with the following in page 132: T (6) If I is regular, then I p ¼ fxR : x 2 TðRÞreg ; xR  Ig is principal if and only if gcdðcSÞ exists for all regular c 2 R. If these conditions hold, then one has I p ¼ ðgcdðSÞÞ and gcdðcSÞ ¼ cgcdðSÞ for all regular c 2 R. (7) If I is regular, then I v is principal if and only if I v ¼ I p and I p is principal, if and only if I v ¼ I p and gcdðdSÞ exists for all regular d 2 R. Statements (1)–(7) of Proposition 2.8.19 has been corrected with the following in page 135: (1) R is a weak GCD ring and I p ¼ I v for every ideal I of R generated by finitely many regular elements of R. (2) I p is principal and I p ¼ I v for every ideal I of R generated by finitely many regular elements of R. (3) I v is principal for every ideal I of R generated by finitely many regular elements of R.

The updated version of the book can be found at https://doi.org/10.1007/978-3-030-24401-9 © Springer Nature Switzerland AG 2020 J. Elliott, Rings, Modules, and Closure Operations, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-24401-9_7

C1

C2

Correction to: Rings, Modules, and Closure Operations

(4) gcdða; bÞ exists and ða; bÞv ¼ ða; bÞp for all regular a; b 2 R (in which case gcdðca; cbÞ ¼ cgcdða; bÞ for all regular c 2 R). (5) ða; bÞv is principal for all regular a; b 2 R. (6) The intersection of any two regular principal ideals of R is principal. (7) The lcm of any two regular elements of R exists. Proposition 2.10.8 has been replaced with the following in page 148. Proposition 2.10.8. Let R be a ring, and let I be a unital generalized semistar operation on R. (1) (2) (3) (4) (5) (6) (7) (8)

R is GCD if and only if R is p-Bézout. R is strong GCD if and only if R is a p-PIR. R is weak GCD if and only if R is weak p-Bézout. R is weak strong GCD if and only if R is a weak p-PIR. R is H-Bézout if and only if R is a GCD ring such that Ht ’ pt . R is a H-PIR if and only if R is a strong GCD ring such that H ’ p. R is H-Bézout if and only if R is a weak GCD ring such that Ht ’rg pt . R is a weak H-PIR if and only if R is a weak strong GCD ring such that H ’rg p.

Statement (6) of Proposition 2.10.20 has been corrected with the following in page 151. (6) R is weak GCD ring and Ht ’wk pt (or Ht ’wk t ’wk pt ). Replacement in the Lemma: Statement (2) of Lemma 2.8.18 has been corrected with the following in page 135. (2) ða; bÞv ¼ ða; bÞp and gcdðca; cbÞ exists for all regular c 2 R (in which case gcdðca; cbÞ ¼ cgcdða; bÞ). Replacement in the Theorem: Theorem 2.10.16 has been replaced with the following in page 150. Theorem 2.10.16. Let R be a ring. The following conditions are equivalent. (1) R is factorial. (2) R is a weak pt -PIR, that is, I pt is principal for all regularly generated ideals I of R. (3) R is a weak p-PIR and p ’rg pt . (4) R is a weak strong GCD ring and p ’rg pt . (5) R is a weak GCD ring and satisfies ACCRP. (6) R is an r-atomic weak GCD ring.

