Rings, Monoids and Module Theory: AUS-ICMS 2020, Sharjah, United Arab Emirates, February 6–9 (Springer Proceedings in Mathematics & Statistics, 382) 9811684219, 9789811684210

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Springer Proceedings in Mathematics & Statistics

Ayman Badawi Jim Coykendall   Editors

Rings, Monoids and Module Theory AUS-ICMS 2020, Sharjah, United Arab Emirates, February 6–9

Springer Proceedings in Mathematics & Statistics Volume 382

This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including data science, operations research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today.

More information about this series at https://link.springer.com/bookseries/10533

Ayman Badawi · Jim Coykendall Editors

Rings, Monoids and Module Theory AUS-ICMS 2020, Sharjah, United Arab Emirates, February 6–9

Editors Ayman Badawi Department of Mathematics American University of Sharjah Sharjah, United Arab Emirates

Jim Coykendall Department of Mathematical Sciences Clemson University Clemson, SC, USA

ISSN 2194-1009 ISSN 2194-1017 (electronic) Springer Proceedings in Mathematics & Statistics ISBN 978-981-16-8421-0 ISBN 978-981-16-8422-7 (eBook) https://doi.org/10.1007/978-981-16-8422-7 Mathematics Subject Classification: 13Axx, 13Bxx, 13Cxx, 13Dxx, 13Exx, 13Hxx, 13Gxx, 13Jxx, 13Pxx, 05Cxx, 16Yxx, 18Mxx, 13-06 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 This work is subject to copyright. All rights are solely and exclusively licensed 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 Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Dedicated to Daniel D. Anderson

Preface

Ayman Badawi and Jim Coykendall dedicate these proceedings of the special session on Rings, Monoids, and Module Theory that was organized during the 3rd International Conference on Mathematics and Statistics (AUS-ICMS’20) held between February 6–9, 2020 at the American University of Sharjah, United Arab Emirates, to the work of the distinguished algebraist Daniel D. Anderson. Many participants and colleagues from around the world felt it appropriate to acknowledge his broad and sweeping contributions to research in algebra by writing an edited volume in his honor. The editors have tried to present a balanced mix of survey papers, which will enable expert and non-expert alike to get a good overview of developments across a range of areas of algebra. Every effort has been made to make these research papers easily accessible in their introductory sections. We would hope that the material is of interest to both beginning graduate students and experienced researchers alike. The topics covered are, inevitably, a cross-section of the vast expanse that is modern algebra. The book is divided into two sections—surveys and recent research developments—with each section hopefully offering symbiotic utility to the reader. Finally, we would like to express our sincere thanks to the colleagues who contributed papers so enthusiastically, to the many experts who acted as referees for all the papers, to the professional staff at Springer, and in particular to Shamim Ahmad, for their help in producing a volume which we hope is an appropriate recognition of our friend Daniel D. Anderson. Sharjah, United Arab Emirates Clemson, USA

Ayman Badawi Jim Coykendall

vii

Students of Daniel D. Anderson

Name

School

Year

Pascual-Garcia, Joaquin

University of Iowa

1986

Kang, Byung

University of Iowa

1987

Spellerberg, Richard

University of Iowa

1990

Knopp, Kent

University of Iowa

1991

LeDocq, Rebecca

University of Iowa

1991

Alarcon, Francisco

University of Iowa

1992

Naseer, Muhammad

University of Iowa

1992

Park, Jeanam

University of Iowa

1992

Valdes-Leon, Silvia

University of Iowa

1993

LaGrassa, Susan

University of Iowa

1995

Mullins, Bernadette

University of Iowa

1995

Quintero-Contreras, Roy

University of Iowa

1997

Stickles, Jr, Joe

University of Iowa

1998

Cook, Sylvia

University of Iowa

1999

Axtell, Michael

University of Iowa

2000

Smith, Eric

University of Iowa

2001

Ahn, Myung-Sook

University of Iowa

2003

Clarke, Sharon

University of Iowa

2003

Robeson, John

University of Iowa

2003

Winders, Michael

University of Iowa

2004

Ganatra, Amit

University of Iowa

2005

Descendants 5

(continued)

ix

x

Students of Daniel D. Anderson (continued)

Name

School

Year

Bataineh, Malik

University of Iowa

2006

Frazier, Andrea

University of Iowa

2006

Hamon, Suzanne

University of Iowa

2007

Chun, Sangmin

University of Iowa

2008

Ortiz-Albino, Reyes

University of Iowa

2008

Kintzinger, John

University of Iowa

2009

Preisser, Jonathan

University of Iowa

2009

McKinney, Colin

University of Iowa

2010

Reinkoester, Jeremiah

University of Iowa

2010

Florescu, Alina

University of Iowa

2013

Juett, Jason

University of Iowa

2013

Mooney, Christopher

University of Iowa

2013

Edmonds, Ranthony

University of Iowa

2018

Hasse, Erik

University of Iowa

2018

Bombardier, Kevin

University of Iowa

2019

Descendants

Contents

Dan Anderson and His Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . David F. Anderson

1

Bounded and Finite Factorization Domains . . . . . . . . . . . . . . . . . . . . . . . . . . David F. Anderson and Felix Gotti

7

Factorization and Irreducibility in Modules . . . . . . . . . . . . . . . . . . . . . . . . . . A. A. Altidor, H. E. Bruch, and J. R. Juett

59

On -potent Domains and -homogeneous Ideals . . . . . . . . . . . . . . . . . . . . . Muhammad Zafrullah

89

On the Set of Molecules of Numerical and Puiseux Monoids . . . . . . . . . . . 111 Marly Gotti and Marcos M. Tirador Where Some Inert Minimal Ring Extensions of a Commutative Ring Come from, II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 David E. Dobbs A Survey on EM Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Emad Abuosba and Manal Ghanem Some Remarks on the D + M Construction . . . . . . . . . . . . . . . . . . . . . . . . . . 143 David F. Anderson On a Problem About Lowest Terms Domains Posed by D. D. Anderson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Roy O. Quintero Contreras Regularity and Related Properties in Tensor Products of Algebras Over a Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 S. Kabbaj and F. Suwayyid Tame-Wild Dichotomy for Commutative Noetherian Rings—A Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 Lee Klingler, Roger Wiegand, and Sylvia Wiegand xi

xii

Contents

On the Characterization of τ(n) -Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 A. Hernández-Espiet and R. M. Ortiz-Albino Bounded Periodic Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 D. D. Anderson and P. V. Danchev On Gracefully and Harmoniously Labeling Zero-Divisor Graphs . . . . . . 239 Christopher P. Mooney A Survey on Genus of Selected Graphs from Commutative Rings . . . . . . 261 T. Asir, K. Mano, and M. Subathra A Computational Approach to Shephard Groups . . . . . . . . . . . . . . . . . . . . . 287 Dong-Il Lee BZS Near-Rings and Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 Mark Farag and Ralph P. Tucci

About the Editors

Daniel D. Anderson is one of the leading algebraists of his generation. He was born in Fort Dodge, Iowa, and grew up in Gowrie, Iowa, a town of about 1,000 people located 80 miles northwest of Des Moines. His father was a rural mail carrier and his mother was an elementary school teacher. His father’s cousin was E. F. Lindquist, an education professor at the University of Iowa who helped develop the ACT test and the GED exam. He graduated from Prairie Community High School in 1967 along with 58 others, including another future mathematician (his twin brother David F. Anderson now retired and Emeritus Professor at the University of Tennessee). Daniel D. Anderson attended the University of Iowa, where he received his BA and MS degrees. In 1971, Anderson enrolled at the University of Chicago on an NSF Fellowship and received his Ph.D. in 1974 under the supervision of the remarkable algebraist Irving Kaplansky. In 1977, he joined the University of Iowa where he spent his career as a teacher, researcher, and stints as graduate program director and chair. Daniel D. Anderson published more than 230 papers in different branches of algebra (with emphasis on Commutative Algebra) and is a noted advisor of students. His current total of 36 Ph.D. students and 41 descendants makes him one of the most prolific advisors of his time. At the outset, let us stress that it is impossible in a few pages to give a detailed overview of the many research contributions made by Daniel D. Anderson, but it is clear that his impact will continue to reverberate for generations to come. A striking feature of Daniel xiii

xiv

About the Editors

D. Anderson’s research output is the number of coauthors, some 50 in total, but perhaps more surprising is the number of co-authors with whom he wrote multiple papers and the duration of these collaborations. Many co-authors had more than 10 joint papers with him, and these collaborations endured for more than 20 years. Of note, the following authors co-authored more than 10 papers with Dan Anderson: Muhammad Zafrullah (48), David F. Anderson (31), Sangmin Chun (16), and E. W. Johnson (14). Additionally, the work of Dan and his co-authors has had a lasting effect in the field of commutative algebra. Of particular note is the paper “Factorization in Integral Domains”. At this writing, this paper has 94 citations on MathSciNet and has 275 listed on Google Scholar, and this paper arguably was at the forefront of a renaissance of factorization theory which has led to an industry of research as well as an updating of one of the MSC subject classification codes (13F15).

Personal Comments Ayman Badawi I never had the opportunity to be a co-author with Daniel D. Anderson. As a researcher in Commutative Algebra, I admired his huge knowledge and his passion for commutative ring theory. Without a doubt, Daniel D. Anderson’s work has influenced many algebraists. He has had a significant impact on the field of algebra, in particular commutative algebra. James Coykendall I first met Dan Anderson in the spring of 1996. Although I had known his brother David for many years, I had never met Dan until he reached out to help me when I was a young mathematician trying to start his career. Shortly after I graduated and was on my post-doc, Dan graciously invited me to give a talk at a conference he was organizing and this first conference turned out to be quite formative in my career. Over the years that followed, Dan was always supportive and helpful to me in my blossoming career. I continue to call him a colleague and friend and continue to admire him as a mathematician and marvel at his work.

Dan Anderson and His Mathematics David F. Anderson

Abstract In this note, I give some reminiscences about my twin brother Dan Anderson and his mathematics. Keywords Dan Anderson · University of Iowa · Commutative rings 2010 Mathematics Subject Classification 01A70

It is also a great pleasure to write a little about my twin brother Dan, but to me, he will always be Daniel. He has had a lasting effect on the mathematical community through his research, PhD students, and service at the University of Iowa. This was recognized by being named an AMS Fellow in 2018. He has written a detailed narrative ([2]) about us growing up in Gowrie, a small town in north-central Iowa, about 80 miles northwest of Des Moines. So I will not repeat it here, but will just add a few additional things about Dan. I vividly remember family picnics and vacations, riding bikes, going swimming, playing army, playing ping pong, doing “experiments” in our basement chemistry lab, and flying model rockets with Dan (the Estes catalogs were then just mimeographed sheets of paper). We made model planes and cars, gun powder, CO2 match head rockets, dugouts, and club houses. We both had paper routes and that got us interested in coin collecting. Not surprisingly, we still have many of the same interests, own many of the same mutual funds, collect coins, and are avid book collectors, particularly books from the Folio Society. And it seems that he has finally gotten interested in the Great Courses. We did many things with Boy Scouts, including winter camping at Dolliver’s Park, a trip to the Badlands and Black Hills in South Dakota in 1963, and an East Coast tour to the World’s Fair in New York City and the National Jamboree at Valley Forge in 1964. Of course, Dan was an Eagle Scout, the highest rank in Scouting.

D. F. Anderson (B) Department of Mathematics, The University of Tennessee, Knoxville, TN 37996-1300, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. Badawi and J. Coykendall (eds.), Rings, Monoids and Module Theory, Springer Proceedings in Mathematics & Statistics 382, https://doi.org/10.1007/978-981-16-8422-7_1

1

2

D. F. Anderson

We both enjoyed playing neighborhood touch football almost every day, and later played football and ran track in high school. Dan won many ribbons in track. In fact, he placed in the Iowa Indoor State Track meet at Iowa City in the Spring of 1967, running a 220 leg on the mile medley relay. As a little side note, our shop teacher Richard Ryan, who was also the high school girls basketball coach, convinced us to help scrimmage the girls “six-on-six” basketball team. They were better than we. We both grew up big Iowa State fans and always planned to attend college there as it was only about 40 miles away. We would go to a football game each fall in Ames with our dad, and I remember we looked forward to going to the university book store to buy Iowa State souvenirs and look at math books. I particularly remember we saw MacLane/Birkhoff’s new Algebra and found it rather puzzling (Is this really algebra?). In the summer of 1966, we both attended an NSF-sponsored science program for rising seniors at the University of Iowa in Iowa City. Dan fell in love with the University of Iowa, and the rest is history. So he attended Iowa and I attended Iowa State; I think it was good that we attended different universities as undergraduates. However, to “get his goat”, I would sometimes call Iowa City a scruffy little river town. Dan started his undergraduate work at Iowa during the summer of 1967, right after graduation from high school. He took a calculus course (out of Purcell’s Calculus with Analytical Geometry if I remember correctly) as our high school did not offer calculus. I think he also worked in the computer center. He really enjoyed the university and soon started taking graduate math classes. He participated in an REU at the University of Indiana in summer 1969; I believed he worked on knot theory. I recall that I only visited him once at Iowa, and he visited me just once at Iowa State. During his visit, some of my friends confused him for me. Dan graduated from Iowa in 1970 in three years as Phi Beta Kappa, a Collegiate Scholar, and Woodrow Wilson Fellow. That summer, he got married (he and Kathy just celebrated their 50th wedding anniversary this July), had a honeymoon in Europe, and then went to the University of Chicago on a NSF Graduate Fellowship. Their daughter Caity was born the next year. I didn’t actually see a lot of Dan at Chicago, and the only course I remember we took together was Murthy’s year long course on commutative algebra (more on that later) and projective modules during 1972-73. He was married and intent on graduating as quickly as possible, while I was more interested in playing bridge and “Chicago” slowpitch softball (and during 1973-74, in my future wife Konnie, who was an undergrad at Iowa State at the time). However, I do remember when I was first going through Kaplansky’s Commutative Rings and being impressed with the way Dan could rattle off all the definitions and equivalences, while I was having trouble just remembering the difference between Dedekind, Bézout, and Prüfer domains. Dan graduated from UC in 1974 under Irving Kaplansky; his thesis title was Multiplicative Lattices. His first job was as a visiting assistant professor at the University of Iowa for 1974-75. His goal was always to return to the University of Iowa. And things worked out rather well; after being an assistant professor for a year at Virginia Tech (1975-76) and then at the University of Missouri (1976–1977) with Jim Huckaba, he again returned to Iowa in the fall of 1977 as an assistant professor.

Dan Anderson and His Mathematics

3

He quickly rose through the ranks to full professor in 1983, and then to Associate Department Head and finally Department Head. Along the way, he served on several university committees; I particularly remember one to determine instate/out-of-state tuition, and he was also the university’s NCAA representative for several years. You may know that Dan is a very serious collector of all US coins, half cents through gold, but you probably didn’t know that his real expertise is in the less well known area of (Iowa) trade tokens. He is a leading expert in this area and has also written several scholarly articles on tokens (for instance, see [1]). Moreover, he won an NTCA 2018 Literary Award for Variously Spelled Towns in Iowa Trade Tokens. In fact, Dan seems to collect just about anything and everything. Dan has had 36 Ph.D. students, starting in 1986. That’s averaging a bit more than one per year! I consider this to be his most impressive achievement; really, really impressive. Although their theses cover many different topics, the recurring theme has been factorization and its generalizations, particularly, in rings with zero-divisors. He has coauthored about 75 papers with 30 of these students. Math Reviews lists over 230 publications (and still counting!) for Dan, starting in 1973, with over 2600 citations. These publications run the full gamut, including short notes, very long articles, survey/expository articles, lectures notes, and edited conference proceedings. He has had over 70 coauthors, the most frequent being Muhammad Zafrullah, with me being number two. Most of these publications are in commutative algebra, but there are some in lattices and ordered structures, group theory, and even K-theory. Several of the early ones, influenced by his undergraduate days at Iowa, were in lattices (abstract ideal theory). Again, the recurring themes concern divisibility, factorization, and rings with zero-divisors. His first two references in Math Reviews are for notes he wrote up for the Chicago Lecture Series on P. M. Murthy’s commutative algebra course given at UC in fall 1972 and winter 1973. Volume I appeared in 1973 and Volume II in 1976. However, without a doubt, Dan’s most profitable math-related publication (but not listed in Math Reviews) is his Student Solution Manual for Stewart’s Calculus (Volume I). It has sold thousands of copies and gone through many editions. Dan’s top three cited articles are (1) Armendariz rings and Gaussian rings (with Vic Camillo), Comm. Algebra 26(1988), 2265–2272; (2) Beck’s coloring of a commutative ring (with M. Naseer), J. Algebra 159(1993), 500–514; and (3) Factorization in integral domains (with Muhammad Zafrullah and me), J. Pure Appl. Algebra 69(1990), 1–19. Over 40 of his papers have been cited 20 or more times. It would take way too long to decribe his research and publications in detail; so I will just paint a broad picture. Most of his research has been in standard topics in commutative algebra (commutative rings with identity and modules, but with a much more ideal-theoretic flavor rather than homological) and lattice theory, with briefer forays into semirings, semigroups, and rings without identity. As mentioned above, his early research often focused on lattices (abstract ideal therory). He has returned to this subject off and on and has published about 30 papers in this area. He has 14 papers with his undergraduate “lattice mentor”, and later colleague, E. W. Johnson at the University of Iowa. By far, his most popular research topic is all aspects of divisibility and factorization in both integral domains

4

D. F. Anderson

and commutative rings with zero-divisors. Here, “divisibility” includes gcds, lcms, factorizations, divisorial and invertible ideal, primes and atoms, groups of divisibility, Bézout, Prüfer, Dedekind domain, and Krull domains, and their many generalizations. He has been particularly interested in extending classical theories for integral domains to rings with zero-divisors. In this case, the usual equivalent conditions for integral domains often become very different when zero-divisors are allowed. Over 70 of Dan’s papers concern divisibility topics in some form or another, with about 15 on rings with zero-divisors. He also has several important papers on graded rings and star operations. I would like to mention two series of seminal papers on factorization, all written about the same time in the early 1990s. The first three are with Muhammad Zafurullah and me: the already mentioned (1) Factorization in integral domains, J. Pure Appl. Algebra 69(1990), 1–19; along with (2) Factorization in integral domains, II, J. Algebra 152(1992), 78–93; and (3) Rings between D[X ] and K [X ], Houston J. Math. 17(1991), 109–129. These papers studied properties weaker than unique factorization in arbitrary integral domains, introduced such concepts as bounded factorization domains (BFDs) and finite factorization domains (FFDs), and investigated relationships between atomic domains, domains satisfying ACCP, BFDs, FFDs, and HFDs. Of course, these concepts have also been throughly investigated for commutative rings with zero-divisors in later papers by Dan and his students. The next three papers, also with me, are on the elasticity of factorizations in integral domains: (4) Elasticity of factorizations in integral domains, J. Pure Appl. Algebra 80(1992), 217–235; (5) Elasticity of factorizations in integral domains, II, Houston J. Math. 20(1994), 1–515; and (6) Rational elasticity of factorizations in integral domains (also with Scott Chapman and Bill Smith), Proc. Amer. Math. Soc. 117(1993), 37– 43. I would like to think that these papers, particularly (1) and (4), helped kindle interest and guide the direction of research in these areas. One could argue that Beck’s coloring of a commutative ring (with M. Naseer), J. Algebra 159(1993), 500–514, the single paper Dan wrote on zero-divisor graphs, has had the greatest “ripple” effect. This paper was the bridge between Beck’s original paper, Coloring of commutative ring, J. Algebra 116(1988), 208–226, and my paper with Phil Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217(1999), 434–447. Dan’s paper got Phil and me interested in this area as a possible thesis topic, and I remember having his paper on my desk when we tossed about things to work on. Finally, I certainly cannot overestimate the importance of Dan’s work with Muhammad Zafrullah and its influence on commutative ring theory. It covered many topics, but mostly related to factorization and divisibility in integral domains. Many papers with Muhammad were on “almost whatever” domains, where the usual “whatever” Property P for an integral domain using elements x, y, . . . is replaced by, there is a positive integer n such that Property P holds for x n , y n , . . .. Some of my most enjoyable collaborations have been with Daniel, particularly our work on graded integral domains in the early 1980s, factorization in integral domains starting in the 1990s, and related work with Muhammad Zafrullah. We seem to complement each other very well. In retrospect, I wish we had worked on

Dan Anderson and His Mathematics

5

zero-divisor graphs together. But, as Dan mentioned in his article about me, we very rarely talk mathematics. I guess we just prefer to think about things on our own. Both he and David Dobbs at Tennessee influenced me to shift from algebraic K-theory and projective modules to commutative ring theory; thank you.