Correction to: Rings, Modules, and Closure Operations

C3

Moreover, each of the equivalent conditions above in conjunction with the condition that p ’rg v (or pt ’rg t) is equivalent to each of the following conditions, all of which are equivalent. (7) R is factorial and p ’rg v. (8) R is a weak t-PIR, that is, I t is principal for all regularly generated ideals I of R. (9) R is a weak Mori weak v-PIR. (10) R is a weak TV weak v-PIR. (11) R is a weak Krull ring R with Cltwk ðRÞ ¼ 0. Replacement in Corrolary: Statement (4) of Corollary 2.10.21 has been corrected with the following in page 151. (4) R is a weak GCD ring and pt ’wk t. Corrections in Exercises: Part (b) of Exercise 13 has been deleted (and relabelled the remaining parts) in page 177. In Exercise 10. “Use Exercise 2.7.13” has been changed into “Use Exercise 2.7.14” in page 180. “(See Exercise 3.7.6.)” has been changed into “(See Exercise 3.7.7.)” in page 229. “(See also Exercise 3.7.12.)” has been changed into “(See also Exercise 3.7.13.)” in page 230. In Exercise 15. “totally ordered group” has been changed into “totally ordered abelian group” in page 277. Exercise 25. “Exercise 2.7.13” has been changed into “Exercise 2.7.14” in page 278. Exercise 16. “(See Exercise 3.3.10.)” has been changed into “(See Exercise 3.4.10.)” in page 283. Exercise 8. “Use Corollary 3.8.5” has been changed into “Use Proposition 3.8.3” in page 284. Exercise 7. “Exercise 2.7.13” has been changed into “Exercise 2.7.14”. “Exercise 2.7.18” has been changed into “Exercise 2.7.19” in page 288. Exercise 10. “Exercise 3.2.3.11” has been changed into “Exercise 3.2.11” in page 288. Exercise 17. “Exercise 3.6.22” has been changed into “Exercise 3.6.25” in page 290. Exercise 9. “Exercise 2.7.1” has been changed into “Exercise 2.7.18” in page 291. Exerise 29(e). “Use part (d) to complete Exercise 24” has been changed into “Use part (d) to complete Exercise 28” in page 358.

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Index

Symbols M-surprime operation, 377, 379 N -subprime operation, 377, 379 Q-semistar operation, 431, 433 T -semistar operation, 213, 215, 330, 332 L-injective envelope, 430, 432 -Bézout, 129 -Dedekind, 105 -GCD, 136 -Marot, 128 -Noetherian, 117, 118 -PID, 129 -PIR, 129 -Prüfer, 105, 312, 314 -class group, 109 -closed, 11 -coherent, 119, 120 -finite, 118 -ideal, 303, 305 -invertible, 104, 309, 311 -large-locally Prüfer, 258–260 -maximal, 303, 305 -multiplication, 11, 97 -prime, 187, 189, 303, 305 -stable, 303, 305 d-operation, 95 s-operation, 99 t-UFR, 145, 146 t-compatible extension, 190, 192 t-linked extension, 190, 192 t-localizing prime, 206, 208 t-operation, 99 v-Marot ring, 127 v-closure, 86 v-domain, 89 v-operation, 86 w-compatible extension, 282–284

w-divisorial ring, 114 w-envelope of a module, 433, 435 w-linked extension, 282–284 w-operation, 99 r-closed ideal, 408, 410 r-dense ideal, 407, 409 s-maximal, 302, 304 s-prime, 302, 304

A Absolutely flat, 72 ACCP, 140 ACCRP, 139, 140 Additively regular, 124 Algebra homomorphism, 6 Algebraic integer, 82, 92 Algebraic poset, 461, 463 Algebra over a ring, 6 Almost Dedekind domain, 198, 200 Almost Dedekind ring, 230, 232 Almost integral element, 83 Almost PVMR, 259–261 Almost t-Dedekind domain, 198, 200 Antistable semiprime operation, 388, 390 Approximation theorem for Krull rings, 280–282 Arithmetical ring, 283–285 Ascending chain condition, 117, 118 Associate, 137, 138

B Basic closure system of operations on a category of modules, 402, 404 Basic interior system of operations on a category of modules, 402, 404

© Springer Nature Switzerland AG 2019 J. Elliott, Rings, Modules, and Closure Operations, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-24401-9

485

486 Bdcpo, 458, 459, 461 Bézout domain, 78 Bézout ring, 80 BG-UFR, 145 Bounded complete, 70 Bourbaki associated prime, 192, 194 C Canonical topology, 431, 433 Chain, 10 Characteristic, 6 Characteristic of a ring, 6 Closed under direct products, 408, 410 Closed under direct sums, 408, 410 Closed under isomorphisms, 408, 410 Closed under quotient modules, 408, 410 Closed under submodules, 408, 410 Closed under taking extensions, 413, 415 Closure operation, 11 Closure system, 370, 372 Closure system on a category of modules, 401, 403 Cohereditary preradical, 399, 401 Coherent ordered magma, 463, 464, 466 Compact, 460, 461, 463 Compatible operations on modules, 368, 370 Complete, 10 Complete integral closure, 83 Complete lattice, 10, 66 Completely Integrally Closed (CIC), 84 Completely r-integrally closed, 144, 145 Conductor ideal, 193, 195 Content of a polynomial, 239, 241 Contractive, 11 D Dcpo, 458, 459, 461 Dedekind domain, 78 Dedekind ring, 79 Dedekind–Mertens number, 240, 242 Dense left ideal, 422, 424 Dense topology, 394, 396, 421–424 Directed, 12 Directed complete poset, 458, 459, 461 Directed poset, 458, 459, 461 Discrete rank at most one valuation ring, 220, 222 Discrete rank one valuation domain, 78 Discrete rank one valuation on a ring, 210, 212 Discrete rank one valuation pair, 220, 222 Discrete rank one valuation ring, 220, 222