References 1. D.D. Anderson, A token from my grandfather’s general store. Talkin’ Tokens 64, 52–53 (2002) 2. D.D. Anderson, David Anderson and his mathematics, Advances in commutative algebra, 1–6 (Trends Math. Birkhäuser/Springer, Singapore, 2019)

Bounded and Finite Factorization Domains David F. Anderson and Felix Gotti

Abstract An integral domain is atomic if every nonzero nonunit factors into irreducibles. Let R be an integral domain. We say that R is a bounded factorization domain if it is atomic and for every nonzero nonunit x ∈ R, there is a positive integer N such that for any factorization x = a1 · · · an of x into irreducibles a1 , . . . , an in R, the inequality n ≤ N holds. In addition, we say that R is a finite factorization domain if it is atomic and every nonzero nonunit in R factors into irreducibles in only finitely many ways (up to order and associates). The notions of bounded and finite factorization domains were introduced by D. D. Anderson, D. F. Anderson, and M. Zafrullah in their systematic study of factorization in atomic integral domains. In this chapter, we present some of the most relevant results on bounded and finite factorization domains. Keywords BFD · Bounded factorization domain · FFD · Finite factorization domain · Factorization · D+M construction · HFD · ACCP · Atomic domain · Monoid domain 2010 Mathematics Subject Classification Primary: 13A05 · 13F15 · Secondary: 13A15 · 13G05

Dedicated to Daniel D. Anderson on his retirement. D. F. Anderson Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA e-mail: [email protected] F. Gotti (B) Department of Mathematics, MIT, Cambridge, MA 02139, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. Badawi and J. Coykendall (eds.), Rings, Monoids and Module Theory, Springer Proceedings in Mathematics & Statistics 382, https://doi.org/10.1007/978-981-16-8422-7_2

7

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D. F. Anderson and F. Gotti

1 Introduction During the last three decades, the study of factorizations based on Diagram (1.1) has earned significant attention among researchers in commutative algebra and semigroup theory. This diagram of classes of integral domains satisfying conditions weaker than unique factorization was introduced by Anderson, Anderson, and Zafrullah in [4]. We proceed to recall the definitions of the atomic classes in Diagram (1.1). Let R be an integral domain. Following Cohn [22], we say that R is atomic if every nonzero nonunit of R can be factored into irreducibles. In addition, R satisfies the ascending chain condition on principal ideals (or ACCP) if every ascending chain of principal ideals of R eventually stabilizes. If an integral domain satisfies ACCP, then it is atomic; however, there are atomic domains that do not satisfy ACCP (the first example was constructed by Grams in [36]). On the other hand, R is called a half-factorial domain (or an HFD) if R is atomic and any two factorizations of the same nonzero nonunit of R have the same number of irreducibles (counting repetitions). The term “half-factorial domain” was coined by Zaks in [49]. For a survey on half-factorial integral domains, see [18]. UFD

HFD

FFD

BFD

ACCP domain

atomic domain

(1.1) We say that R is a bounded factorization domain (or a BFD) if it is atomic and for every nonzero nonunit x ∈ R, there is a positive integer N such that x = a1 · · · an for irreducibles a1 , . . . , an ∈ R implies that n ≤ N . In addition, we say that R is a finite factorization domain (or an FFD) if it is atomic and every nonzero nonunit of R factors into irreducibles in only finitely many ways (up to order and associates). The notions of a BFD and an FFD were introduced in [4] as part of Diagram (1.1). The purpose of this chapter is to survey some of the fundamental results related to bounded and finite factorization domains that have been established in the last three decades, indicating for the interested reader the sources where the most relevant results originally appeared. Although the rings we consider here have no nonzero zero-divisors, it is worth pointing out that the bounded and finite factorization properties have been extensively investigated in the context of commutative rings with zero-divisors by D. D. Anderson and his students; see [7] for more details and references. This chapter is organized as follows. In Sect. 2, we recall some definitions and settle down the notation we will use throughout this paper. In Sect. 3, we give a few results about the bounded and finite factorization properties in the abstract context of monoids. Our treatment of monoids is brief as we only present results that will be useful later in the context of integral domains. Then in Sect. 4, we turn our attention to bounded and finite factorization domains, providing several characterizations and showing, among other results, that Noetherian domains and Krull domains are BFDs and FFDs, respectively. We also consider the popular D + M construction. In Sect. 5,

Bounded and Finite Factorization Domains

9

we explore conditions under which the bounded and finite factorization properties are inherited by subrings or passed to ring extensions; we put particular emphasis on ring extensions by localization and pullback constructions. Directed unions are also considered. In Sect. 6, we treat integral domains somehow related to rings of polynomials and rings of power series. We put special emphasis on the class of monoid domains. Finally, in Sect. 7, we briefly explore an abstraction of the finite factorization property introduced by D. D. Anderson and the first author in [2], where factorizations in an integral domain are identified up to a given arbitrary equivalence relation on the set of irreducibles (not necessarily that of being associates).

2 Preliminary In this section, we briefly review some notation and terminology we will use throughout this chapter. For undefined terms or a more comprehensive treatment of nonunique factorization theory, see [27] by A. Geroldinger and F. Halter-Koch.

2.1 General Notation As is customary, Z, Z/nZ, Q, R, and C will denote the set integers, integers modulo n, rational numbers, real numbers, and complex numbers, respectively. We let N and N0 denote the set of positive and nonnegative integers, respectively. In addition, we let P denote the set of primes. For p ∈ P and n ∈ N, we let F pn be the finite field of cardinality p n . For a, b ∈ Z with a ≤ b, we let [[a, b]] denote the set of integers between a and b, i.e., [[a, b]] = {n ∈ Z | a ≤ n ≤ b}. In addition, for S ⊆ R and r ∈ R, we set S≥r = {s ∈ S | s ≥ r } and S>r = {s ∈ S | s > r }.

2.2 Factorizations Although a monoid is usually defined to be a semigroup with an identity element, here we will additionally assume that all monoids are cancellative and commutative. Let M be a monoid. We say that M is torsion-free provided that for all a, b ∈ M, if a n = bn for some n ∈ N, then a = b. The quotient group gp(M) of a monoid M is the set of quotients of elements in M (i.e., the unique abelian group gp(M) up to isomorphism satisfying that any abelian group containing a homomorphic image of M will also contain a homomorphic image of gp(M)). The group of invertible elements of M is denoted by U (M). The monoid M is reduced if |U (M)| = 1. An element a ∈ M \U (M) is an irreducible (or an atom) if whenever a = uv for some u, v ∈ M, then either u ∈ U (M) or v ∈ U (M). The set of irreducibles of M is denoted by I (M). The monoid M is atomic if every non-invertible element factors

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into irreducibles. A subset I of M is an ideal of M provided that I M = I (or, equivalently, I M ⊆ I ). The ideal I is principal if I = bM for some b ∈ M. The monoid M satisfies the ascending chain condition on principal ideals (or ACCP) if every ascending chain of principal ideals of M eventually stabilizes. It is clear that the monoid M is atomic if and only if its quotient monoid Mred = M/U (M) is atomic. Let Z (M) denote the free (commutative) monoid on I (Mred ), and let π : Z (M) → Mred be the unique monoid homomorphism fixing a for every a ∈ I (Mred ). If z = a1 · · · a ∈ Z (M), where a1 , . . . , a ∈ I (Mred ), then  is the length of z and is denoted by |z|. For every b ∈ M, we set Z (b) = Z M (b) = π −1 (bU (M)) and L(b) = L M (b) = {|z| | z ∈ Z (b)}. If M is atomic and |Z (b)| < ∞ for every b ∈ M, then we say that M is a finite factorization monoid (or an FFM). On the other hand, if M is atomic and |L(b)| < ∞ for every b ∈ M, then we say that M is a bounded factorization monoid (or a BFM). Clearly, every FFM is a BFM. The monoid M is a unique factorization monoid (or a UFM) if Z (b) is a singleton for every b ∈ M, and M is a half-factorial monoid (or an HFM) if L(b) is a singleton for every b ∈ M. It is clear that every UFM is both an FFM and an HFM and that every HFM is a BFM. Let R be an integral domain. We let R ∗ denote the multiplicative monoid of R, i.e., ∗ R = R \ {0}. We set Z (R) = Z (R ∗ ), and for every x ∈ R ∗ , we set Z (x) = Z R ∗ (x) and L(x) = L R ∗ (x). It is clear that R is a BFD (resp., an FFD, an HFD, or a UFD) if and only if R ∗ is a BFM (resp., an FFM, an HFM, or a UFM). As we did for monoids, we let U (R) and I (R) denote the group of units and the set of irreducibles of R, respectively. In addition, we let P(R) denote the set of primes of R. The quotient field of R is denoted by qf(R). An overring of R is a subring of qf(R) containing R. The abelian group qf(R)∗ /U (R), written additively, is the group of divisibility of R and is denoted by G(R). The group G(R) is partially ordered under the relation xU (R) ≤ yU (R) if and only if y ∈ x R; we let G(R)+ denote the monoid consisting of all the nonnegative elements of G(R). Even before we consider the bounded and finite factorization properties on monoid domains (in Sect. 6.3), many of the examples that we construct here will involve such rings. For an integral domain R and a monoid M, we let R[X ; M] denote the ring of polynomial expressions with coefficients in R and exponents in M. Following R. Gilmer [28], we will write R[M] instead of R[X ; M]. When M is torsion-free, R[M] is an integral domain by [28, Theorem 8.1] and the group of units of R[M] is U (R[M]) = {u X m | u ∈ U (R) and m ∈ U (M)} by [28, Theorem 11.1]. A detailed study of monoid rings is given by Gilmer in [28].

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3 Bounded and Finite Factorization Monoids In this section, we briefly present some basic results related to both the bounded and finite factorization properties in the abstract context of monoids. Diagram (1.1) also holds for the more general class consisting of monoids (see Diagram (3.1) below). UFM

HFM

FFM

BFM

ACCP monoid

atomic monoid

(3.1) The last two implications in Diagram (3.1) are the only ones that are not immediate from definitions. We argue these two implications in this section (Corollary 3.2 and Remark 3.8) and obtain, as a result, Diagram (3.1). As this survey focuses on integral domains, we will give a result in the context of monoids only if it is needed in Sects. 4–7.

3.1 The Bounded Factorization Property To begin with, we characterize BFMs in terms of the existence of certain “length functions”. Let M be a monoid. A function  : M → N0 is called a length function of M if it satisfies the following two properties: (i) (u) = 0 if and only if u ∈ U (M); (ii) (bc) ≥ (b) + (c) for every b, c ∈ M. The following characterization of a BFM will prove useful at several later points. Proposition 3.1 ([38, Theorem 1]) A monoid M is a BFM if and only if there is a length function  : M → N0 . Proof Suppose first that M is a BFM. Then define a function  : M → N0 by (b) = max L(b). Condition (i) in the definition of a length function follows immediately. In addition, it is clear that max L(bc) ≥ max L(b) + max L(c) for every b, c ∈ M, from which we obtain condition (ii). Conversely, suppose that  : M → N0 is a length function. Take b ∈ M \ U (M) such that b = a1 · · · am for some a1 , . . . , am ∈ M \ U (M), and set b j = a1 · · · a j for every j ∈ [[1, m]]. As (bm ) > (bm−1 ) > · · · > (b1 ), the inequality m ≤ (b) holds. Now observe that if we take m as large as it can possibly be, then the maximality of m guarantees that a1 , . . . , am ∈ I (M). Hence M is atomic. Since sup L(b) ≤ (b) for every b ∈ M \ U (M), we conclude that M is a BFM.  As we mentioned in the introduction, every BFD satisfies ACCP. This actually holds in the more general context of monoids, as the next corollary indicates.

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Corollary 3.2 ([38, Corollary 1]) If M is a BFM, then M satisfies ACCP. Proof By Proposition 3.1, there is a length function  : M → N0 . Suppose that (bn M)n∈N is an ascending chain of principal ideals of M. For every n ∈ N, the inclusion bn ∈ bn+1 M ensures that (bn ) ≥ (bn+1 ). Hence there is an n 0 ∈ N such that (bn ) = (bn+1 ) for every n ≥ n 0 . This implies that bn ∈ bn+1 U (M) for every  n ≥ n 0 , and so (bn M)n∈N must stabilize. Thus, M satisfies ACCP. The reverse implication of Corollary 3.2 does not hold in general. The following example, which is a fragment of [4, Example 2.1], corroborates our observation. Example 3.3 1 Let M be the additive submonoid of Q≥0 generated by the set {1/ p | p ∈ P}. It can be readily checked that A (M) = {1/ p | p ∈ P}. In addition, it is not hard to verify that for every q ∈ M, there is a unique N (q) ∈ N0and a unique sequence of nonnegative integers (c p (q)) p∈P such that q = N (q) + p∈P c p (q) 1p .  Set S(q) = p∈P c p (q). It is clear that if q ∈ q + M for some q ∈ M, then N (q ) ≤ N (q). Also, if q is a proper divisor of q in M, then N (q ) = N (q) ensures that S(q ) < S(q). Thus, every sequence (qn )n∈N in M satisfying that qn ∈ qn+1 + M for every n ∈ N must stabilize, and so M must satisfy ACCP. Finally, we can see that M is not a BFM because P ⊆ L(1). The bounded factorization property is inherited by those submonoids that preserve invertible elements. Proposition 3.4 ([38, Theorem 3]) Let M be a BFM. Then every submonoid N of M satisfying U (N ) = U (M) ∩ N is also a BFM. Proof Let N be a submonoid of M such that U (N ) = U (M) ∩ N . Since M is a BFM, there is a length function  : M → N0 by Proposition 3.1. As U (N ) = U (M) ∩ N , the equality (u) = 0 holds for u ∈ N if and only if u ∈ U (N ). This, along with the fact that (bc) ≥ (b) + (c) for every b, c ∈ N , guarantees that  is still a length function when restricted to N . Hence N is a BFM.  The reduced monoids in the following example will be useful later to construct monoid domains that are BFDs with further desired properties. Example 3.5 Let M be an additive submonoid of Q≥0 such that 0 is not a limit point of M \ {0}. Then it follows from [33, Proposition 4.5] that M is a BFM.

3.2 The Finite Factorization Property We now turn to give two characterizations of an FFM. To do so, we use Dickson’s Lemma, a standard result in combinatorics stating that for every k ∈ N, a subset of Nk0 contains only finitely many minimal elements under the usual product ordering. The atomicity and factorizations of additive submonoids of Q≥0 (known as Puiseux monoids) have been systematically studied in the last few years (see [19] for a recent survey).

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Proposition 3.6 ([38, Theorem 2 and Corollary 2]) Let M be a monoid. Then the following statements are equivalent. (a) M is an FFM. (b) Every element of M has only finitely many non-associate divisors. (c) M is atomic and every element of M is divisible by only finitely many nonassociate irreducibles. Proof We assume, without loss of generality, that M is reduced. (a) ⇒ (b): Suppose that M is an FFM, and fix b ∈ M. If d is a divisor of b in M, then every factorization of d is a subfactorization of some factorization of b. This, together with the fact that Z (b) is finite, implies that b has only finitely many divisors in M. (b) ⇒ (c): Assume that every element of M has only finitely many divisors. Note that M must satisfy ACCP by Corollary 3.2. Suppose, by way of contradiction, that M is not atomic. Then the set S consisting of all the elements of M that do not factor into irreducibles is nonempty. Since M satisfies ACCP and S is nonempty, there is a b ∈ S such that the ideal bM is maximal among all principal ideals of M generated by elements of S. Because b ∈ S, there are b1 , b2 ∈ M \ U (M) with b = b1 b2 such that b1 ∈ S or b2 ∈ S. So bM is strictly contained in either b1 M or b2 M, which contradicts the maximality of bM. Thus, M is atomic. The second part of the statement is clear. (c) ⇒ (a): Suppose that M is atomic and every element of M is divisible by only finitely many irreducibles. Take b ∈ M, and let Ab be the set of irreducibles in M dividing b. Because Z (b) is a subset of the finite-rank free commutative monoid F on Ab , it follows from Dickson’s Lemma that Z (b) has only finitely many minimal elements with respect to the order induced by division in F. This, along with the fact that any two factorizations in Z (b) are incomparable as elements of F, implies that |Z (b)| < ∞. Thus, M is an FFM.  As a consequence of Proposition 3.6, finitely generated monoids are FFMs. Corollary 3.7 Every finitely generated monoid is an FFM. Proof It suffices to prove the corollary for reduced monoids. Let M be a finitely generated reduced monoid that is minimally generated by a1 , . . . , am . It readily follows that I (M) = {a1 , . . . , am }, and therefore, M is atomic. Thus, M is an FFM by Proposition 3.6.  In the proof of Proposition 3.6, we have incidentally argued the following remark. Remark 3.8 Every monoid satisfying ACCP is atomic. In contrast to what we have already seen for BFMs, an FFM M can have a submonoid N satisfying U (N ) = U (M) ∩ N that is not an FFM.

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Example 3.9 Let M be the additive monoid Z × N0 . Then it is easy to verify that M is atomic with U (M) = Z × {0} and I (M) = Z × {1}. Since Mred ∼ = N0 , the monoid M is a UFM and, in particular, an FFM. Now consider the submonoid N = {(0, 0)} ∪ (Z × N) of M. Note that N is reduced with I (N ) = Z × {1}. As a result, U (N ) = U (M) ∩ N . However, N is not an FFM as (0, 2) = (−n, 1) + (n, 1) for every n ∈ N. We record the following proposition, whose proof is straightforward. Proposition 3.10 ([38, Corollary 3]) Every submonoid of a reduced FFM is an FFM. To conclude this section, we give some examples of FFMs that will be used later to construct monoid domains that are FFDs and have further algebraic properties. Example 3.11 Let (qn )n∈N be an increasing sequence of positive rational numbers, and consider the additive submonoid M = qn | n ∈ N of Q≥0 . It is not hard to argue that M is an FFM; indeed, it follows from [33, Theorem 5.6] that any additive submonoid of the nonnegative cone of an ordered field F is an FFM provided that such a monoid can be generated by an increasing sequence of F.

4 Bounded and Finite Factorization Domains In this section, we provide characterizations and give various examples and classes of BFDs and FFDs.

4.1 Characterizations of BFDs and (Strong) FFDs There are several other useful ways to rephrase what it means for an integral domain to be a BFD. The following proposition illustrates this observation. Proposition 4.1 ([4, Theorem 2.4]) The following statements are equivalent for an integral domain R. (a) R is a BFD. (b) There is a length function  : R ∗ → N0 . (c) For every x ∈ R ∗ , there is a positive integer n such that every (strictly) ascending chain of principal ideals starting at x R has length at most n. (d) For every x ∈ G(R)+ , there is a positive integer n such that x is the sum of at most n (minimal) positive elements in G(R)+ .

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Proof (a) ⇔ (b): This is a direct consequence of Proposition 3.1. ∗ . As a result, for (a) ⇔ (d): It is clear that G(R)+ = {xU (R) | x ∈ R ∗ } = Rred ∗ every x ∈ R \ U (R), the set L(x) has an upper bound n ∈ N if and only if x is the sum of at most n positive elements in G(R)+ . (b) ⇒ (c): Let  : R ∗ → N0 be a length function. Take an x ∈ R ∗ and set n = (x). Let x0 R  x1 R  · · ·  xk R be a strictly ascending chain of principal ideals of R such that x0 = x. It is clear that x0 , . . . , xk are pairwise non-associates in R, and also that xi−1 ∈ xi R ∗ for every i ∈ [[1, k]]. Therefore n = (x0 ) > (x1 ) > · · · > (xk ). This implies that the length of x0 R  x1 R  · · ·  xk R is at most n. (c) ⇒ (b): Define the function  : R ∗ → N0 by taking (x) to be the smallest n ∈ N0 such that every ascending chain of principal ideals of R starting at x R has length at most n. If x ∈ U (R), then x R = R and so (x) = 0. In addition, if for x0 , y0 ∈ R, we take two ascending chains of principal ideals x0 R  x1 R  · · ·  x j R and y0 R  y1 R  · · ·  yk R, then the ascending chain of principal ideals x0 y0 R  x1 y0 R  · · ·  x j y0 R ⊆ y0 R  y1 R  · · ·  yk R starts at x0 y0 R and has length at least j +  k. Hence (x y) ≥ (x) + (y) for all x, y ∈ R ∗ . Thus,  is a length function. The elasticity,2 introduced by R. Valenza [48] in the context of algebraic rings of integers, is an arithmetic invariant that allows us to measure how far an atomic integral domain is from being an HFD. Given an atomic integral domain R, its elasticity is defined as follows:    sup L(x)  ∗ x ∈ R \ U (R) ρ(R) = sup min L(x)  when R is not a field, and ρ(R) = 1 when R is a field. Clearly, 1 ≤ ρ(R) ≤ ∞ and ρ(R) = 1 if and only if R is an HFD. For a survey on the elasticity of integral domains, see [11]. Following [1], we say that R is a rational bounded factorization domain (or an RBFD) if R is atomic and ρ(R) < ∞. Observe that HFD ⇒ RBFD ⇒ BFD. Moreover, none of these implications is reversible and being an FFD does not imply being an RBFD. The following example sheds some light upon these observations.  Example 4.2 For every r ∈ R≥1 {∞}, [1, Theorem 3.2] guarantees the existence of a Dedekind domain (with torsion divisor class group) whose elasticity is r . Let D1 be a Dedekind domain such that ρ(D1 ) = 3/2. Since ρ(D1 ) > 1, the domain D1 is not an HFD. Thus, not every RBFD is an HFD. On the other hand, let D2 be a Dedekind domain such that ρ(D2 ) = ∞. As we will see in Corollary 4.16, every Dedekind domain is an FFD. As a result, D2 is an FFD that is not an RBFD. Therefore not every BFD is an RBFD. Following A. Grams and H. Warner [37], we say that an integral domain R is an idf-domain if every nonzero element of R has at most finitely many non-associate irreducible divisors. We next give several useful characterizations of an FFD. 2

Although R. Valenza coined the term elasticity and introduced it in the context of algebraic rings of integers, it is worth noting that J. L. Steffan [47] also studied elasticity about the same time in the more general context of Dedekind domains.

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Proposition 4.3 ([4, Theorem 5.1] and [10, Theorem 1]) The following statements are equivalent for an integral domain R. (a) (b) (c) (d)

R is an FFD. R is an atomic idf-domain. Every element of R ∗ has only finitely many non-associate divisors. Every nonzero principal ideal of R is contained in only finitely many principal ideals.  (e) For any infinite family {xi R | i ∈ I } of principal ideals, i∈I xi R = {0}. (f) For every xU (R) ∈ G(R)+ , the interval [0, xU (R)] of the ordered monoid G(R)+ is finite. Proof (a) ⇔ (b) ⇔ (c): It follows directly from Proposition 3.6. (c) ⇔ (d): It is clear as, for all x, y ∈ R ∗ , it follows that x divides y in R ∗ if and only if y R ⊆ x R. (d) ⇔ (e): This is straightforward. ∗ , we only need to observe that, for every y ∈ R ∗ , (c) ⇔ (f): Since G(R)+ = Rred  the inclusion yU (R) ∈ [0, xU (R)] holds if and only if y divides x in R ∗ . Remark 4.4 Graph-theoretic characterizations of an FFD have been given by J. Coykendall and J. Maney in [25, Proposition 3.1] and, more recently, by J. D. LaGrange in [44, Theorem 13]. Following D. D. Anderson and B. Mullins [10], we say that an integral domain R is a strong finite factorization domain (or an SFFD) if every nonzero element of R has only finitely many divisors, and we say that R is a strong idf-domain if every nonzero element of R has only finitely many divisors which are either units or irreducibles. We can characterize SFFDs as follows. Proposition 4.5 ([10, Theorem 5]) The following statements are equivalent for an integral domain R.

(a) R is an SFFD. (b) R is an atomic strong idf-domain. (c) R is an FFD and U (R) is finite. Proof (a) ⇒ (b): Consider the map  : R ∗ → N0 defined by letting (x) be the number of nonunit divisors of x in R. Clearly, (u) = 0 for every u ∈ U (R). If x, y ∈ R ∗ , then every nonunit divisor of x divides x y, and for every nonunit divisor d of y, we see that xd divides x y but does not divide x; whence (x y) ≥ (x) + (y). As a result,  is a length function. Since R is a BFD by Proposition 4.1, it must be atomic. In addition, it is clear that R is a strong idf-domain. (b) ⇒ (c): That R is an FFD follows from Proposition 4.3. In addition, U (R) is the set of divisors of 1, and therefore, it must be finite. (c) ⇒ (a): Since R is an FFD, every element of R ∗ has only finitely many nonassociate divisors. In addition, every element of R ∗ has only finitely many associates because U (R) is finite. Hence every element of R ∗ must admit only finitely many divisors. Thus, R is an SFFD. 

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Not every FFD is an SFFD; indeed, there are integral domains with all its subrings being FFDs that are not SFFD. The following example was given in [10, Remark 3]. Example 4.6 For p ∈ P and m ∈ N, let F pm be the finite field of cardinality p m . Since for every n ∈ N, the field F p2n contains a copy of F p2n−1 as a subfield, we  can consider the field F = n∈N0 F p2n . Although F is an infinite field, every proper subring of F is a finite field and so an SFFD. However, F is not an SFFD because |U (F)| = |F∗ | = ∞. Lastly, observe that every subring of F is a field, and therefore, an FFD.

4.2 Some Relevant Classes of BFDs and FFDs In this subsection, we identify some relevant classes of BFDs and FFDs. It is clear that every HFD is a BFD. Observe, however, that a BFD need not be an HFD; for instance, the BFD Q[X 2 , X 3 ] is not an HFD because (X 2 )3 = (X 3 )2 . Similarly, although every FFD is a BFD, there are BFDs that are not FFDs; indeed, the integral domain R + X C[X ] is a BFD (by Theorem 4.9) that is not an FFD (see Example 4.10). As we illustrate in the next example, for every q ∈ Q>0 , the monoid domain Q[Mq ], where Mq = {0} ∪ Q≥q , is a BFD that is neither an HFD nor an FFD. The monoid domain Q[M1 ] seems to be used first by Gilmer in [28, page 189] as an example of an integral domain satisfying ACCP with a localization not satisfying ACCP. The same monoid domain was used in [4, Example 2.7(a)] as an example of a one-dimensional BFD with a localization that is not a BFD (cf. Example 5.21). The fact that Q[M1 ] is a BFD that is not an FFD was implicit in [4, Example 4.1(b)] and later observed in [41, Example 3.26]. Example 4.7 For q ∈ Q>0 , let Mq denote the additive monoid {0} ∪ Q≥q . Note that Mq is one of the monoids in Example 3.5, and so it is a BFM. By a simple degree consideration, one can verify that the monoid domain Q[Mq ] is a BFD (cf. [32, Theorem 4.3(2)]). It is clear that I (Mq ) = [q, 2q) ∩ Q. Then for every n ∈ N with n > 2/q, both q + q2 + n1 and q + q2 − n1 are irreducibles in Mq and 3q = q + q2 +  1 + q + q2 − n1 . Since |Z (3q)| = ∞, the monoid Mq is not an FFM. Therefore n part (1) of Proposition 6.28 guarantees that Q[Mq ] is not an FFD. Finally, we check that Q[Mq ] is not an HFD. To do this, take q1 , q2 ∈ I (Mq ) such that q1 = q2 , and then write q1 = a1 /b1 and q2 = a2 /b2 for some a1 , a2 , b1 , b2 ∈ N such that gcd(a1 , b1 ) = gcd(a2 , b2 ) = 1. Observe that X a1 a2 = (X q1 )a2 b1 = (X q2 )a1 b2 . Since a2 b1 = a1 b2 , we see that (X q1 )a2 b1 and (X q2 )a1 b2 are factorizations of X a1 a2 with different lengths. Thus, Q[Mq ] is not an HFD (for a more general result in this direction, see [34, Theorem 4.4]). As a consequence of Corollary 3.2, every BFD satisfies ACCP. The reverse implication of this observation does not hold in general, as we proceed to illustrate with an example of a monoid domain first given in [4, Example 2.1].

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Example 4.8 We have seen in Example 3.3 that the additive monoid M = 1/ p | p ∈ P satisfies ACCP but is not a BFM. In addition, we have seen that I (M) = {1/ p | p ∈ P}. Now consider the monoid domain Q[M]. From the fact that M satisfies ACCP, we can easily argue that Q[M] also satisfies ACCP. However, Q[M] is not a BFD; indeed, for every p ∈ P, there is a length- p factorization of X , namely, X = (X 1/ p ) p . Noetherian domains are among the most important examples of BFDs. Theorem 4.9 ([4, Proposition 2.2]) Every Noetherian domain is a BFD. Proof Let R be a Noetherian domain, and take x ∈ R ∗ \ U (R). We know that there are only finitely many height-one prime ideals over x R in R, namely, P1 , . . . , Pn . By / the Krull Intersection Theorem, for every i ∈ [[1, n]], there is a ki ∈ N such that x ∈ Piki . Set k = max{ki | i ∈ [[1, n]]}. We claim that max L R (x) < kn. Suppose, by way of contradiction, that x = x1 · · · xm for some m ≥ kn and x1 , . . . , xm ∈ R \ U (R). Since x R contains a power of P1 · · · Pn , for every j ∈ [[1, m]] the inclusion x R ⊆ x j R ensures that x j ∈ P for some P ∈ {P1 , . . . , Pn }. Therefore there is an i ∈ [[1, n]] such that x ∈ Pik . However, this contradicts that x ∈ / Piki . Thus, the set of lengths of every nonzero nonunit of R is bounded, and so R is a BFD.  We will see that integrally closed Noetherian domains are FFDs in Corollary 4.16, and we will characterize Noetherian FFDs in Proposition 5.19. For now, it is worth noting that not every Noetherian domain is an FFD. Example 4.10 (cf. Propositions 4.23 and 6.19) Consider the integral domain R = R + X C[X ]. It is not hard to verify that R is a Noetherian domain, although it is a direct consequence of [15, Theorem 4]. For every p ∈ P, let ζ p be a primitive p-root of unity. Since U (R) = R∗ , it is clear that distinct primes yield non-associate primitive roots of unity. Then {(ζ p X )(ζ p−1 X ) | p ∈ P} is a set consisting of infinitely many factorizations of X 2 in R. Hence R is not an FFD. As a final note, observe that R is an HFD by [6, Theorem 5.3]. ∈ I } be a family of subrings of the same integral For a nonempty set I , let {Ri | i domain. The integral domain R = i∈I Ri is said to be the locally finite intersection / U (Ri )} is finite. As the next of the Ri ’s if for every x ∈ R ∗ , the set {i ∈ I | x ∈ proposition illustrates, we can produce BFDs by taking locally finite intersections of BFDs. Proposition 4.11 ([4, p. 17]) For a nonempty set I , let {Ri | i ∈ I } be a family of subrings of an integral domain. If Ri is a BFD for every i ∈ I , then the locally finite intersection i∈I Ri is a BFD.  Proof Set R = i∈I Ri . By Proposition 4.1, for every i ∈ I , there is a length function i : Ri∗ → N0 . Since R is a locally finite intersection, the function  =  ∗ i∈I i : R → N0 is well defined. From the definition of , it immediately follows that  is a length function. As a result, Proposition 4.1 guarantees that R is a BFD. 

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We proceed to identify some relevant classes of FFDs. It is clear that every UFD is an FFD, but it is not hard to verify that Q[X 2 , X 3 ] (resp., (Z/2Z)[X 2 , X 3 ]) is an FFD (resp., an SFFD) that is not even an HFD (in Example 4.10, we have seen an HFD that is not an FFD). A Cohen-Kaplansky domain (or a CKD) is an atomic domain with finitely many non-associate irreducibles. These integral domains were first investigated by I. S. Cohen and I. Kaplansky in [21] and then by D. D. Anderson and J. L. Mott in [9]. It follows from Proposition 4.3 that every CKD is an FFD. By Theorem 4.9, every Noetherian domain is a BFD. It turns out that every onedimensional Noetherian domain is an FFD provided that each of its residue fields is finite. Proposition 4.12 ([10, Example 1]) Every one-dimensional Noetherian domain whose residue fields are finite is an FFD. Proof Let R be a one-dimensional Noetherian domain whose residue fields are finite. It follows from [45, Theorem 2.7] that R/I is finite for every nonzero proper ideal I of R. Fix x ∈ R ∗ \ U (R). Clearly, two distinct principal ideals y R and y R of R containing the ideal x R yield distinct subgroups y R/x R and y R/x R of the additive group R/x R. Since |R/x R| < ∞, the principal ideal x R can only be contained in finitely many principal ideals of R. Hence it follows from Proposition 4.3 that R is an FFD.  Throughout this survey, an integral domain is said to be quasilocal if it has exactly one maximal ideal, while it is said to be local if it is Noetherian and quasilocal. The following corollary, which is a direct consequence of Proposition 4.12, was first observed in [10]. Corollary 4.13 Every one-dimensional local domain with finite residue field is an FFD. Let R be a one-dimensional local domain with maximal ideal M. Since by Corollary 4.13 we know that R is an FFD provided that R/M is finite, we may wonder what happens in the case where R/M is infinite. Under the assumption that R/M is infinite, it follows that R is an FFD if and only if R is integrally closed; this is [10, Corollary 4]. As for BFDs, we can produce new FFDs by considering locally finite intersections of FFDs. Proposition 4.14 ([10, Theorem 2]) For a nonempty set I , let {Ri | i ∈ I } be a family of subrings of an integral domain. If Ri is an FFD for every i ∈ I , then the  locally finite intersection i∈I Ri is an FFD.  Proof Set R = i∈I Ri . Take a nonunit x ∈ R ∗ , and set J = {i ∈ I | x ∈ / U (Ri )}. It follows from Proposition 4.3 that, for every j ∈ J , the ideal x R j is contained in only finitely many principal ideals of R j . We claim that the ideal x R is contained in only finitely many principal ideals of R. To verify this, take y ∈ R such that x R ⊆ y R. It / U (R j ) implies that j ∈ J . is clear that x Ri ⊆ y Ri for every i ∈ I . As a result, y ∈ Since J is finite, x R is contained in only finitely many principal ideals of R. Thus, using Proposition 4.3 once again, we conclude that R is an FFD. 

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An integral domain is called a valuation domain if any two principal ideals are comparable. In particular, DVRs are valuation domains. An integral domain R has finite character if there is a family {Vi | i ∈ I } of valuation overrings of R indexed by a nonempty set I such that R = i∈I Vi is a locally finite intersection. An integral domain R is a Krull domain if its localization at any height-one prime ideal is a DVR and R has finite character with respect to the family consisting of such DVRs. Dedekind domains are the Krull domains of dimension 1. The class of Krull domains is an important source of FFDs. Theorem 4.15 ([4, Proposition 2.2] and [37, Proposition 1]) Every Krull domain is an FFD, and thus a BFD. Proof Let R be a Krull domain. Mimicking the proof of Theorem 4.9, we can show that R is a BFD, and therefore, an atomic domain. Now let X be the set of all heightone prime ideals of R. Then R has finite character with respect to the family of DVRs {R P | P ∈ X }. As a result, it follows from [37, Proposition 1] that R is an idf-domain. Since R is an atomic idf-domain, Proposition 4.3 guarantees that R is an FFD. Finally, we observe that Proposition 4.14 can be used to give an alternative proof.  Corollary 4.16 Integrally closed Noetherian domains and, in particular, Dedekind domains and rings of algebraic integers are FFDs.

4.3 The D + M Construction This subsection is devoted to the D + M construction, which is a rich source of examples in commutative ring theory. Let T be an integral domain, and let K and M be a subfield of T and a nonzero maximal ideal of T , respectively, such that T = K + M. For a subdomain D of K , set R = D + M. This construction was introduced and studied by Gilmer [29, Appendix II] for valuation domains, and then it was investigated by J. Brewer and E. A. Rutter [15] for arbitrary integral domains. To begin with, we consider units and irreducibles in D + M. Recall that an integral domain is quasilocal if it has a unique maximal ideal. When we work with the D + M construction, we will often denote an element of T by α + m, tacitly assuming that α ∈ K and m ∈ M. Lemma 4.17 Let T be an integral domain, and let K and M be a subfield of T and a nonzero maximal ideal of T , respectively, such that T = K + M. For a subdomain D of K , set R = D + M. Then the following statements hold. (1) U (R) = U (T ) ∩ R if and only if D is a field. (2) If T is quasilocal, then U (T ) = {α + m | α ∈ K ∗ } and U (R) = {α + m | α ∈ U (D)}.

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Proof (1) For the direct implication, take α ∈ D ∗ . As α ∈ K ∗ , it follows that α ∈ U (T ) ∩ R = U (R), and so α −1 ∈ K ∩ R = D. Hence D is a field. Conversely, assume that D is a field. It is clear that U (R) ⊆ U (T ) ∩ R. To argue the reverse inclusion, take α + m 1 ∈ U (T ) ∩ R, and then let β + m 2 be the inverse of α + m 1 in T . Since α belongs to the field D and (α + m 1 )(β + m 2 ) = 1, we see that β = α −1 ∈ D. Therefore β + m 2 ∈ R, and so α + m 1 ∈ U (R). Thus, U (T ) ∩ R ⊆ U (R). (2) The first equality is an immediate consequence of the fact that in a quasilocal domain the units are precisely the elements outside the unique maximal ideal. To check the second equality, take α + m 1 ∈ U (R) and let β + m 2 ∈ R be the inverse of α + m 1 in R. As in the previous part, α −1 = β ∈ D, and so α ∈ U (D). Conversely, any α + m 1 ∈ R with α ∈ U (D) is a unit in T , and its inverse β + m 2 is such that  β ∈ D, whence α + m 1 ∈ U (R). Lemma 4.18 Let T be an integral domain, and let K and M be a subfield of T and a nonzero maximal ideal of T , respectively, such that T = K + M. For a subdomain D of K , set R = D + M. Then I (R) ⊆ U (T ) ∪ I (T ). Moreover, the following statements hold. (1) If D is a field, then I (R) ⊆ I (T ). (2) If T is quasilocal and D is a field, then I (R) = I (T ) ⊆ M. Proof Take a = d + m in I (R), where d ∈ D and m ∈ M. If m = 0, then a ∈ D ∗ ⊆ U (T ). So assume that m = 0. Take x, y ∈ T such that a = x y. If d = 0, then either x ∈ M or y ∈ M. Assume that x ∈ M and write a = (k −1 x)(ky) for some k ∈ K ∗ such that ky ∈ R. Because a is irreducible in R, either k −1 x ∈ U (R) or ky ∈ U (R). Since x ∈ M, it follows that ky ∈ U (R) ⊆ U (T ). Thus, y ∈ U (T ), and so a ∈ I (T ). If d = 0, then there are k1 , k2 ∈ K ∗ with k1 k2 = d and m 1 , m 2 ∈ M such that x = k1 (1 + m 1 ) and y = k2 (1 + m 2 ). As a = d(1 + m 1 )(1 + m 2 ), either d(1 + m 1 ) ∈ U (R) ⊆ U (T ) or 1 + m 2 ∈ U (R) ⊆ U (T ). Hence either x or y (or both) belongs to U (T ), and so a ∈ U (T ) ∪ I (T ). As a consequence, I (R) ⊆ U (T ) ∪ I (T ). (1) If D is a field, then it follows from part (1) of Lemma 4.17 that I (R) ∩ U (T ) is empty. This, along with I (R) ⊆ U (T ) ∪ I (T ), implies the desired inclusion. (2) By part (1), I (R) ⊆ I (T ), and by part (2) of Lemma 4.17, I (T ) ⊆ M. As I (T ) ⊆ R, if an irreducible a in T factors as a = x y for x, y ∈ R, then it follows from part (1) of Lemma 4.17 that either x ∈ U (T ) ∩ R = U (R) or y ∈ U (T ) ∩ R = U (R), and so a ∈ I (R). Thus, I (T ) ⊆ I (R).  Remark 4.19 With the notation as in Lemma 4.18, the inclusion I (R) ⊆ U (T ) ∪ I (T ) may be proper. For instance, taking R = Z + X Q[X ] and T = Q[X ], we can see that 4 ∈ U (T ) \ I (R) and X + 2 ∈ I (T ) \ I (R). Moreover, the inclusion I (R) ⊆ I (T ) may be proper even when D is a field. To see this, it suffices to take R = Q + X R[X ] and T = R[X ], and then observe that X + π ∈ I (T ) \ I (R). We proceed to examine when the D + M construction yields BFDs and FFDs.