Index Discrete rank zero, 220, 222 Discrete valuation ring, 78 Divisorial closure, 86 Divisorial closure semiprime operations, 393, 395 Divisorial closures, 100, 101 Divisorial Kaplansky fractional ideal, 86 Divisorial left nuclei, 454–458 Divisorial nuclei, 454–458 Divisorial right nuclei, 454–458 Divisorial ring, 114 Dominates, 244, 246 Downward closed, 12 DVR, 78 DW ring, 114 E Enough -maximal ideals, 303, 305 Enough regular -prime ideals, 233, 235 Essential domain, 186, 188 Essential extension of a module, 418, 420 Essential ideal, 394, 396, 421–424 Essential prime, 181, 183, 237, 239 Essential ring, 237, 239 Exact system of operations on a category of modules, 401, 403 Expansive, 11 F Factorial ring, 138 Faithful, 8 Few zerodivisors, 124 Field of fractions, 65 Finitary closure, 460, 462 Finite character, 196, 198 Finite t-character, 196, 198 Finite type, 12 Finite type ideal filter, 389, 391 Finite type left exact preradical, 420, 422 Finite type linear topology, 389, 391 Finite-type semiprime operation, 319, 321 Finite-type semistar operation, 301, 303 Finite type semistar operation, 97 Finite-type star operation, 294, 296 Finitely regularly generated, 146 FJS -invertible, 309, 311 FJS -Prüfer, 312, 314 Flat module, 75 Fractional, 68 Fractional ideal, 68 Fractional star operation, 292, 294 Frame, 445, 446, 448

Index

487

Functorial system of operations on a category of modules, 401, 403

Invertible, 73, 74 Irreducible element, 137

G Gabriel filter on a ring, 384, 386 Gabriel topology, 385, 387 Galois connection, 366, 368 Gaussian polynomial, 240, 242 Gaussian ring, 240, 242 GCD ring, 132, 133 Generalized -Prüfer, 155, 156 Generalized GCD domain, 177, 178 Glaz–Vasconcelos ideal, 432, 434 Goldie dense ideal, 422, 424 Goldie topology, 394, 396, 422, 424 Greatest common divisor, 131

J Join semilattice, 10

H Hereditary class of modules, 411, 413 Hereditary preradical, 399, 401 Hereditary pretorsion theory, 416, 418 Hereditary torsion theory cogenerated by a class of modules, 423, 425 Hereditary torsion theory generated by a class of modules, 423, 425 Heyting algebra, 445, 447 Hilbert basis theorem, 172, 173 H ring, 122

I Ideal class group, 108 Ideal filter on a ring, 384, 386 Idealization, 125, 126 Idempotent hull, 321, 323 Idempotent preradical, 399, 401 Idempotent pretorsion theory, 408, 410 Idempotent radical associated to a preradical, 415, 417 Idempotent self-map, 11 Integer-valued polynomials, 79 Integral closure, 83 Integral closure of a submodule, 327, 329 Integral element, 82 Integrally Closed (IC), 83 Integrally closed submodule, 327, 329 Integral over a submodule, 327, 329 Interior operation, 11 Interior system on a category of modules, 401, 403 Invariant system of operations on a category of modules, 401, 403