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Proposition 4.20 ([4, Proposition 2.8]) Let T be an integral domain, and let K and M be a subfield of T and a nonzero maximal ideal of T , respectively, such that T = K + M. For a subdomain D of K , set R = D + M. Then R is a BFD if and only if T is a BFD and D is a field. Proof For the direct implication, suppose that R is a BFD. Assume, by way of contradiction, that D is not a field, and take d ∈ D ∗ such that d −1 ∈ / D. In this case, d ∈ / U (R), and for every m ∈ M the decomposition m = d(d −1 m) ensures that m ∈ / I (R). Then no element of M \ {0} factors into irreducibles, contradicting that R is atomic. Thus, D must be a field. We proceed to argue that T is a BFD. Fix x ∈ T ∗ \ U (T ), and take k ∈ K ∗ such that xk −1 ∈ R. As R is atomic, xk −1 factors into irreducibles in R, and so Lemma 4.18 ensures that x factors into irreducibles in T . Therefore T is atomic. We can readily check that for every m ∈ M, the element m (resp., 1 + m) is irreducible in R if and



only if m (resp., 1 + m) is irreducible in T . Suppose that xk −1 = ri=1 m i sj=1 (α j + m j ) for irreducibles m 1 , . . . , m r ∈ M and α1 + m 1 , . . . , αs + m s ∈ K ∗ + M of T .

Set α = sj=1 α j . If r = 0, then α ∈ D and so xk −1 factors as a product of r + s irre

ducibles in R as xk −1 = α(1 + α1−1 m 1 ) sj=2 (1 + α −1 0, then xk −1 still j m j ). If r >



r factors as a product of r + s irreducibles in R as xk −1 = (αm 1 ) i=2 m i sj=1 (1 + −1 −1 α −1 j m j ). Hence L T (x) = L T (xk ) ⊆ L R (xk ). Thus, T is also a BFD. For the reverse implication, suppose that T is a BFD and D is a field. Fix x ∈ R ∗ \

r U (R), and write x = i=1 m i sj=1 (α j + m j ) for irreducibles m 1 , . . . , m r ∈ M and

α1 + m 1 , . . . , αs + m s ∈ K ∗ + M of T . As before, set α = sj=1 α j . Observe that

if r = 0, then α ∈ R, and so x factors into irreducibles in R as x = α sj=1 (1 +

r α −1 j m j ). If r > 0, then x still factors into irreducibles in R as x = (αm 1 ) i=2 m i

s −1 j=1 (1 + α j m j ). Hence R is atomic. Finally, observe that since D is a field, the inclusion I (R) ⊆ I (T ) holds by Lemma 4.18. This guarantees that L R (x) ⊆  L T (x). Thus, R is also a BFD. Remark 4.21 With the notation as in Proposition 4.20, we have also proved that R is atomic if and only if T is atomic and D is a field. The same assertion holds if one replaces being atomic by satisfying ACCP (see [4, Proposition 1.2]). Two of the most important special cases of the D + M construction are the following. Example 4.22 Let F1  F2 be a proper field extension. (1) Consider the subring R1 = F1 + X F2 [X ] of the ring of polynomials F2 [X ]. Since F2 [X ] is a UFD, it is also a BFD. As X F2 [X ] is a nonzero maximal ideal of F2 [X ], it follows from Proposition 4.20 that R1 is a BFD. Observe that R1 is not a UFD because, for instance, X is an irreducible that is not prime. (2) On the other hand, consider the subring R2 = F1 + X F2 [[X ]] of the ring of power series F2 [[X ]]. As in the previous case, it follows from Proposition 4.20 that R2 is a BFD that is not a UFD.

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Finally, it is worth noting that certain ring-theoretic properties of R1 and R2 only depend on the field extension F1  F2 . For instance, both R1 and R2 are Noetherian if and only if [F2 : F1 ] < ∞ [15, Theorem 4], while both R1 and R2 are integrally closed if and only if F1 is algebraically closed in F2 (cf. [15, p. 35]). Now we give a result for FFDs that is parallel to Proposition 4.20. Proposition 4.23 ([4, Proposition 5.2]) Let T be an integral domain, and let K and M be a subfield of T and a nonzero maximal ideal of T , respectively, such that T = K + M. For a subdomain D of K , set R = D + M. Then R is an FFD if and only if T is an FFD, D is a field, and K ∗ /D ∗ is finite. Proof For the direct implication, suppose that R is an FFD. Since R is in particular a BFD, D must be a field by Proposition 4.20. We proceed to verify that K ∗ /D ∗ is finite. Take m ∈ M \ {0}. Note that in any factorization of m into irreducibles of R, one of the irreducibles must belong to M. After replacing m by such an irreducible, we can assume that m belongs to both I (R) and I (T ). Observe that for α, β ∈ K ∗ the elements αm and βm are irreducibles in both R and T , and they are associate elements in R if and only if α and β determine the same coset of K ∗ /D ∗ . On the other hand, the set {αm | α ∈ K ∗ } ⊆ R has only finitely many non-associate elements because it consists of divisors of m 2 in R. As a result, K ∗ /D ∗ is a finite group. By Proposition 4.3, proving that T is an FFD amounts to verifying that every x ∈ T has finitely many non-associate irreducible divisors. After replacing x by αx . . , xn form for some α ∈ K ∗ , we can assume that x ∈ R. Suppose that x1 , .

a maximal set of non-associate irreducible divisors of x in R. Let x = ri=1 m i sj=1 (α j + m j ) be a factorization of x into irreducibles of T , where α1 , . . . , αs ∈ K ∗ and m 1 , . . . , m r , m 1 , . . . , m s ∈ M. If x ∈ M, then r > 0, and therefore, the elements m 1 , . . . , m r and 1 + α1−1 m 1 , . . . , 1 + αs−1 m s are irreducible divisors of x in R. . , xn in T . On the other Hence they are associate to some of the elements x1 , . .

hand, if x ∈ / M, then r = 0. Therefore we can write x = α sj=1 (1 + α −1 j m j ), where

s α = j=1 α j ∈ D ∗ . In this case, the elements 1 + α1−1 m 1 , . . . , 1 + αs−1 m s are irreducible divisors of x in R, and as a result, they are associate to some of the elein T . So in any case, the irreducible factors on the right-hand side ments x

1 , . . . , x n

of x = ri=1 m i sj=1 (α j + m j ) are associate to some of the elements x1 , . . . , xn . Thus, x has finitely many non-associate irreducible divisors. For the reverse implication, take x ∈ R ∗ . We will verify that x has only finitely many non-associate irreducible divisors in R. Assume that x1 , . . . , xn ∈ R form a maximal set of non-associate irreducible divisors of x in T , and let α1 D ∗ , . . . , αm D ∗ be the cosets of D ∗ in K ∗ . If d ∈ R is an irreducible divisorof x  in R, then d ∈ I (T )  m n ∗ because D is a field, and therefore, d ∈ nj=1 x j K ∗ = i=1 j=1 αi x j D . Thus, every irreducible divisor of x in R is associate to some element in {αi x j | (i, j) ∈ [[1, m]] × [[1, n]]}. It now follows from Proposition 4.20 and Proposition 4.3 that R is an FFD.  Remark 4.24 With the notation as in Proposition 4.23, when D is a field it follows from Brandis’ Theorem [14] that K ∗ /D ∗ is finite if and only if K is finite or D = K .

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We conclude this section revisiting Example 4.22. Example 4.25 Let F1  F2 be a field extension, and set R1 = F1 + X F2 [X ] and R2 = F1 + X F2 [[X ]]. As with the properties of being Noetherian and integrally closed, whether R1 and R2 are FFDs only depends on the field extension F1  F2 . Indeed, because F2 [X ] and F2 [[X ]] are both FFDs, Proposition 4.23 guarantees that R1 and R2 are FFDs if and only if F2∗ /F1∗ is finite. Since F1 = F2 , it follows from Brandis’ Theorem [14] that F2∗ /F1∗ is finite if and only if F2 is finite. Finally, if F2 is finite, then |U (R1 )| = |F1∗ | < ∞, and it follows from Proposition 4.5 that R1 is, in fact, an SFFD.

5 Subrings, Ring Extensions, and Localizations In this section, we study when being a BFD (resp., an FFD) transfers from an integral domain to its subrings and overrings. We pay special attention to ring extensions by localization.

5.1 Inert Extensions Let A ⊆ B be a ring extension. Following Cohn [22], we call A ⊆ B an inert extension if x y ∈ A for x, y ∈ B ∗ implies that ux, u −1 y ∈ A for some u ∈ U (B). Let A ⊆ B be an inert extension of integral domains. Take x, y ∈ B such that x y = a ∈ I (A) \ U (B), and then write a = (ux)(u −1 y) for some u ∈ U (B) such that ux, u −1 y ∈ A. So either ux or u −1 y belongs to U (A), and therefore, either x or y belongs to U (B). Thus, a ∈ I (B). As a result, I (A) ⊆ U (B) ∪ I (B). We record this last observation, which was first given in [5, Lemma 1.1]. Remark 5.1 If A ⊆ B is an inert extension of integral domains, then I (A) ⊆ U (B) ∪ I (B). As a result of the previous remark, one can easily check that if A ⊆ B is an inert extension of integral domains with U (A) = U (B) ∩ A, then I (A) = I (B) ∩ A. Example 5.2 Let R be an integral domain. It is clear that the extension R ⊆ R[X ] is inert. On the other hand, consider the extension R[X 2 ] ⊆ R[X ]. Clearly, U (R[X 2 ]) = U (R). Observe, in addition, that X X = X 2 ∈ R[X 2 ] even though uX ∈ / R[X 2 ] for any u ∈ U (R). Hence the extension R[X 2 ] ⊆ R[X ] is not inert. The extensions D ⊆ R = D + M and R ⊆ T = K + M in the D + M construction are both inert. Lemma 5.3 Let T be an integral domain, and let K and M be a subfield of T and a nonzero maximal ideal of T , respectively, such that T = K + M. For a subdomain D of K , set R = D + M. Then the extensions D ⊆ R and R ⊆ T are both inert.

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Proof First, we prove that D ⊆ R is inert. Take α1 + m 1 and α2 + m 2 in R with nonzero α1 and α2 such that (α1 + m 1 )(α2 + m 2 ) ∈ D. Then (α1 + m 1 )(α2 + m 2 ) = α1 α2 , and therefore, 1 + α1−1 m 1 and 1 + α2−1 m 2 are units in R that are inverses of each other. After setting u = 1 + α2−1 m 2 , we obtain u(α1 + m 1 ) = α1 ∈ D and u −1 (α2 + m 2 ) = α2 ∈ D. Hence the extension D ⊆ R is inert. To show that R ⊆ T is also inert, suppose that x y ∈ R for some x, y ∈ T , and write x = α1 + m 1 and y = α2 + m 2 . If α1 = α2 = 0, then ux ∈ R and u −1 y ∈ R for u = 1. So we assume that α1 = 0 or α2 = 0. If α1 = 0, then ux ∈ R and u −1 y ∈ R for u = α2 . Similarly, if α2 = 0, then ux ∈ R and u −1 y ∈ R for u = α1−1 . Finally, if α1 α2 = 0, then x y ∈ R implies that α1 α2 ∈ D, and so ux ∈ R and u −1 y ∈ R for  u = α2 . The following example shows that extensions by localization are not necessarily inert. Example 5.4 Let K be a field and consider the integral domain R = K [X 2 , X 3 ]. First, we observe that the subset S = {1, X 2 , X 3 , . . .} of R is a multiplicative set and R S = K [X, X −1 ]. In addition, U (R S ) = {α X n | α ∈ K ∗ and n ∈ Z}. Because (1 − X )(1 + X ) = 1 − X 2 ∈ R, in order for the extension R ⊆ R S to be inert, there must be an integer n such that X n (1 − X ) ∈ R and X −n (1 + X ) ∈ R, which is not possible. Hence the extension R ⊆ R S is not inert. However, localizing at certain special multiplicative sets yields inert extensions. Following [5], we say that a saturated (i.e., divisor-closed) multiplicative set S of an integral domain R is a splitting multiplicative set if we can write every x ∈ R as x = r s for some r ∈ R and s ∈ S with r R ∩ s R = r s R for every s ∈ S. Lemma 5.5 ([5, Proposition 1.5]) Let R be an integral domain, and let S be a splitting multiplicative set of R. Then R ⊆ R S is an inert extension. Proof Take x, y ∈ R S such that x y = r ∈ R. As S is a splitting multiplicative set, there are a, b ∈ R and s, t, u, v ∈ S with x = ast −1 and y = buv −1 such that a R ∩ s R = as R and b R ∩ s R = bs R for every s ∈ S. Since absu = r tv ∈ b R ∩ vt R = bvt R, there is a c ∈ R satisfying asu = cvt. Taking w = u/v ∈ U (R S ), we see that wx = c and w−1 y = b, which are both in R. Hence the extension R ⊆ R S is inert.  Multiplicative sets generated by primes are always saturated. The next proposition characterizes the multiplicative sets generated by primes that are splitting multiplicative sets. Lemma 5.6 ([5, Proposition 1.6]) Let R be an integral domain, and let S be a multiplicative set of R generated by primes. Then the following statements are equivalent. (a)  S is a splitting multiplicative set. n (b) p R = n∈N n∈N pn R = {0} for every prime p ∈ S and every sequence ( pn )n≥1 of non-associate primes in S.

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(c) For every nonunit x ∈ R ∗ , there is an n x ∈ N such that x ∈ p1 · · · pn R for p1 , . . . , pn ∈ S implies that n ≤ n x . Proof (b) ⇔ (c): It follows easily. (a) ⇒ (c): Take a nonunit x ∈ R ∗ . As S is generated by primes, we can write x = r p1 · · · pn x for some n x ∈ N and (possibly repeated) primes p1 , . . . , pn x ∈ S such that r R ∩ s R = r s R for every s ∈ S. Observe that none of the primes p in S divides r as, otherwise, r p R = r R ∩ p R = r R, which would imply that p is a unit. As a result, if x ∈ p1 · · · pn R for some p1 , . . . , pn ∈ S, then n ≤ n x . (c) ⇒ (a): Fix a nonunit x ∈ R ∗ , and then take the smallest n x ∈ N among those satisfying (c). So there are (possibly repeated) primes p1 , . . . , pn x ∈ S such that s = p1 · · · pn x ∈ S divides x. Take a ∈ R such that x = as. It is clear that no prime in S can divide a. Now if y = ar ∈ a R ∩ s R for some r ∈ R and s ∈ S, then the fact that S is generated by primes (none of them dividing a) guarantees that s divides r , and so y ∈ as R. Thus, a R ∩ s R = as R for every s ∈ S, and we conclude that S is a splitting multiplicative set.  Corollary 5.7 ([5, Corollary 1.7]) Let R be an atomic domain. Then every multiplicative set of R generated by primes is a splitting multiplicative set. Proof Let S be a multiplicative set of R generated by primes. Suppose that x is a nonunit in R ∗ . Because R is atomic, x = a1 · · · an for some a1 , . . . , an ∈ I (R). If x ∈ p1 · · · pm R for some primes p1 , . . . , pm ∈ S, then there exists a permutation σ : [[1, n]] → [[1, n]] such that pi and aσ (i) are associates in R for every i ∈ [[1, m]]. As a result, m ≤ n. It then follows from Lemma 5.6 that S is a splitting multiplicative set.  In general, multiplicative sets generated by primes are not always splitting multiplicative sets. On the other hand, there are splitting multiplicative sets that are not generated by primes. The following examples, which can be found in [5, Example 1.8] confirm these observations. Example 5.8 (a) Let R be a two-dimensional valuation domain with height-one prime ideal P and principal maximal ideal  M = p R. In addition, let S be the multiplicative set of R generated by p. Since n∈N p n R = P = {0}, it follows from Lemma 5.6 that S is not a splitting multiplicative set. Finally, note that p can be chosen so that V [1/ p] is a DVR. (b) Consider the integral domain R = Z + X Q[[X ]], and set S = Z∗ . It is clear that S is a multiplicative set generated by primes and R S = Q[[X ]]. However, 5.6 guarantees that S is not a splitting multiplicative set because  Lemma n n∈N p R = X Q[[X ]]  = {0} for every prime p in S and n∈N pn R = X Q[[X ]] = {0} for every sequence ( pn )n∈N of non-associate primes in S.

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(c) Let R be a GCD-domain that is not a UFD (for instance, a non-discrete onedimensional valuation domain), and consider the integral domain R[X ]. It is clear that S = R ∗ is a multiplicative set of R[X ]. Since R is a GCD-domain, every p(X ) ∈ R[X ]∗ can be written as c( p) p (X ), where c( p) ∈ S is the content of p(X ) and p (X ) ∈ R[X ] has content 1. For s ∈ S, take p (X )q(X ) ∈ p (X )R[X ] ∩ s R[X ], and note that c(q) ∈ s R by Gauss’ Lemma. This implies that p (X )q(X ) ∈ sp (X )R[X ], and so p (X )R[X ] ∩ s R[X ] = sp (X )R[X ]. As a result, S is a splitting multiplicative set. Finally, observe that S cannot be generated by primes because R is not a UFD. As for the case of splitting multiplicative sets, multiplicative sets generated by primes yield inert extensions. Lemma 5.9 ([5, Proposition 1.9]) Let R be an integral domain, and let S be a multiplicative set of R generated by primes. Then R ⊆ R S is an inert extension. Proof Take x, y ∈ R S∗ such that x y ∈ R. Now write x = a( p1 · · · pm )−1 and y = b(q1 · · · qn )−1 for elements a, b ∈ R and primes p1 , . . . , pm , q1 , . . . , qn in S such that / q j R for any j ∈ [[1, n]]. Because x y ∈ R, it fola∈ / pi R for any i ∈ [[1, m]] and b ∈ lows that b ∈ p1 · · · pm R and a ∈ q1 · · · qn R. Then after setting u = p1 · · · pm (q1 · · · qn )−1 ∈ U (R S ), we see that ux = a(q1 · · · qn )−1 ∈ R and u −1 y = b( p1 · · · pm )−1 ∈  R. Thus, R ⊆ R S is an inert extension. Some of the results we will discuss in the next two subsections have been generalized by D. D. Anderson and J. R. Juett in [8] to the context of inert extensions A ⊆ B of integral domains that satisfy U (A) = U (B) ∩ A and B = AU (B).

5.2 Subrings In general, the properties of being a BFD or an FFD are not inherited by subrings. Example 5.10 Let A = Z + X Q[X ] ⊆ B = Q[X ]. Since B is a UFD, it is in particular a BFD and an FFD. However, as Z is not a field, it follows from Proposition 4.20 that A is neither a BFD nor an FFD. The following proposition is an immediate consequence of Proposition 3.4. Proposition 5.11 ([4, p. 9]) Let A ⊆ B be an extension of integral domains with U (A) = U (B) ∩ A. If B is a BFD, then A is also a BFD. In particular, if an integral extension of an integral domain A is a BFD, then A is also a BFD. As the following example shows, the converse of Proposition 5.11 does not hold. Example 5.12 Consider the extension of integral domains A = Q[X ] ⊆ B = Q[M], where M is the additive submonoid of Q≥0 generated by the set {1/ p | p ∈ P}. Since M is a reduced monoid, U (A) = Q∗ = U (B) = U (B) ∩ A. It is clear that A is a BFD. However, we have seen in Example 4.8 that B is not a BFD.