K Kaplansky fractional ideal, 66 Krull domain, 106 Krull ring, 106 L Large localization, 202, 204 Lattice, 10 Lattice-ordered, 10, 70 Least common multiple, 131 Left exact preradical, 399, 401 Left exact system of operations on a category of modules, 401, 403 Left functorial system of operations on a category of modules, 401, 403 Left ideal system on, 371, 373 Left nucleus, 12 Left residuated, 12 Left semiexact system of operations on a category of modules, 401, 403 Left semiprime operation, 367, 369 Left stable, 465, 467 Left sup-spanning subset, 449, 450, 452 Left T -semistar operation, 424, 426 Left unital magma, 442, 444 Linear ring topology, 385, 387 Locale, 445, 446, 448 Localization of a module with respect to a Gabriel filter, 430, 432 Localizing system on a ring, 385, 387 Locally finite collection of algebras, 195, 197 Locally Prüfer ring, 247, 249 Local ring, 74 M Marot ring, 123, 124 Maximally Prüfer ring, 247, 249 McCoy ring, 395, 397 Meet semilattice, 10 Modular closure operation, 370, 372 Module closure operation on a module, 381, 383 Module of quotients with respect to a Gabriel filter, 430, 432

488 Module system, 470, 472, 474 Moore family, 370, 372 Mori ring, 117, 118 Multiplicative lattice, 445, 446, 448 Multiplicative semilattice, 474, 475, 477 Multiplicative subset, 6

N Nagata transform, 288–290 Near multiplicative lattice, 446, 447, 449 Near prequantale, 446, 447, 449 Near PVMR, 259–261 Near quantale, 446, 447, 449 Near sup-complete poset, 441, 443 Near sup-preserving, 442, 444 Nonsingular ring, 422, 424 Nonunital star operation, 297, 299 Normalized discrete rank one valuation, 184, 186 Northcott attached prime, 192, 194 Nucleus, 11

Index Prenucleus, 321, 323 Preorder, 10 Pre-pretorsion class, 408, 410 Pre-pretorsion-free class, 408, 410 Prequantale, 445, 446, 448 Preradical, 398, 400 Pretorsion class, 408, 410 Pretorsion-free class, 408, 410 Pretorsion theory, 408, 410 Prime element, 137 Principal Ideal Domain (PID), 78 Principal Ideal Ring (PIR), 80 P-ring, 237, 239 Product of submodules, 66 Prüfer domain, 78 Prüfer ring, 79 Prüfer valuation pair, 246, 248 Prüfer v-Multiplication Domains (PVMD), 106 Prüfer v -Multiplication Ring (PVMR), 106 Pure ideal, 159, 160

O Operation, 11 Ordered group, 11 Ordered magma, 11 Ordered monoid, 11 Ordered semigroup, 11 Order in a number field, 115, 116 Order-preserving, 11 Overring, 65

Q QMD, 313, 315 Quantale, 445, 446, 448 Quantale module, 435, 437 Quasi-coherent ring, 120 Quasi -invertible, 308, 310 Quasi-Marot, 124, 125 Quasi -Prüfer, 312, 314 Quotient field, 65

P Paravaluation, 209, 211 Paravaluation ring, 244, 246 Partially ordered set, 10 Partial ordering, 10 P-domain, 237, 239 PF ring, 266–268 Picard group, 171, 172 Plus closure, 360, 362 PMD, 105, 313, 315 PMR, 105 Poset, 10 Preclosure, 11 Preclosure system on a category of modules, 401, 403 Precoherent ordered magma, 463, 464, 466 Preinterior operation, 11 Preinterior system on a category of modules, 401, 403

R Radical, 399, 401 Radical pretorsion theory, 408, 410 r-atomic ring, 139, 140 r-Bézout ring, 79 Reduced map of posets, 11, 394, 396 Reduction of a submodule, 329, 331 Regular, 68 Regular element, 68 Regular Kaplansky fractional ideal, 68 Regular locally Prüfer ring, 162, 163, 247, 249 Regularly generated, 123, 124 Regularly prime element, 138, 139 Regular semiprime operation, 322, 324 Residuated, 12, 67 Residue field, 5 r-finite type semistar operation, 103 r-GCD ring, 110

Index Right exact system of operations on a category of modules, 401, 403 Right functorial system of operations on a category of modules, 401, 403 Right nucleus, 12 Right residuated, 12 Right semiexact system of operations on a category of modules, 401, 403 Right semiprime operation, 367, 369 Right stable, 465, 467 Right sup-spanning subset, 449, 450, 452 Right T -semistar operation, 424, 426 Right unital magma, 442, 444 Rigid closure system, 403, 405 Rigid semiprime operation, 321, 323 Ring topology, 385, 387 r-integrally closed, 144, 145 r-Noetherian, 81 r-PIR, 79 r-prime, 138, 139 r-spectral semistar operation, 289–291 r-stable semistar operation, 103 r-UFR, 110