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By strengthening the hypothesis of Proposition 5.11, we can obtain a version for FFDs. Proposition 5.13 ([10, Theorem 3]) Let A ⊆ B be an extension of integral domains. Suppose that (U (B) ∩ qf(A)∗ )/U (A) is finite. If B is an FFD, then A is also an FFD. Proof Take x ∈ A∗ , and let D ⊆ A∗ be the set of divisors of x in A∗ . Since B is an FFD, Proposition 4.3 ensures that x has only m finitely many non-associate xi U (B). Because (U (B) ∩ divisors in B, namely, x1 , . . . , xm . Clearly, D ⊆ i=1 qf(A)∗ )/U (A) is finite, only finitely many cosets gU (A) of U (B) ∩ qf(A)∗ inter(A). For each pair (i, j) ∈ [[1, m]] × [[1, n]], sect D. Let them be g1 U (A), . . . , gn U  take xi, j ∈ D such that xi, j ∈ xi U (B) g j U (A). Now take y ∈ A∗ with x ∈ y A. Since y ∈ D, there must be a pair (i, j) ∈ [[1, m]] × [[1, n]] such that y ∈ xi U (B) ∩ g j U (A) ⊆ qf(A), and so yxi,−1j ∈ U (B) ∩ qf(A) ∩ U (A) = U (A). Hence every divisor of x in A∗ is an associate of one of the elements xi, j . Thus, A is an FFD by Proposition 4.3.  The converse of Proposition 5.13 does not hold, as the next example illustrates. Example 5.14 Consider the extension of integral domains A = Q[X ] ⊆ B = Q[M], where M is the additive monoid {0} ∪ Q≥1 . Since M is reduced, U (A) = Q∗ = U (B) = U (B) ∩ qf(A)∗ . Also, A is an FFD because it is a UFD. However, we have already verified in Example 4.7 that the monoid domain B is not an FFD. Let R be an integral domain, and let S be a multiplicative set of R. The fact that R S satisfies either the bounded or the finite factorization property does not imply that R does. For instance, the quotient field of every (non-BFD) integral domain is trivially an FFD. The next “Nagata-type” theorem provides a scenario where the bounded and finite factorization properties are inherited by an integral domain from some special localization. Theorem 5.15 ([5, Theorems 2.1 and 3.1]) Let R be an integral domain, and let S be a splitting multiplicative set of R generated by primes. Then the following statements hold. (1) If R S is a BFD, then R is a BFD. (2) If R S is an FFD, then R is an FFD. Proof Assume first that R S is atomic. Fix a nonunit x ∈ R ∗ . As S is a splitting multiplicative set generated by primes, we can write x = r s for some r ∈ R and s ∈ S such that s is a product of primes and no prime in S divides r in R. Take a1 , . . . , an ∈ I (R S ) such that r = a1 · · · an . Since no prime in S divides r , we can assume that a1 , . . . , an ∈ I (R). Hence x can be written in R as a product of irreducibles. Thus, R is atomic. (1) Now assume that R S is a BFD. By the conclusion of the previous paragraph, x = a1 · · · an for some a1 , . . . , an ∈ I (R). Assume, without loss of generality, that there is a j ∈ [[0, n]] such that a j+1 , . . . , an are the elements among a1 , . . . , an that

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belong to S and, therefore, are primes. Set a = a1 · · · a j and s = a j+1 · · · an . Because R S is a BFD, there is an  ∈ N such that each factorization of a in R S has at most  irreducible factors. As each irreducible in I (R) \ S dividing a remains irreducible in R S , the set L R (a) is bounded by . Suppose now that x = b1 · · · bm is another factorization of x in R. Then there are exactly n − j irreducibles (counting repetitions) in b1 , . . . , bm that are primes in S; let them be bm−n+ j+1 , . . . , bm . Set b = b1 · · · bm−n+ j and t = bm−n+ j+1 · · · bm . It is clear that t R = s R, and so b R = a R. In particular, max L R (b) = max L R (a) ≤ , and therefore, max L R (x) ≤  + n − j. Thus, R is a BFD. (2) Finally, assume that R S is an FFD. Take a nonunit x ∈ R ∗ , and write x = r p1 · · · pn for primes p1 , . . . , pn ∈ S so that no prime in S divides r . Since R S is an idf-domain by Proposition 4.3, r has only finitely many irreducible divisors in R S , namely, a1 , . . . , am . As we did in the first paragraph, we can assume that a1 , . . . , am ∈ I (R). Now suppose that y ∈ I (R) divides x in R. Then either y is an associate of some of the primes in S or y ∈ I (R S ). In the latter case, sy = tai for some i ∈ [[1, m]] and s, t ∈ S. Then y and ai are associates if y is not an associate of some prime in S. As a result, a1 , . . . , am , p1 , . . . , pn account, up to associates, for all irreducible divisors of x in R. Hence R is an atomic idf-domain, and thus an FFD by Proposition 4.3.  Remark 5.16 Theorem 5.15 still holds if we replace BFD by either ACCP or UFD.

5.3 Ring Extensions and Overrings Let A ⊆ B be an extension of integral domains. Often B is not a BFD (resp., an FFD) even when A is a BFD (resp., an FFD) and U (A) = U (B) ∩ qf(A). Example 5.17 Consider the extension of integral domains A = R[X ] ⊆ B = R[Q≥0 ]. Because A is a UFD, it is, in particular, a BFD. On the other hand, B is not even atomic because the additive monoid Q≥0 is not atomic. Finally, we observe that U (A) = R∗ = U (B), from which the equality U (A) = U (B) ∩ qf(A) follows. However, if we require the ideal [A : A B] = {r ∈ A | r B ⊆ A} to be nonzero, then the property of being an FFD passes from A to B. Proposition 5.18 ([10, Theorem 4]) Let A ⊆ B be an extension of integral domains, and suppose that [A : A B] is nonzero. If A is an FFD, then the group U (B)/U (A) is finite and B is an FFD. Proof Let x be a nonzero nonunit in [A : A B]. Observe that for every u ∈ U (B), the fact that ux, u −1 x ∈ A implies that x 2 = (ux)(u −1 x) ∈ ux A. As A is an FFD, it follows from Proposition 4.3 that the set {ux A | u ∈ U (B)} is finite, and therefore, we can take u 1 , . . . , u n ∈ U (B) such that for every u ∈ U (B), the equality ux A = u i x A holds for some i ∈ [[1, n]]. Then for every u ∈ U (B), we can take i ∈ [[1, n]] and

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v ∈ U (A) such that u = u i v, whence uU (A) = u i vU (A) = u i U (A). As a result, the group U (B)/U (A) is finite. We proceed to argue that B is an FFD. As before, let 0 = x ∈ [A : A B]. Let b ∈ B, and suppose that y is a divisor of b in B. Then (x y)(x y ) = x 2 b for some y ∈ B, and both x y and x y belong to A. Therefore x 2 b A ⊆ x y A, and so xb A ⊆ y A. As A is an FFD and xb ∈ A, Proposition 4.3 guarantees that the set {y A | y divides b in B} is finite, and thus, the set {y B | y divides b in B} is also finite. As a consequence, B is an FFD.  Next we characterize Noetherian FFDs. Proposition 5.19 ([10, Theorem 6]) The following statements are equivalent for a Noetherian domain R. (a) R is an FFD. (b) If S is an overring of R that is a finitely generated R-module, then U (S)/U (R) is finite. (c) There is an FFD overring T of R that is integral over R such that if S is an intermediate domain of the extension R ⊆ T that is a finitely generated Rmodule, then U (S)/U (R) is finite. Proof (a) ⇒ (b): It follows from Proposition 5.18. (b) ⇒ (c): Take T to be the integral closure of R. Since R is a Noetherian domain, it follows from the Mori-Nagata Theorem that T is a Krull domain. As a consequence, it follows from Theorem 4.15 that T is an FFD. (c) ⇒ (a): Let T be an overring of R satisfying the conditions in (c). Suppose towards a contradiction that R is not an FFD, and take a nonunit r ∈ R ∗ with infinitely many non-associate divisors. Since every divisor of r in R is also a divisor of r in T , the fact that T is an FFD guarantees the existence of a sequence (rn )n∈N of nonassociate divisors of r in R such that r1 T = rn T for every n ∈ N. Let I be the ideal generated by the terms of the sequence (rn )n∈N . Since R is Noetherian, I is generated by r1 , . . . , rm for some m ∈ N. Consider the overring S = R[{r j r1−1 | j ∈ [[2, m]]}] of R. For every n ∈ N, the equality r1 T = rn T implies that rn r1−1 ∈ U (T ). Therefore S is an intermediate domain of the extension R ⊆ T . Because S is a finitely generated R-module, the group U (S)/U (R) is finite by (c). This, together with the fact that rn r1−1 ∈ S ∩ U (T ) = U (S) for every n ∈ N, ensures the existence of i, j ∈ N≥2 such that ri r1−1 U (R) = r j r1−1 U (R). However, this implies that ri U (R) = r j U (R), which is a contradiction.  Corollary 5.20 ([38, Theorem 7]) Let R be a Noetherian domain whose integral closure T is a finitely generated R-module. Then R is an FFD if and only if the group U (T )/U (R) is finite. Now we consider whether the bounded and finite factorization properties transfer from an integral domain to its extensions by localization. As the following example illustrates, such transfers do not happen in general.

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Example 5.21 Let R = R[M] be the monoid domain of M = 1 − 1/n | n ∈ N over R, and let S be the multiplicative set {X q | q ∈ M}. The generating sequence (1 − 1/n)n∈N is increasing, so M is one of the FFMs in Example 3.11. In addition, as M can be generated by an increasing sequence, it is not hard to argue that R[M] is an FFD (this is similar to the proof of [32, Theorem 4.3.3]). In particular, R[M] is a BFD. On the other hand, it follows from [26, Proposition 3.1] that gp(M) = Q, and therefore, R S = R[Q≥0 ] is not even atomic. Hence R S is not a BFD (or an FFD). If an extension of an integral domain by localization is inert, then both the bounded and finite factorization properties transfer. Theorem 5.22 ([5, Theorems 2.1 and 3.1]) Let R be an integral domain, and let S be a multiplicative set of R such that R ⊆ R S is an inert extension. Then the following statements hold. (1) If R is a BFD, then R S is a BFD. (2) If R is an FFD, then R S is an FFD. Proof Suppose first that R is atomic. Take a nonzero nonunit x in R S and write it as x = r/s, where r ∈ R and s ∈ S. Since R is atomic, there are a1 , . . . , an ∈ I (R) such that r = a1 · · · an . As the extension R ⊆ R S is inert, in light of Remark 5.1 we can assume that a1 , . . . , a j ∈ I (R S ) and a j+1 , . . . , an ∈ U (R S ) for some j ∈ [[0, n]], and so a1 · · · a j ∈ Z R S (x). Thus, R S is an atomic domain. (1) Assume now that R is a BFD. To argue that R S is indeed a BFD, suppose that the principal ideal x R S of R S is strictly contained in the principal ideal y R S . Assume, without loss of generality, that x, y ∈ R. Take r ∈ R and s ∈ S such that x = y(r/s). Since the extension R ⊆ R S is inert, there is a u ∈ U (R S ) such that uy and u −1 (r/s) are both elements of R. Setting y = uy, we see that x R = (uy)(u −1 (r/s))R  uy R, where the inclusion is strict because r/s ∈ / U (R S ). Hence x R is properly contained in uy R, and uy R S = y R S . Since R is a BFD, it follows from Proposition 4.1 that R S is also a BFD. (2) Finally, assume that R is an FFD. Take a nonzero nonunit r/s ∈ R S , and let r1 , . . . , rn form a maximal set of non-associate divisors of r in R. Let y ∈ R S be a divisor of r in R S , and write r = yy for some y ∈ R S . As the extension R ⊆ R S is inert, there is a u ∈ U (R S ) such that uy and u −1 y belong to R. Then y = u −1 vri for some i ∈ [[1, n]] and v ∈ U (R). As a result, r1 , . . . , rn form a maximal set of non-associate divisors of r/s in R S . Hence Proposition 4.3 guarantees that R S is an FFD.  Combining Theorem 5.22 and Lemmas 5.5 and 5.9, we obtain the following corollary. Corollary 5.23 ([5, Corollary 2.2]) Let R be an integral domain, and let S be a multiplicative set of R such that S is either generated by primes or a splitting multiplicative set. If R is a BFD (resp., an FFD), then R S is a BFD (resp., an FFD). Remark 5.24 Theorems 5.15 and 5.22 hold if we replace being a BFD by being an atomic domain, satisfying ACCP, or being a UFD (see [5, Theorems 2.1 and 3.1]).

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The algebraic closures of an integral domain are among the most useful and investigated overrings. We conclude this section illustrating that neither the bounded nor the finite factorization property ascend to the seminormal, integral, or complete integral closures. Example 5.25 Consider the submonoid M = (3/2)n | n ∈ N0  of Q≥0 . By [26, Propostion 3.1], the seminormal closure of M is M = Z[1/2] ∩ Q≥0 , where Z[1/2] denotes the localization of Z at the multiplicative set {2n | n ∈ N0 }. Now consider the monoid domain R = Q[M]. It follows from [32, Theorem 4.3] that R is an FFD (cf. = R = Q[M ], Example 5.21), while it follows from [32, Theorem 5.3] that R = R  where R , R, and R are the seminormal, root, and complete integral closures of R, respectively. Since M is not finitely generated, it follows from [26, Proposition 3.1] that M is antimatter (i.e., contains no irreducibles). Therefore X cannot be written  as a product of irreducibles in R . Then although R is an FFD (and so a BFD), R ( R or R) is not even atomic. We conclude this subsection with a few words about directed unions of integral domains in connection to the bounded and finite factorization properties. Recall that a partial order is a directed set if for all α, β ∈ , there is a θ ∈ such that α ≤ θ and β ≤ θ . A family (Rγ )γ ∈ of integral domains indexed by a nonempty directed set is called a directed  family of integral domains if Rα is a subring of Rβ whenever α ≤ β. In this case, γ ∈ Rγ is called the directed union of (Rγ )γ ∈ . As the next theorem states, the property of being a BFD (or an FFD) passes from the members of a directed family of integral domains to its directed union provided that every extension in the directed family is inert. Lemma 5.26 Let (Rγ )γ ∈ be a directed family of integral domains such that every extension Rα ⊆ Rβ is inert. If R is the directed union of (Rγ )γ ∈ , then the extension Rγ ⊆ R is inert for every γ ∈ . Proof Fix γ ∈ , and consider the extension Rγ ⊆ R. Take x, y ∈ R ∗ such that x y ∈ Rγ . Then x ∈ Rα and y ∈ Rβ for some α, β ∈ . Since (Rγ )γ ∈ is a directed family, there is a θ ∈ such that x, y ∈ Rθ and Rγ ⊆ Rθ . As the extension Rγ ⊆ Rθ is inert, ux, u −1 y ∈ Rγ for some u ∈ U (Rθ ) ⊆ U (R). Thus, the extension Rγ ⊆ R is also inert.  Theorem 5.27 ([5, Theorem 5.2]) Let (Rγ )γ ∈ be a directed family of integral domains such that every extension Rα ⊆ Rβ is inert. Then the following statements hold.  (1) If Rγ is a BFD for every γ ∈ , then the directed union γ ∈ Rγ is a BFD. (2) If Rγ is an FFD for every γ ∈ , then the directed union γ ∈ Rγ is an FFD.  Proof (1) Suppose that Rγ is a BFD for every γ ∈ , and set R = γ ∈ Rγ . Take a nonunit x ∈ R ∗ . Since x ∈ Rγ for some γ ∈ , and Rγ is atomic, there are a1 , . . . , an ∈ I (Rγ ) such that x = a1 · · · an . As the extension Rγ ⊆ R is inert by Lemma 5.26, it follows from Remark 5.1 that a1 , . . . , an are either irreducibles or units in R. Hence R is an atomic domain.

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Now take x, y ∈ R such that x R  y R, and write x = yr for some r ∈ R ∗ \ U (R). Since x ∈ Rα for some α ∈ and the extension Rα ⊆ R is inert, there is / U (R), we see that u −1 r ∈ / U (Rα ). a u ∈ U (R) with uy, u −1 r ∈ Rα . Because r ∈ −1 So y R = yu R and x Rα = yr Rα = yu(u r )Rα  yu Rα . Hence for any ascending chain of principal ideals of R starting at x R, we can construct an ascending chain of principal ideals of Rα starting at x Rα and having the same length. Since Rα is a BFD, it follows from Proposition 4.1 that R is also a BFD. (2) Now suppose that Rγ is an FFD for every γ ∈ . Fix x ∈ R, and take α ∈ such that x ∈ Rα . By Proposition 4.3, there is a largest (finite) list x1 , . . . , xm of non-associate divisors of x in Rα . Let y be a divisor of x in R and write x = yr for some r ∈ R. Because the family (Rγ )γ ∈ is directed, there is a β ∈ such that Rα ⊆ Rβ and y, r ∈ Rβ . Since Rα ⊆ Rβ is an inert extension, yu, r u −1 ∈ Rα for some u ∈ U (Rβ ). As yu divides x in Rα , there exists v ∈ U (Rα ) ⊆ U (R) such that yu = x j v for some j ∈ [[1, m]]. Hence y ∈ x j U (R). Therefore every divisor of x in R must be associate to some of the elements x1 , . . . , xm in R. Thus, R is an FFD by Proposition 4.3.  Remark 5.28 A similar version of Theorem 5.27 holds if one replaces being a BFD (or an FFD) by satisfying ACCP, being an HFD, or being a UFD; see [5, Theorem 5.2].

5.4 Pullback Constructions We conclude this section by studying the bounded and finite factorization properties for integral domains given by certain pullbacks that generalize the D + M construction. To formalize this, consider an integral domain T with a nonzero maximal ideal M, and let ϕ : T → K be the natural projection on the residue field K = T /M. For a subring D of K , we call R = ϕ −1 (D) the pullback of D by ϕ. Observe that the D + M construction is a special case of a pullback: indeed, if k is a subfield of T such that T = k + M, then K = T /M can be identified with k canonically, and so any subring D of k can be thought of as an actual subring of K . When T is quasilocal, the results that we have already established for the D + M construction extend to pullbacks, as we will see in Propositions 5.32 and 5.33. First, we prove the following lemmas. Lemma 5.29 ([12, Lemma 6.1]) Let R ⊆ T be an extension of integral domains, and let I be a nonzero ideal of both R and T . If R is atomic and I is a prime ideal of R, then U (R) = U (T ) ∩ R. Proof Since U (R) is contained in U (T ) ∩ R, it suffices to show that every element of U (T ) ∩ R is a unit of R. Take x ∈ U (T ) ∩ R. Since I is a nonzero prime ideal of the atomic domain R, there must be an irreducible a of R contained in I . Because I is also an ideal of T , it follows that x −1 a ∈ I ⊆ R \ U (R), and so the equality  a = x(x −1 a) ensures that x ∈ U (R). Hence U (R) = U (T ) ∩ R.

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Lemma 5.30 ([12, Lemma 6.2]) Let T be an integral domain with a nonzero maximal ideal M, and let ϕ : T → K be the natural projection on K = T /M. In addition, let D be a subring of K , and set R = ϕ −1 (D). Then the following statements hold. (1) U (R) = U (T ) ∩ ϕ −1 (U (D)), and so U (R) = U (T ) ∩ R when D is a field. (2) If T is quasilocal, then U (R) = U (T ) ∩ R if and only if D is a field. Proof (1) It is clear that U (R) ⊆ U (T ) ∩ ϕ −1 (U (D)). In order to argue the reverse inclusion, take x ∈ ϕ −1 (U (D)) with x −1 ∈ T . Since ϕ(x) ∈ U (D), it follows that ϕ(x −1 ) = ϕ(x)−1 ∈ U (D). As a result, x −1 ∈ ϕ −1 (U (D)) ⊆ R, and so x ∈ U (R). Hence U (T ) ∩ ϕ −1 (U (D)) ⊆ U (R). The second statement is an immediate consequence of the first. (2) Proving this part amounts to noting that when T is quasilocal, restricting ϕ to  U (T ) yields a surjective group homomorphism U (T ) → K ∗ . Note that in the pullback construction, R is quasilocal with maximal ideal M if and only if D is a field. Lemma 5.31 Let T be a quasilocal integral domain with nonzero maximal ideal M, and let ϕ : T → K be the natural projection on K = T /M. In addition, let D be a subring of K , and set R = ϕ −1 (D). If D is a field, then I (R) = I (T ) ⊆ M. Proof To argue that I (R) ⊆ I (T ), take m ∈ I (R), and suppose that m = x y for some x, y ∈ T . Since M ⊆ R and m ∈ I (R), the elements x and y cannot be contained in M simultaneously. Therefore either x ∈ T \ M = U (T ) or y ∈ T \ M = U (T ), and so m ∈ I (T ). To argue the reverse inclusion, take m ∈ I (T ), and suppose that m = x y for some x, y ∈ R. Since m ∈ I (T ), either x ∈ U (T ) or y ∈ U (T ). Because T is quasilocal and D is a field, it follows from part (2) of Lemma 5.30 that U (R) = U (T ) ∩ R. Therefore x ∈ U (R) or y ∈ U (R), and so m ∈ I (R). Thus, I (R) = I (T ). Finally, the fact that T is quasilocal ensures that I (T ) ⊆ M.  Proposition 5.32 ([12, Proposition 6.3]) Let T be a quasilocal integral domain with nonzero maximal ideal M, and let ϕ : T → K be the natural projection on K = T /M. In addition, let D be a subring of K , and set R = ϕ −1 (D). Then R is a BFD if and only if T is a BFD and D is a field. Proof Suppose first that R is an atomic domain. Since M is a maximal ideal of T contained in R, it follows that M is a nonzero prime ideal of R. This, along with Lemma 5.29, guarantees that U (R) = U (T ) ∩ R. Because T is quasilocal, D is a field by part (2) of Lemma 5.30, and so it follows from Lemma 5.31 that I (R) = I (T ). Therefore every element in M factors into irreducibles in R if and only if it factors into irreducibles in T . Hence T must be atomic. On the other hand, assume that T is atomic and D is a field. As D is a field, U (R) = U (T ) ∩ R by part (1) of Lemma 5.30, which implies that U (R) = R \ M. Therefore every nonzero nonunit of R can be written as a product of elements in I (T ) because T is atomic. As I (T ) = I (R) by Lemma 5.31, the atomicity of R follows.

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Assuming that D is a field, it is not hard to verify that for every nonzero x, y ∈ M the inclusion x T  yT holds if and only if the inclusion x R  y R holds. As a result, it follows from Proposition 4.1 that R is a BFD if and only if T is a BFD.  Parallel to Proposition 5.32, we proceed to give a result for the finite factorization property in pullback constructions. Proposition 5.33 ([12, Propositions 6.3 and 6.7]) Let T be an integral domain with a nonzero maximal ideal M, and let ϕ : T → K be the natural projection on K = T /M. In addition, let D be a subring of K , and set R = ϕ −1 (D). Then the following statements hold. (1) R is an FFD if and only if T is an FFD and the group U (T )/U (R) is finite. (2) If T is quasilocal, then R is an FFD if and only if T is an FFD, D is a field, and the group K ∗ /D ∗ is finite. Proof (1) For the direct implication, suppose that R is an FFD. Since M is a nonzero ideal of T that is contained in R, the nonempty set M \ {0} is contained in [R : R T ] = {r ∈ R | r T ⊆ R}. As a result, it follows from Proposition 5.18 that T is an FFD and U (T )/U (R) is finite. Conversely, suppose that T is an FFD and the group U (T )/U (R) is finite. Let F be the quotient field of R inside qf(T ). Since (U (T ) ∩ F ∗ )/U (R) is a subgroup of U (T )/U (R), the former must be finite. Thus, R is an FFD by Proposition 5.13. (2) Suppose that R is an FFD. It follows from the previous part that T is an FFD. In addition, it follows from Proposition 5.32 that D is a field, and so U (R) = U (T ) ∩ R by Lemma 5.30. Because T is quasilocal, the map ϕ : U (T ) → K ∗ obtained by restricting ϕ to U (T ) is a surjective group homomorphism. By composing this map with the natural projection K ∗ → K ∗ /D ∗ , we obtain a surjective group homomorphism U (T ) → K ∗ /D ∗ , whose kernel is U (R) because U (R) = U (T ) ∩ R. Hence U (T )/U (R) ∼ = K ∗ /D ∗ , and so the previous part ensures that K ∗ /D ∗ is finite. For the reverse implication, assume that T is an FFD, D is a field, and K ∗ /D ∗ is finite. As in the previous part, U (T )/U (R) ∼ = K ∗ /D ∗ . Hence U (T )/U (R) is finite, and it also follows from the previous part that R is an FFD.  The condition of T being quasilocal in Proposition 5.32 and in part (2) of Proposition 5.33 is not superfluous, as we show in the following example, which is part of [12, Example 6.6]. Example 5.34 (1) Set T = Q[π ] + X R[X ] and consider the ring homomorphism ϕ : T → C defined by ϕ( f ) = f (i). Since ϕ is surjective, ker ϕ is a nonzero maximal ideal of T , and we can think of ϕ as the natural projection T → T /M, where M = ker ϕ. Take D = Q and R = ϕ −1 (D). Because R[X ] is a BFD and U (R[X ]) ∩ R = Q \ {0} = U (R), it follows from Proposition 5.11 that R is a BFD. However, Q[π ] is not a field. In addition, the fact that Q[π ] is not a field, along with Remark 4.21, ensures that T is not even atomic. In particular, T is not a BFD. (2) Let D be a subring of a field K . Consider a family of indeterminates indexed by K , namely, {X k | k ∈ K }, and set T = Z[{X k | k ∈ K }]. Now let ϕ : T → K be

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the ring homomorphism determined by the assignments X k → k for every k ∈ K . As ϕ is surjective, we can assume that ϕ is the natural projection T → T /M, where M = ker ϕ. Now take any subring D and set R = ϕ −1 (D). Because T is a UFD and U (R) = U (T ) = {±1}, it follows from Proposition 5.13 that R is an FFD regardless of our choice of D.

6 Polynomial-Like Rings In this section, we study conditions under which the bounded and finite factorization properties transfer between an integral domain and its “polynomial-like rings”. We put special emphasis on integral domains of the form A + X B[X ] and A + X B[[X ]], where A ⊆ B is an extension of integral domains, and the generalized case obtained by replacing the single extension A ⊆ B by the possibly-infinite tower of integral domains A1 ⊆ A2 ⊆ · · · .

6.1 Bounded Factorization Subdomains of R[X] and R[[X]] Let A ⊆ B be an extension of integral domains. As in [12], we say that B is a bounded factorization domain with respect to A or an A-BFD if for every nonzero nonunit x ∈ B, there is an n 0 ∈ N such that if x = b1 · · · bn for some nonunits b1 , . . . , bn ∈ B, then at most n 0 of the bi ’s belong to A. Theorem 6.1 ([12, Proposition 2.1]) Let A ⊆ B be an extension of integral domains. Then the following statements are equivalent. (a) A + X B[X ] is a BFD. (b) A + X B[[X ]] is a BFD. (c) B is an A-BFD and U (A) = U (B) ∩ A. In addition, if B is a BFD, then (c) can be replaced by the statement (c ) U (A) = U (B) ∩ A, and if qf(A) ⊆ B, then (c) can be replaced by the statement (c ) A is a field. Proof Set R = A + X B[X ] and T = A + X B[[X ]]. (a) ⇒ (b): By Proposition 4.1, there is a length function  R : R ∗ → N0 . Now  ∞ ∗ i =  R (an X n ) + n for every T define i=n a i X ∞ thei function T : T → N0 by  ∞ i = 0 if  and only if n = 0 and i=n ai X with an  = 0. Clearly, T  i=n ai X ∞ ∞ ai X i and g = i=m bi X i in T ∗ with a0 ∈ U (A). In addition, for all f = i=n an = 0 and bm = 0, the fact that  R is a length function guarantees that T ( f g) =  R (an bm X n+m ) + n + m ≥  R (an X n ) +  R (bm X m ) + n + m = T ( f ) + T (g). Hence T is a length function, and so T is a BFD by Proposition 4.1.

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(b) ⇒ (c): It is clear that U (A) ⊆ U (B) ∩ A. For the reverse inclusion, take u ∈ A such that u −1 ∈ B. Since T is a BFD, it satisfies ACCP, and therefore, the ascending chain of principal ideals (u −n X T )n∈N must stabilize. Then u −n X T = u −(n+1) X T for some n ∈ N, from which we obtain u ∈ U (T ) ∩ A = U (A). Thus, U (A) = U (B) ∩ A. To show that B is an A-BFD, let b be a nonzero nonunit of B. Since T is a BFD, there is an n 0 ∈ N such that bX cannot be the product of more than n 0 nonunits in T . Write b = a1 · · · am b1 · · · bn , where a1 , . . . , am are nonunits of A and b1 , . . . , bn are nonunits in B \ A. Clearly, a1 , . . . , am are nonunits in T . Then bX = a1 · · · am (b1 · · · bn X ), and so m ≤ n 0 − 1. Hence B is an A-BFD. (c) ⇒ (a): Assume now that B is an A-BFD satisfying U (A) = U (B) ∩ A. It n bi X i with bn = 0 to be a immediately follows that A is a BFD. Take f = i=0 nonzero nonunit of R. If n = 0, then f = b0 ∈ A and so there is an n 0 ∈ N such that f cannot be the product of more than n 0 nonunits of R. On the other hand, suppose that n ≥ 1. As B is an A-BFD, there is an upper bound n 1 ∈ N for the number of nonunit factors in A of a factorization of bn in B. Then a factorization of f in R has at most n 1 + n nonunit factors. Thus, R is a BFD. (c) ⇔ (c ) when B is a BFD: This is clear as B is also an A-BFD. (c) ⇔ (c ) when qf(A) ⊆ B: For the direct implication, it suffices to note that qf(A)∗ ⊆ U (B) implies that A∗ ⊆ U (B) ∩ A = U (A). The reverse implication follows from the fact that every extension of the field A is an A-BFD.  Corollary 6.2 ([4, Proposition 2.5], [5, Corollary 2.2], and [42, Corollary 3.1]) The following statements are equivalent for an integral domain R. (a) (b) (c) (d) (e)

R is a BFD. R[X ] is a BFD. R[[X ]] is a BFD. R[X, X −1 ] is a BFD. The ring of formal Laurent series R((X )) is a BFD.

Proof (a) ⇔ (b) ⇔ (c): These equivalences follow by taking B = A = R in Theorem 6.1. (b) ⇔ (d): Observe that the ring of Laurent polynomials R[X, X −1 ] is the localization of R[X ] at the multiplicative set S = {u X n | u ∈ U (R) and n ∈ N0 } generated by the prime X . Then Lemma 5.6 guarantees that S is a splitting multiplicative set, while Lemma 5.9 guarantees that the extension R[X ] ⊆ R[X ] S = R[X, X −1 ] is inert. As a consequence, it follows from Theorems 5.15 and 5.22 that R[X ] is a BFD if and only if R[X, X −1 ] is a BFD. (c) ⇔ (e): After observing that R((X )) = R[[X ]] S , where S = {u X n | u ∈ U (R) and n ∈ N0 }, we can simply repeat the argument given in the previous paragraph.  Corollary 6.3 ([4, Proposition 2.6]) Let R be a BFD, and let {X i | i ∈ I } be a family of indeterminates for some set I . Then every subring of R[{X i | i ∈ I }] containing R is a BFD.

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Proof Set R I = R[{X i | i ∈ I }], and let T be a subring of R I containing R. Take f to be a nonunit of R I , and then take a finite subset J of I such that f ∈ R J = R[{X j | j ∈ J }]. As R is a BFD and |J | < ∞, it follows from Corollary 6.2 that R J is a BFD. The equality Z R I ( f ) = Z R J ( f ) ensures that |L R I ( f )| = |L R J ( f )| < ∞. Hence R I is a BFD. Since R ⊆ T ⊆ R I , it follows that U (T ) = U (R) = U (R I ). Thus, Proposition 5.11 guarantees that T is a BFD.  With the notation as in Theorem 6.1, the integral domain A is a BFD if A + X B[X ] is a BFD. However, the converse of this implication does not hold in general. Example 6.4 Consider the integral domain R = Z + X Q[X ]. Clearly, Z is a BFD. Observe, on the other hand, that R is a particular case of the D + M construction, where D = Z is not a field. Thus, R is not a BFD by Proposition 4.20. We would also like to emphasize that even if A + X B[X ] and A + X B[[X ]] are both BFDs, B may not be a BFD. The following example is [4, Example 2.7]. Example 6.5 Let Z¯ be the ring of algebraic integers. Since the ascending chain n ¯ does not satisfy ACCP, and ¯ n∈N does not stabilize, Z of principal ideals (21/2 Z) ¯ ] is a BFD. so it is not a BFD. However, the integral domain R = Z + X Z[X ∗ To verify this, let  : Z → N0 be a length function, and define  R : R ∗ → N0 by  R ( f ) = ( f (0)) + deg f . It is clear that  R ( f ) = 0 if and only if ( f (0)) = deg f = 0, which happens precisely when f ∈ {±1} = U (R). In addition, it is clear that  R ( f g) ≥  R ( f ) +  R (g) when f, g ∈ R ∗ . Hence  R is a length function, and so Proposition 4.1 guarantees that R is a BFD. It follows from Theorem 6.1 that ¯ Z + X Z[[X ]] is also a BFD. With the notation as in Theorem 6.1, if B is taken to be the quotient field of A, then the property of being a BFD transfers from A to any intermediate integral domain of the extension A[X ] ⊆ A + X B[X ] provided that we impose a certain condition. Proposition 6.6 ([6, Theorem 7.5]) Let R be an integral domain with quotient field K , and let T be an integral domain such that R[X ] ⊆ T ⊆ R + X K [X ]. In addition, assume that for every n ∈ N0 , there is an rn ∈ R ∗ such that rn f ∈ R[X ] for every f ∈ T with deg f ≤ n. Then T is a BFD if and only if R is a BFD. Proof For the direct implication, suppose that T is a BFD. It is clear that U (R) = U (T ). Therefore R is a BFD by Proposition 5.11. For the reverse implication, suppose that R is a BFD. Take a nonzero nonunit f ∈ T , and write f = c1 · · · ck g, where c1 , . . . , ck ∈ R \ U (R) and g ∈ T \ R. Let n and c be the degree and the leading coefficient of f , respectively. As rn f ∈ R[X ], it follows that rn c, the leading coefficient of rn f , belongs to R. Similarly, the fact that rn g ∈ R[X ] ensures that the leading coefficient c of rn g belongs to R. Because R is a BFD containing rn c, the equality rn c = c1 · · · ck c implies that k ≤ max L R (rn c),  and so L T ( f ) is bounded by n + max L R (rn c). Hence T is a BFD.

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Let R be an integral domain with quotient field K . The set Int(R) = { f ∈ K [X ] | f (R) ⊆ R} is easily seen to be a subring of R + X K [X ] containing R[X ]. In particular, Int(R) is an integral domain, and it is called the ring of integer-valued polynomials on R. A survey on rings of integer-valued polynomials has been given by P. J. Cahen and J. L. Chabert in [17], and the reader can find considerably more information about Int(R) in the book [16], written by the same authors. It is not hard to verify that there exists a sequence (rn )n∈N whose terms are nonzero elements of R such that rn f ∈ R[X ] for every f ∈ Int(R) with deg f ≤ n (see [16, Proposition I.3.1]). As a result, we obtain the following consequence of Proposition 6.6. Corollary 6.7 ([6, Corollary 7.6]) Let R be an integral domain. Then the ring Int(R) is a BFD if and only if R is a BFD. An integral domain is called a Prüfer domain if every nonzero finitely generated ideal is invertible. Dedekind domains and valuation domains are examples of Prüfer domains. The following example is [4, Example 2.7(b)]. Example 6.8 Since Z is a BFD, the ring Int(Z) of integer-valued polynomials on Z is also a BFD. In addition, Int(Z) is a two-dimensional completely integrally closed Prüfer domain that satisfies ACCP. However, the localization of Int(Z) at any height-two maximal ideal is a two-dimensional valuation domain that is not even atomic. Next we turn our attention to certain integral domains that generalize subrings of the form A + B[X ] and A + B[[X ]]. For the rest of this subsection, we let (An )n∈N0 be an ascending chain of integral domains contained in a field L (that is, An is a subring of An+1 for every n ∈ N0 ), and we set A = n∈N0 An . Observe that A is a subring of L. In addition, we set A[X ] =

n∈N0

An X n

and

A[[X ]] =



An X n .

(6.1)

n∈N0

It is clear that A[X ] and A[[X ]] are subrings of A[X ] and A[[X ]], respectively. Parallel to Theorem 6.1, we will give a necessary and sufficient condition for the integral domains A[X ] and A[[X ]] to be BFDs. The results about A[X ] and A[[X ]] we have included in this section are from the unpublished Ph.D. dissertation of P. L. Kiihne [40]. Before proceeding, we emphasize that even if An is a BFD for every n ∈ N0 , the integral domain A may not be a BFD; for this, see Example 6.11 below. Theorem 6.9 ([40, Theorem 3.3.5]) The following statements are equivalent. (a) A[X ] is a BFD. (b) A[[X ]] is a BFD. (c) U (A0 ) = U (A) ∩ A0 , and An is an A0 -BFD for every n ∈ N0 . In addition, if A[X ] (or, equivalently, A[[X ]]) is a BFD, then A[X ] and A[[X ]] are BFDs.