S Scott closed, 12 Scott continuous, 12 Scott open, 12 Scott-topological, 459, 461 Scott topology, 12 Self-map, 11 Semihereditary ring, 266–268 Semilocal ring, 74 Semimultiplicative lattice, 447, 449 Semiprequantale, 446, 447, 449 Semiprime operation, 291, 293, 367, 369 Semiprime operation on a module, 370, 372 Semiquantale, 446, 447, 449 Semiregular ideal, 395, 397 Semistable semiprime operation, 388, 390 Semistar -class group, 109 Semistar operation, 94, 292, 294 Semistar plus closure, 360, 362 Semistar t-compatible extension, 190, 192 Semistar tight closure, 347, 349 Semistar t-linked extension, 190, 192 Special principal ideal ring, 145 Special Principal Ideal Ring (SPIR), 145 Spectral semistar operation, 254–256, 307, 309 Stable closure operation on a module, 382, 384

489 Stable nucleus, 465, 467 Stable order-preserving operation on a module, 381, 383 Stable semiprime operation, 319, 321 Stable semistar operation, 97, 301, 303 Stable star operation, 294, 296 Stable system of order-preserving operations, 418, 420 Standard preclosure, 323, 325 Star equivalent semistar operations, 102 Star operation, 101, 292, 294 Strictly -finite, 118 Strictly -invertible, 309, 311 Strictly -Prüfer, 312, 314 Strong GCD ring, 132, 133 Strong Krull prime, 192, 194 Strongly essential prime, 256–258 Strongly essential ring, 259–261 Strongly hereditary class of modules, 411, 413 Strong Mori domain, 121, 122 Submodule closure system, 371, 373 Sup-preserving, 442, 444 Sup-spanning subset, 449, 450, 452 Symbolic power, 225, 227 System of operations on a category of modules, 401, 403

T TD ring, 114 Tight closure, 344, 346 Tightly closed, 344, 346 Topologizing ideal filter, 384, 386 Torsion, 8, 399, 401 Torsion class, 414, 416 Torsion-free, 8, 399, 401 Torsion-free class, 414, 416 Torsion semiprime operation, 415, 417 Torsion submodule, 8 Torsion theory, 414, 416 Torsion theory cogenerated by a class of modules, 415, 417 Torsion theory generated by a class of modules, 415, 417 Total integral closure, 343, 345 Totally integrally closed, 343, 345 Totally ordered set, 10 Total quotient ring, 6, 63, 64 Total ring of fractions, 64 Tree, 261–263 Trivial extension, 125, 126 Trivial semiprime operation, 318, 320

490 Trivial semistar operation, 95 TV ring, 114 TW ring, 114

U UFR, 145, 146 UMT domain, 178, 179 Uniform ideal filter, 384, 386 Unique factorization ring, 145, 146 Unique r-factorization ring, 110 Universal semistar operation, 113, 114

V Valuation, 209, 211 Valuation domain, 181, 183 Valuation on a field, 183, 185 Valuation pair, 210, 212 Valuation ring, 212, 214 Valuative T -semistar operation, 214, 216 Value group of a valuation, 183, 185, 210, 212 Vector space, 7 Von Neumann regular, 72

Index W Weak -Bézout ring, 146, 147 Weak -class group, 149 Weak -Dedekind ring, 146, 147 Weak -Noetherian ring, 146, 147 Weak -PIR, 146, 147 Weak -Prüfer ring, 146, 147 Weak Bézout ring, 146, 147 Weak Bourbaki associated prime, 192, 194 Weak Dedekind ring, 146, 147 Weak GCD ring, 132, 133 Weak global dimension at most one, 266– 268 Weak global dimension zero, 72 Weak ideal system, 470, 471, 473 Weak Krull ring, 146, 147 Weak Mori ring, 146, 147 Weak PIR, 146, 147 Weak Prüfer ring, 146, 147 Weak PVMR, 146, 147 Weak strong GCD ring, 132, 133 Weak TV ring, 146, 147 Well behaved prime, 206, 208