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Proof (a) ⇒ (c): Suppose that A[X ] is a BFD. It is clear that U (A0 ) ⊆ U (A) ∩ A0 . For the reverse inclusion, take u ∈ A0 with u −1 ∈ A. Take m ∈ N0 with u −1 ∈ Am . As A[X ] satisfies ACCP by Corollary 3.2, the ascending chain of principal ideals (u −n X m A[X ])n∈N of A[X ] must stabilize. Consequently, there is an n ∈ N such that u −n X m A[X ] = u −(n+1) X m A[X ], from which we deduce that u ∈ U (A[X ]) ∩ A0 = U (A0 ). Hence U (A) ∩ A0 ⊆ U (A0 ). To prove the second statement, fix k ∈ N0 , and then take a nonunit b ∈ A∗k . Since A[X ] is a BFD, there is an n 0 ∈ N such that bX k cannot be the product of more than n 0 nonunits in A[X ]. Write b = a1 · · · am b1 · · · bn , where a1 , . . . , am are nonunits of A0 and b1 , . . . , bn are nonunits in Ak \ A0 . Then bX k = a1 · · · am (b1 · · · bn X k ), and since a1 , . . . , am are nonunits in A[X ], the inequality m ≤ n 0 − 1 holds. Hence Ak is an A0 -BFD. (c) ⇒ (a): Assume that U (A0 ) = U (A) ∩ A0 and An is an A0 -BFD for every n ∈ N0 . Because U (A0 ) = U (Ad ) ∩ A0 and Ad is an A0 -BFD, Theorem 6.1 guarantees that A0 + X Ad [X ] is a BFD. Since f ∈ A0 + X Ad [X ] and every pair of non-associate divisors of f in A[X ] is also a pair of non-associate divisors of f in A0 + X Ad [X ], the fact that A0 + X Ad [X ] is a BFD implies that L A[X ] ( f ) is finite. Hence A[X ] is a BFD. (b) ⇒ (c): It follows mimicking the argument we used to prove the implication (a) ⇒ (c). (A) ∩ A0 and An is an A0 -BFD for every (c) ⇒ (b): Assume ∞ that iU (A0 ) = U bi X ∈ A[[X ]]∗ be a nonunit, and assume that bm = 0. Since n ∈ N0 . Let f = i=m Am is an A0 -BFD, there is an n 0 ∈ N such that any factorization of bm in Am involves at most n 0 factors in A0 . Now suppose that f = f 1 · · · f k g1 · · · g in A[[X ]], where f 1 , . . . , f k are nonunits with order 0 and g1 , . . . , g are nonunits of order at least 1. It is clear that  ≤ m. On the other hand, comparing the coefficients of the monomials with degree m in both sides of the equality f = f 1 · · · f k g1 · · · g , we see that bm = c1 · · · ck c in Am , where c1 , . . . , ck are nonunits in A0 . Therefore k ≤ n 0 , and so  max L A[[X ]] ( f ) ≤ m + n 0 . We conclude that A[[X ]] is a BFD. Corollary 6.10 ([40, Corollary 3.3.9]) If A0 is a field, then A[X ] and A[[X ]] are BFDs. Proof It is an immediate consequence of Theorem 6.9 since when A0 is a field both statements of part (c) of Theorem 6.9 trivially hold.  In the spirit of Theorem 6.9, observe that if there is an N ∈ N0 such that An = A N for every n ≥ N , then A[X ] is a BFD (or A[[X ]] is a BFD) if and only if A N is an A0 BFD and U (A N ) ∩ A0 = U (A0 ). On the other hand, the converse of the last statement of Theorem 6.9 does not hold, as we proceed to illustrate using [40, Example 3.3.12]. Example 6.11 Let F be a field, and for every n ∈ N, let Mn be the additive monoid {0} ∪ Q≥1/n . Now set A0 = F and An = F[Mn ] for every n ∈ N. We have seen in Example 4.7 that the integral domain An is a BFD for every n ∈ N. However,  A = n∈N0 An = F[Q≥0 ] is not a BFD because it is not even atomic. As A is not a BFD, it follows from Corollary 6.2 that neither A[X ] nor A[[X ]] are BFDs. On the other hand, since A0 is a field, both A[X ] and A[[X ]] are BFDs by Corollary 6.10.

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6.2 Finite Factorization Subdomains of R[X] and R[[X]] The main purpose of this subsection is to characterize when the integral domains A + X B[X ] and A + X B[[X ]] are FFDs for a given extension of integral domains A ⊆ B. Unfortunately, the equivalences in Theorem 6.1 do not hold if we replace BFD by FFD. However, some of the equivalences in Corollary 6.2 are still true for the finite factorization property. Theorem 6.12 ([4, Proposition 5.3] and [42, Corollary 4.2]) The following statements are equivalent for an integral domain R. (a) R is an FFD. (b) R[X ] is an FFD. (c) R[X, X −1 ] is an FFD. Proof (a) ⇒ (b): Assume that R is an FFD, and let K be the quotient field of R. Suppose, by way of contradiction, that R[X ] is not an FFD. It follows from Proposition 4.3 that there is a nonzero nonunit f ∈ R[X ] having infinitely many non-associate divisors in R[X ]. Take ( f n )n∈N to be a sequence of non-associate divisors of f in R[X ]. Let c be the leading coefficient of f , and let cn ∈ R be the leading coefficient of f n for every n ∈ N. Since cn is a divisor of c for every n ∈ N, and R is an FFD, after replacing ( f n )n∈N by a subsequence, we can assume that c1 and cn are associates in R for every n ∈ N. In addition, after replacing f n by c1 cn−1 f n for every n ∈ N≥2 , we can assume that all polynomials in the sequence ( f n )n∈N have the same leading coefficient, namely, c1 . Since each f n divides f in K [X ], which is an FFD, there are distinct i, j ∈ N such that f i and f j are associates in K [X ]. As f i and f j have the same leading coefficient, they must be equal, which contradicts that they are non-associates in R[X ]. (b) ⇒ (c): Suppose that R[X ] is an FFD. Since the extension R[X ] ⊆ R[X ] S = R[X, X −1 ] is inert for the multiplicative set S = {u X n | u ∈ U (R) and n ∈ N0 } (see the proof of Corollary 6.2), it follows from Theorem 5.22 that R[X, X −1 ] is an FFD. (c) ⇒ (a): Suppose that R[X, X −1 ] is an FFD, and let K be the quotient field of R. Because U (R[X, X −1 ]) = {u X n | u ∈ U (R) and n ∈ Z}, the group (U (R[X, X −1 ]) ∩ K ∗ )/U (R) is trivial. As a result, it follows from Proposition 5.13 that R is an FFD.  Corollary 6.13 ([10, Examples 2, 4, and 7]) For an FFD R and a set {X i | i ∈ I } of indeterminates, the following statements hold. (1) R[{X i | i ∈ I }] is an FFD. (2) R[{X i , X i−1 | i ∈ I }] is an FFD. (3) If R is either a finite field or Z, then every subring of R[{X i | i ∈ I }] is an SFFD, and hence an FFD. Proof (1) Set R I = R[{X i | i ∈ I }]. Take a nonunit f in R ∗I , and let J be a finite subset of I such that f ∈ R J = R[{X j | j ∈ J }]. The integral domain R J is an FFD

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by Theorem 6.12. In addition, every divisor of f in R I is also a divisor of f in R J . As U (R J ) = U (R) = U (R I ), the fact that f has only finitely many non-associate divisors in R J implies that f has only finitely many non-associate divisors in R I . Hence R I is an FFD. (2) The integral domain R[{X i , X i−1 | i ∈ I }] is the localization of R[{X i | i ∈ I }] at the multiplicative set S generated by {X i | i ∈ I }. Since X i is a prime element in R[{X i | i ∈ I }] for every i ∈ I , it follows from Lemma 5.9 that the extension R[{X i | i ∈ I }] ⊆ R[{X i , X i−1 | i ∈ I }] is inert. As R[{X i | i ∈ I }] is an FFD by part (1), part (2) of Theorem 5.22 guarantees that R[{X i , X i−1 | i ∈ I }] is an FFD. (3) Let R be either a finite field or Z. As in part (1), set R I = R[{X i | i ∈ I }]. Let T be a subring of R I , and then let f be a nonunit in T ∗ . Take a finite subset J of I such that f belongs to R J = R[{X j | j ∈ J }]. By part (1), R J is an FFD. Moreover, since |U (R J )| = |U (R)| < ∞, it follows from Proposition 4.5 that R J is an SFFD. As every divisor of f in T is also a divisor of f in R J , the element f has only finitely many divisors in T . Thus, T is an SFFD, and therefore, an FFD.  In contrast to Theorem 6.12, the ring of power series R[[X ]] is not necessarily an FFD when R is an FFD. To illustrate this observation with an example, we need the following proposition. Proposition 6.14 ([10, Corollary 2]) Let R be an integral domain. If R[[X ]] is an FFD, then R is completely integrally closed. Therefore when R is Noetherian, R[[X ]] is an FFD if and only if R is integrally closed. Proof Suppose towards a contradiction that R is not completely integrally closed, and then take an almost integral element t ∈ qf(R) \ R over R. As a result, the ideal [R : R R[t]] is nonzero, and therefore, R[[X ]] : R[[X ]] R[t][[X ]] = {0}. Since R[[X ]] is an FFD, it follows from Proposition 5.18 that the group U (R[t][[X ]])/ U (R[[X ]]) is finite. Hence we can choose m, n ∈ N so that m = n and (1 + t X m )U (R[[X ]]) = (1 + t X n )U (R[[X ]]), which implies that (1 + t X m )(1 + t X n )−1 ∈ R[[X ]]. However, this contradicts that (1 + t X m )(1 + t X n )−1 = 1 − t X m + · · · and −t ∈ / R. Thus, R is completely integrally closed. The direct implication of the second statement follows directly from the first statement because every completely integrally closed domain is integrally closed. The reverse implication is also immediate because every Noetherian integrally closed domain is a Krull domain, which is an FFD by Theorem 4.15.  We are now in a position to illustrate that R[[X ]] may not be an FFD even if R is an FFD. The following example is [10, Remark 2]. Example 6.15 Let F1  F2 be a field extension of finite fields, and consider the integral domain R = F1 + Y F2 [[Y ]]. Since Y F2 [[Y ]] is a nonzero maximal ideal of F2 [[Y ]], the ring R has the form of a D + M construction, where T = F2 [[Y ]]. Since F2 [[Y ]] is an FFD and F2∗ /F1∗ is a finite group, it follows from Proposition 4.23 that R is an FFD. On the other hand, note that every element of F2 \ F1 is an almost integral element over R. Hence R is not completely integrally closed. Thus, Proposition 6.14 guarantees that R[[X ]] is not an FFD.

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Next we characterize when the construction A + X B[X ] of an extension A ⊆ B of integral domains yields FFDs. To do this, we need the finiteness of the group U (B)/U (A), which can be easily verified to be stronger than the condition U (A) = U (B) ∩ A. Proposition 6.16 ([12, Proposition 3.1]) Let A ⊆ B be an extension of integral domains. Then A + X B[X ] is an FFD if and only if B is an FFD and U (B)/U (A) is a finite group. Proof Set R = A + X B[X ]. For the direct implication, suppose that R is an FFD. Since X B[X ] is a nonzero common ideal of R and B[X ], it follows that [R : R B[X ]] < ∞. Hence Proposition 5.18 guarantees that the group U (B[X ])/U (R) = U (B)/U (A) is finite and B[X ] is an FFD. As a consequence, B is an FFD. For the reverse implication, suppose that B is an FFD and U (B)/U (A) is finite. Since B is an FFD, so is B[X ] by Theorem 6.12. Since B[X ] is an FFD and (U (B[X ]) ∩ qf(R))/U (R) = U (B)/U (A) is finite, it follows from Proposition 5.13 that R is an FFD.  We have characterized SFFDs in Sect. 4. We are now in a position to give two more characterizations. Proposition 6.17 ([10, Theorem 5]) The following statements are equivalent for an integral domain R. (a) R is an SFFD. (b) For any set of indeterminates {X i | i ∈ I } over R, every subring of the polynomial ring R[{X i | i ∈ I }] is an SFFD. (c) For any set of indeterminates {X i | i ∈ I } over R, every subring of the polynomial ring R[{X i | i ∈ I }] is an FFD. (d) Every subring of R[X ] is an FFD. Proof (a) ⇒ (b): Let {X i | i ∈ I } be a nonempty set of indeterminates over R, and let T be a subring of R I = R[{X i | i ∈ I }]. Take f ∈ T ∗ . Since R I is an FFD by Corollary 6.13 and U (R I ) = U (R) is finite by Proposition 4.5, there are only finitely many divisors of f in R I . Therefore f has only finitely many divisors in T . Thus, T is an SFFD. (b) ⇒ (c): This is clear. (c) ⇒ (d): This is clear. (d) ⇒ (a): Since R is a subring of R[X ], it is an FFD. Set S = R0 + X R[X ], where R0 is the prime subring of R. As S is a subring of R[X ], it is an FFD. In addition, because X ∈ [S : S R[X ]], it follows from Proposition 5.18 that U (R[X ])/U (S) is finite. Since U (R[X ]) = U (R) and U (S) = U (R0 ), the group U (R)/U (R0 ) is finite. Now the fact that U (R0 ) is finite immediately implies that U (R) is finite. Thus, R is an SFFD by Proposition 4.5.  The power series analog of Proposition 6.16 does not hold, as the following example illustrates.

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Example 6.18 Let F1  F2 be an extension of finite fields. We have seen in Example 6.15 that F1 + Y F2 [[Y ]] is an FFD. Take A = B = F1 + Y F2 [[Y ]]. Although B is an FFD and the group U (B)/U (A) is finite, A + X B[[X ]] = B[[X ]] is not an FFD, as shown in Example 6.15. However, we can characterize when A + X B[[X ]] is an FFD using the condition that B[[X ]] is an FFD, which is stronger than B being an FFD. Proposition 6.19 ([12, Proposition 3.3]) Let A ⊆ B be an extension of integral domains. Then A + X B[[X ]] is an FFD if and only if B[[X ]] is an FFD and U (B)/U (A) is a finite group. Proof For the direct implication, suppose that R = A + X B[[X ]]is an FFD. Since  X B[[X ]] is a nonzero ideal of both B[[X ]] and R, it follows that R : R B[[X ]] = {0}. Therefore B[[X ]] is an FFD and the group U (B)/U (A) ∼ = U (B[[X ]])/U (R) is finite by Proposition 5.18. For the reverse implication, suppose that B[[X ]] is an FFD and the group U (B)/U (A) is finite. Since B[[X ]] is an FFD and the group (U (B[[X ]] ∩ qf(R))/ U (R) = U (B[[X ]])/U (R) ∼ = U (B)/U (A) is finite, it follows from Proposition 5.13  that R is an FFD. Let A ⊆ B be an extension of integral domains. If R = A + X B[X ] is an FFD, then it follows from Proposition 6.16 that U (B)/U (A) is finite, and so U (A) = U (B) ∩ A. Then Proposition 5.13 guarantees that A is also an FFD because U (A) = U (R) ∩ qf(A). Similarly, A is an FFD provided that A + X B[[X ]] is an FFD. We record this observation as a corollary. Corollary 6.20 ([12, Remark 3.5]) Let A ⊆ B be an extension of integral domains. If either A + X B[X ] or A + X B[[X ]] is an FFD, then A is an FFD. We have seen in the previous subsection that, for an integral domain R, the ring of integer-valued polynomials Int(R) is a BFD if and only if R is a BFD. The following proposition, whose proof we omit, gives a parallel result based on the finite factorization property. Proposition 6.21 ([35, Corollary 4.7]) Let R be an integral domain. Then Int(R) is an FFD if and only if R is an FFD. Now we return to study the integral domains A[X ] and A[[X ]] introduced in (6.1). This time, we focus our attention on the finite factorization property. To begin with, we give two sufficient conditions and one necessary condition for A[X ] to be an FFD. Proposition 6.22 ([40, Theorem 3.4.6 and Proposition 3.4.7]) The following statements hold. (1) The integral domain A0 + X A1 + · · · + X n−1 An−1 + X n An [X ] is an FFD for every n ∈ N0 if and only if A[X ] is an FFD.

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(2) If U (A)/U (A0 ) is finite and A[X ] is an FFD, then A[X ] is an FFD. Proof (1) Set Rn = A0 + X Ai + · · · + X n−1 An−1 + X n An [X ] for every n ∈ N0 . For the direct implication, assume that Rn is an FFD for every n ∈ N0 . Take f ∈ A[X ]∗ , and let d be the degree of f . Clearly, every divisor of f in A[X ] belongs to Rd . Since Rd is an FFD and U (Rd ) = U (A0 ) = U (A[X ]), the element f has only finitely many non-associate divisors in A[X ]. Thus, A[X ] is an FFD by Proposition 4.3. For the reverse implication, fix m ∈ N0 and take f ∈ Rm . As in the previous paragraph, two polynomials are non-associate divisors of f in Rm if and only if they are non-associate divisors of f in A[X ]. Because A[X ] is an FFD, so is Rm by Proposition 4.3. (2) Observe that A[X ] is a subring of A[X ], and U (A[X ]) is contained in qf(A[X ]). Therefore (U (A[X ]) ∩ qf(A[X ]))/U (A[X ]) = U (A)/U (A0 ) is finite. As a result, it follows from Proposition 5.13 that A[X ] is an FFD.  If the chain of integral domains (An )n≥0 stabilizes, then we can characterize when A[X ] (or A[[X ]]) is an FFD. Proposition 6.23 ([40, Theorem 3.4.5 and Proposition 3.4.8]) If there is an N ∈ N such that An = A N for every n ≥ N , then the following statements hold. (1) A[X ] is an FFD if and only if A N is an FFD and the group U (A N )/U (A0 ) is finite. (2) A[[X ]] is an FFD if and only if A N [[X ]] is an FFD and the group U (A N [[X ]])/ U (A[[X ]]) is finite. Proof (1) For the direct implication, assume that A[X ] is an FFD. Because A[X ] ⊆   A N [X ] and X N ∈ A[X ] :A[X ] A N [X ] , Proposition 5.18 guarantees that the integral domain A N [X ] is an FFD and the group U (A N [X ])/U (A[X ]) = U (A N )/U (A0 ) is finite. Since A N [X ] is an FFD, so is A N . Conversely, suppose that A N is an FFD and U (A N )/U (A0 ) is finite. The ring of polynomials A N [X ] is an FFD by Theorem 6.12. On the other hand, U (A N [X ]) ⊆ qf(A[X ]), and therefore, (U (A N [X ]) ∩ qf(A[X ]))/U (A[X ]) = U (A N )/U (A0 ) is finite. Thus, it follows from Proposition 5.13 that A[X ] is an FFD. (2) Assume firstthat A[[X ]] is an FFD. Because A[[X ]] is an FFD contained in A N [X ] and X N ∈ A[[X ]] :A[[X ]] A N [[X ]] , it follows from Proposition 5.18 that A N [[X ]] is an FFD and also that the group U (A N [[X ]])/U (A[[X ]]) is finite. For the reverse implication, suppose that A N [[X ]] is an FFD and U (A N [[X ]])/ U (A[[X ]]) is finite. It is easy to verify that the quotient field of A[[X ]] is qf(A N [[X ]]). As a result, we obtain that the group (U (A N [[X ]]) ∩ qf(A[[X ]]))/U (A[[X ]]) = U (A N [[X ]])/U (A[[X ]]) is finite. Since A N [[X ]] is an FFD, Proposition 5.13 guarantees that A[[X ]] is an FFD. 

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6.3 Monoid Domains Let R be an integral domain, and let M be a torsion-free monoid. Since monoids here are assumed to be cancellative and commutative, it follows from [28, Corollary 3.4] that M admits a compatible total order (indeed, every compatible partial order on M extends to a compatible total order on gp(M) [28, Theorem 3.1]). we tacitly n Hence ci X m i ∈ R[M]∗ assume that M is a totally ordered monoid. We say that f = i=1 is represented in canonical form if ci = 0 for every i ∈ [[1, n]] and m 1 > · · · > m n . Observe that any element of R[M]∗ has a unique representation in canonical form. In this case, deg f = m 1 is called the degree of f , while c1 and c1 X m 1 are called the leading coefficient and the leading term of f , respectively. As it is customary for polynomials, we say that f is a monomial if n = 1. Most of the results presented in this subsection were established by H. Kim in [41, 42], where the interested reader can also find similar results concerning atomicity, the ACCP, and the unique factorization property. We start by discussing the bounded factorization property in the context of monoid domains. Proposition 6.24 ([42, Propositions 1.4 and 1.5]) Let R be an integral domain with quotient field K , and let M be a torsion-free monoid. Then the following statements hold. (1) If R[M] is a BFD, then R is a BFD and M is a BFM. (2) If R and K [M] are both BFDs, then R[M] is a BFD. Proof (1) Suppose that the monoid domain R[M] is a BFD. It follows from [28, Theorem 11.1] that U (R[M]) = {u X m | u ∈ U (R) and m ∈ U (M)}. Therefore U (R) = U (R[M]) ∩ R, and it follows from Proposition 5.11 that R is a BFD. To verify that M is a BFM, first note that by virtue of [28, Theorem 11.1], a ∈ I (M) if and only if X a ∈ I (R[M]). As a result, for every b ∈ M \ U (M), the set L M (b) is finite if and only if the set L R[M] (X b ) is finite. This, together with the fact that R[M] is a BFD, implies that M is a BFM. (2) Assume that R and T = K [M] are both BFDs. Proposition 4.1 guarantees the existence of length functions  R : R ∗ → N0 and T : T ∗ → N0 of R ∗ and T ∗ , respectively. Now define the function  : R[M]∗ → N0 by setting ( f ) = T ( f ) +  R (c), where c is the leading coefficient of f . It is clear that every unit u X m of R[M] is a unit of T with u ∈ U (R), and so (u X m ) = T (u X m ) +  R (u) = 0. Also, for polynomial expressions f 1 and f 2 in R[M]∗ with leading coefficients c1 and c2 , respectively, ( f 1 f 2 ) = T ( f 1 f 2 ) +  R (c1 c2 ) ≥ (T ( f 1 ) +  R (c1 )) + (T ( f 2 ) +  R (c2 )) = ( f 1 ) + ( f 2 ). Thus,  is a length function, and it follows from Proposition 4.1 that R[M] is a BFD.  We have just seen that for an integral domain R and a torsion-free monoid M, the fact that R[M] is a BFD guarantees that both R and M satisfy the corresponding property. If every nonzero element of gp(M) has type (0, 0, . . . ), then the reverse implication also holds, as part (3) of the next theorem shows. A nonzero element b of an abelian group G has type (0, 0, . . . ) if there is a largest n ∈ N such that the equation nx = b is solvable in G.

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Theorem 6.25 ([41, Theorems 3.12, Proposition 3.14, and Theorem 3.15]) Let R be an integral domain, F a field, G a torsion-free abelian group whose nonzero elements have type (0, 0, . . . ), and M a torsion-free monoid whose nonzero elements have type (0, 0, . . . ) in gp(M). Then the following statements hold. (1) R[G] is a BFD if and only if R is a BFD. (2) F[M] is a BFD if and only if M is a BFM. (3) R[M] is a BFD if and only if R is a BFD and M is a BFM. Proof (1) If R[G] is a BFD, then it follows from part (1) of Proposition 6.24 that R is a BFD. For the reverse implication, suppose that R is a BFD, and let K be the quotient field of R. The monoid domain K [G] is a UFD by [31, Theorem 7.12]. In particular, K [G] is a BFD, and so part (2) of Proposition 6.24 ensures that R[G] is a BFD. (2) If F[M] is a BFD, then it follows from part (1) of Proposition 6.24 that M is a BFM. Conversely, suppose that M is a BFM. As every nonzero element of gp(M) has type (0, 0, . . . ), the monoid domain T = F[gp(M)] is a UFD, and so a BFD, by [31, Theorem 12]. Since M is a BFM and T is a BFD, Propositions 3.1 and 4.1 guarantee the existence of length functions  M : M → N0 and T : T ∗ → N0 , respectively. Define  : F[M]∗ → N0 by ( f ) = T ( f ) +  M (deg f ). One can easily verify that  is a length function of F[M]∗ (see the proof of part (2) of Proposition 6.24). Hence F[M] is a BFD by Proposition 4.1. (3) The direct implication follows from part (1) of Proposition 6.24. For the reverse implication, suppose that R is a BFD and M is a BFM. The monoid domain R[gp(M)] is a BFD by part (1), while the monoid domain qf(R)[M] is a BFD by part (2). Then it follows from Proposition 4.11 that R[M] = R[gp(M)] ∩ qf(R)[M] is a BFD.  Corollary 6.26 ([41, Corollary 3.17]) Let R be an integral domain, and let M be a finitely generated torsion-free monoid. Then R[M] is a BFD if and only if R is a BFD. Proof Since M is torsion-free and finitely generated, gp(M) is a torsion-free finitely generated abelian group, and so a free abelian group. Hence every nonzero element of gp(M) has type (0, 0, . . . ). In addition, it follows from Corollary 3.7 that M is an FFM, and so a BFM. Hence the corollary is a consequence of part (3) of Theorem 6.25.  In [8], D. D. Anderson and J. R. Juett proved a version of part (3) of Theorem 6.25, where they assume that M is reduced, but not that all nonzero elements of gp(M) have type (0, 0, . . . ). Theorem 6.27 ([8, Theorem 13]) Let R be an integral domain, and let M be a reduced torsion-free monoid. Then R[M] is a BFD if and only if R is a BFD and M is a BFM.

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Proof The direct implication follows from part (1) of Proposition 6.24. To argue the reverse implication, suppose that R is a BFD and M is a BFM. Propositions 3.1 and 4.1 guarantee the existence of length functions  M : M → N0 and  R : R ∗ → N0 , respectively. Define  : R[M]∗ → N0 by ( f ) =  M (deg f ) +  R (c), where c is the leading coefficient of f . As M is reduced, U (R[M]) = U (R) and so ( f ) =  R ( f ) = 0 when f ∈ U (R[M]). In addition, if f 1 and f 2 in R[M]∗ have leading coefficients c1 and c2 , respectively, then ( f 1 f 2 ) =  M (deg f 1 + deg f 2 ) +  R (c1 c2 ) ≥ ( f 1 ) + ( f 2 ). Hence  is a length function, and R[M] is a BFD by Proposition 4.1.  With the notation as in Theorem 6.27, the monoid domain R[M] may be a BFD (in fact, an SFFD) even when not every nonzero element of gp(M) has type (0, 0, . . . ); see, for instance, Example 5.21 and [4, Example 5.4]. Now we turn to discuss the finite factorization property in the context of monoid domains. The following result is parallel to Proposition 6.24. Proposition 6.28 ([42, Propositions 1.4 and 1.5]) Let R be an integral domain with quotient field K , and let M be a torsion-free monoid. Then the following statements hold. (1) If R[M] is an FFD, then R is an FFD and M is an FFM. (2) If R and K [M] are both FFDs, then R[M] is an FFD. Proof (1) Suppose that R[M] is an FFD. Since U (R[M]) ∩ K ∗ = U (R), it follows from Proposition 5.13 that R is an FFD. On the other hand, since a ∈ I (M) if and only if X a ∈ I (R[M]) by [28, Theorem 11.1], we find that |Z M (m)| = |Z R[M] (X m )| < ∞ for every m ∈ M. As a consequence, M is an FFM. (2) Now assume that R and K [M] are both FFDs. Suppose, by way of contradiction, that there is an f ∈ R[M]∗ with infinitely many non-associate divisors in R[M]. Let cX m be the leading term of f . Since every divisor of f in R[M] is also a divisor of f in K [M] and f has only finitely many non-associate divisors in K [M] by Proposition 4.3, there must be a sequence ( f n )n∈N consisting of non-associate divisors of f in R[M] such that f i K [M] = f j K [M] for all i, j ∈ N. For every n ∈ N, let cn X m n be the leading term of f n . As K [M] is an FFD, it follows from part (1) that M is an FFM. Because m ∈ m n + M for every n ∈ N and M is an FFM, after replacing ( f n )n∈N by a suitable subsequence, one can assume that deg f i + M = deg f j + M for all i, j ∈ N. Furthermore, after replacing f n by X u n f n , where u n = deg f 1 − deg f n , one can assume that for every n ∈ N there is a kn ∈ K such that f n = kn f 1 . Clearly, cn divides c in R for every n ∈ N. In addition, if ci and c j are associates in R, then ki /k j ∈ U (R) and so f i and f j are associates in R[M], which implies that i = j. Thus, c has infinitely many non-associate divisors in R, contradicting that R is an FFD.  As for the bounded factorization property, the converse of part (1) of Proposition 6.28 holds provided that every nonzero element of gp(M) has type (0, 0, . . . ).

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Theorem 6.29 ([41, Theorems 3.21, Proposition 3.24, and Theorem 3.25]) Let R be an integral domain, F a field, G a torsion-free abelian group whose nonzero elements have type (0, 0, . . . ), and M a torsion-free monoid whose nonzero elements have type (0, 0, . . . ) in gp(M). Then the following statements hold. (1) R[G] is an FFD if and only if R is an FFD. (2) F[M] is an FFD if and only if M is an FFM. (3) R[M] is an FFD if and only if R is an FFD and M is an FFM. Proof (1) It follows from part (1) of Proposition 6.28 that R is an FFD when the monoid domain R[G] is an FFD. Conversely, assume that R is an FFD, and let K be the quotient field of R. Since the monoid domain K [G] is a UFD by [31, Theorem 7.12], it is an FFD. As a result, R[G] is an FFD by part (2) of Proposition 6.28. (2) By part (1) of Proposition 6.28, M is an FFM provided that F[M] is an FFD. For the reverse implication, suppose that M is an FFM and assume, by way of contradiction, that F[M] is not an FFD. Take an f ∈ F[M]∗ having infinitely many non-associate divisors, and let ( f n )n∈N be a sequence of non-associate divisors of f in F[M]. Since M is an FFM and deg f n is a divisor of deg f in M for every n ∈ N, by virtue of Proposition 3.6 we can assume that deg f n = deg f 1 for every n ∈ N. The monoid domain F[gp(M)] is an FFM by [31, Theorem 7.12]. As f n is a divisor of f in F[gp(M)] for every n ∈ N, Proposition 4.3 guarantees the existence of distinct i, j ∈ N such that f i F[gp(M)] = f j F[gp(M)]. Since deg f i = deg f j , it follows that f j = α f i for some α ∈ F. Hence f i and f j are associates in F[M], which is a contradiction. (3) In light of part (1) of Proposition 6.28, R is an FFD and M is an FFM provided that R[M] is an FFD. To argue the reverse implication, suppose that R is an FFD and M is an FFM. Note that R[gp(M)] is an FFD by part (1) and qf(R)[M] is an FFD by part (2). Therefore Proposition 4.14 guarantees that R[M] = R[gp(M)] ∩ qf(R)[M] is an FFD.  Parallel to Corollary 6.26, we obtain the following corollary, whose proof follows similarly. Corollary 6.30 Let R be an integral domain, and let M be a finitely generated torsion-free monoid. Then R[M] is an FFD if and only if R is an FFD. One can naturally generalize the notion of an SFFD to monoids. A monoid M is called a strong finite factorization monoid (or an SFFM) if every element of M has only finitely many divisors. Clearly, a reduced monoid is an SFFM if and only if it is an FFM. Proposition 6.31 ([42, Propositions 1.4 and 1.5]) Let R be an integral domain with quotient field K , and let M be a torsion-free monoid. Then the following statements hold. (1) If R[M] is an SFFD, then R is an SFFD and M is an SFFM. (2) If R is an SFFD, M is an SFFM, and K [M] is an FFD, then R[M] is an SFFD.

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Proof (1) Assume that R[M] is an SFFD. By Proposition 4.5, U (R[M]) is finite, so U (R) ⊆ U (R[M]) implies that U (R) is also finite. On the other hand, R is an FFD by Proposition 6.28. Hence R is an SFFD by Proposition 4.5. As R[M] is an SFFD, to verify that every element m ∈ M has finitely many divisors in M, it suffices to observe that m ∈ d + M if and only if X m ∈ X d R[M]. (2) Assume that R is an SFFD, M is an SFFM, and K [M] is an FFD. In particular, R and K [M] are FFDs, and so it follows from part (2) of Proposition 6.28 that R[M] is an FFD. On the other hand, U (M) is finite because M is an SFFM, and U (R) is finite by Proposition 4.5. Hence U (R[M]) must be finite. Thus, Proposition 4.5 ensures that R[M] is an SFFD.  As in the case of the bounded and finite factorization properties, we have the following result. Theorem 6.32 ([41, Propositions 3.28 and 3.30]) Let R be an integral domain, F a field, G a torsion-free abelian group, and M a torsion-free monoid whose nonzero elements have type (0, 0, . . . ) in gp(M). Then the following statements hold. (1) R[G] is an SFFD if and only if R is an SFFD and G is the trivial group. (2) F[M] is an SFFD if and only if F is a finite field and M is an SFFM. (3) R[M] is an SFFD if and only if R is an SFFD and M is an SFFM. Proof (1) The reverse implication follows immediately. For the direct implication, assume that R[G] is an SFFD. By Proposition 4.5, the set U (R[G]) is finite, and so G must be a finite group. This, along with the fact that G is torsion-free, ensures that G is the trivial group. Hence R = R[G] is an SFFD. (2) This is an immediate consequence of part (3) below. (3) It follows from part (1) of Proposition 6.31 that if R[M] is an SFFD, then R is an SFFD and M is an SFFM. For the reverse implication, suppose that R is an SFFD and M is an SFFM. Since M is an SFFM, U (M) must be finite. On the other hand, R is an FFD and U (R) is finite by Proposition 4.5. Therefore R[M] is an FFD by part (3) of Theorem 6.29. In addition, as U (R) and U (M) are finite, so is U (R[M]). Thus, R[M] is an SFFD by virtue of Proposition 4.5.  In general, there seems to be no characterization (in terms of R and M) for the monoid domains R[M] that are BFDs (FFDs or SFFDs). In the same direction, the question of whether R[M] satisfies ACCP provided that both R and M satisfy the same condition seems to remain open, although it has been positively answered in [8, Theorem 13] for the case when M is reduced (a result parallel to Theorem 6.27). By contrast, it is known that R[M] need not be atomic provided that both R and M are atomic, even if R is a field or M = N0 (i.e., R[M] = R[X ]); for more details about this last observation, see [24] and [46]. A partially ordered set is Artinian if it satisfies the descending chain condition, and it is narrow if it does not contain infinitely many incomparable elements. For a ring R, a monoid M, and a partial order ≤ compatible with M, the generalized power series  ring R[[X ; M ≤ ]] is the ring comprising all formal sums f = m∈M cm X m whose

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support {m ∈ M | cm = 0} is Artinian and narrow. D. D. Anderson and J. R. Juett have also investigated in [8] when the generalized power series ring R[[X ; M ≤ ]] is a BFD (or satisfies ACCP), obtaining in [8, Theorem 17] a result analogous to Theorem 6.27 but in the context of generalized power series rings.

6.4 Graded Integral Domains We conclude this section by saying a few words about the bounded and finite factorization properties in graded integral domains. An integral domain R is called M-graded for a torsion-free monoid M provided that for every m ∈ M, there is a subgroup Rm of the underlying additive group of R such that the following conditions hold:  (1) R = m∈M Rm is a direct sum of abelian groups, and (2) Rm Rn ⊆ Rm+n for all m, n ∈ M. In this case, a nonzero element of Rm is called homogeneous of degree m. The following proposition generalizes parts (1) of Propositions 6.24 and 6.28 and can be proved in a similar manner. Proposition 6.33 ([43, Proposition 2.1]) Let M be a torsion-free monoid and R =  R be an M-graded integral domain. Then R0 is a BFD (resp., an FFD, an m∈M m SFFD) if R is a BFD (resp., an FFD, an SFFD). Let D be an integral domain with quotient field K , and let I be a proper ideal of D. If t is transcendental over D, then R = D[I t, t −1 ] is called the (generalized) Rees ring of D with respect to I . Observe that the (generalized) Rees ring R is a Z-graded integral domain with quotient field K (t). Various factorization properties of R when I is principal were studied by D. D. Anderson and the first author in [3]. In order to generalize some of the results obtained in [3], H. Kim, T. I. Keon, and Y. S. Park introduced in [43] the notions of graded atomic domain, graded BFD, and graded FFD. Definition 6.34 Let R be a graded integral domain. (1) R is graded atomic if every nonunit homogeneous element of R ∗ is a product of finitely many homogeneous irreducibles in R. (2) R is a graded BFD if R is graded atomic, and for every nonunit homogeneous element of R ∗ , there is a bound on the lengths of factorizations into homogeneous irreducibles. (3) R is a graded FFD if every nonunit homogeneous element of R ∗ has only finitely many non-associate homogeneous irreducible divisors. We are in a position to characterize when a (generalized) Rees ring is a BFD (or an FFD).

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Proposition 6.35 ([43, Proposition 2.5]) For an integral domain D with a proper ideal I , assume that the (generalized) Rees ring R = D[I t, t −1 ] is atomic and t −1 ∈ P(R). Then the following statements are equivalent. (a) R is a BFD (resp., an FFD). (b) R is a graded BFD (resp., a graded FFD). (c) D is a BFD (resp., an FFD). Proof (a) ⇒ (b) ⇒ (c): These implications follow immediately. (c) ⇒ (a): We will only prove the BFD part, as the FFD part follows similarly. Assume that D is a BFD. It follows from Corollary 6.2 that D[t, t −1 ] is also a BFD. Because t −1 is a prime in R, the multiplicative set S it generates in R is a splitting multiplicative set by Lemma 5.6. It is clear that R S = D[t, t −1 ]. Since D[t, t −1 ] is a BFD, part (1) of Theorem 5.15 guarantees that R is also a BFD.  Remark 6.36 The statement of Proposition 6.35 still holds if one simultaneously replaces being a BFD by satisfying ACCP and being a graded BFD by satisfying ACC on homogeneous principal ideals (see [43, Proposition 2.5]).

7 Generalized Bounded and Finite Factorization Domains In this section, we present an abstraction of the unique and finite factorization properties based on an extended notion of a factorization. These ideas were introduced by D. D. Anderson and the first author in [2]. In the same paper, they considered a similar abstraction for half-factoriality and other-half-factoriality (called quasi-factoriality in [2]) that we will not consider here. Let R be an integral domain, and let r be a nonunit of R ∗ . An atomic factorization of r in R is an element a1 · · · an of the free commutative monoid on I (R) (i.e., a formal product of irreducibles up to order) such that a1 · · · an = r in R. Note that, by definition, two atomic factorizations are not identified up to associates. Definition 7.1 Let R be an integral domain, and let ≈ be an equivalence relation on I (R). Then we say that two atomic factorizations a1 · · · am and b1 · · · bn in R are ≈-equivalent if m = n and there is a permutation σ of [[1, m]] such that bi ≈ aσ (i) for every i ∈ [[1, m]]. (1) R is a ≈-CKD if R is atomic and has only finitely many irreducible elements up to ≈-equivalence. (2) R is a ≈-FFD if R is atomic and every nonunit r ∈ R ∗ has only finitely many factorizations in R up to ≈-equivalence. (3) R is a ≈-UFD if R is atomic and for every nonunit r ∈ R ∗ , any two factorizations of r in R are ≈-equivalent. With the notation as in Definition 7.1, observe that when ≈ is the associate relation on I (R), we recover the standard definitions of a CKD, an FFD, and a UFD from

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those of a ≈-CKD, a ≈-FFD, and a ≈-UFD, respectively. The following example is [2, Example 2.6(a)]. Example 7.2 Let R be the ring power series Q[[X ]], and definethe equivaof ∞ ∞ i i lence relation ≈ on I (R) = { i=1 bi X ∈ R | b1  = 0} by setting i=1 bi X ≈ ∞ i i=1 ci X whenever b1 c1 > 0. It can be readily verified that R is a ≈-FFD and a ≈-CKD. In addition, R is a UFD that is not a ≈-UFD. Note that the relation ≈ is strictly contained in the associate relation on I (R). It is clear that if R is a ≈-FFD, then R is a BFD. We record this observation for future reference. Remark 7.3 Let R be an integral domain, and let ≈ be an equivalence relation on I (R). If R is a ≈-FFD, then R is a BFD. Although when ≈ is the associate relation, the definitions of a ≈-FFD and a BFD are not equivalent, they may be equivalent for other choices of ≈. The next example is [2, Example 2.1(c)]. Example 7.4 Let R be an integral domain, and let ≈ be the full equivalence relation on I (R), that is, r ≈ s for all r, s ∈ I (R). Observe that two atomic factorizations of a nonunit in R ∗ are ≈-equivalent if and only if they involve the same number of irreducibles. As a consequence, R is a ≈-FFD if and only if R is a BFD, and R is a ≈-CKD if and only if R is atomic. A CKD (resp., an FFD, a UFD) may not be a ≈-CKD (resp., ≈-FFD, ≈-UFD). To illustrate this, we use [2, Example 2.1(b)]. Example 7.5 Let R be an integral domain, and let ≈ be the diagonal relation on I (R), that is, a ≈ b if and only if a = b for all a, b ∈ I (R). (1) Suppose that R is a ≈-CKD. Because R is atomic and I (R) is finite, the multiplicative monoid R ∗ is finitely generated, and it follows from [39] that R ∗ is finite. In this case, R is a field. Thus, a CKD containing an irreducible cannot be a ≈-CKD. (2) Suppose now that R contains at least one irreducible. Then it is clear that R is a ≈-UFD if and only if R is a UFD and U (R) = {1}. Similarly, R is a ≈-FFD if and only if R is an FFD and U (R) is finite (i.e., R is an SFFD). If R is an integral domain and ≈ is an equivalence relation on I (R), then every implication in Diagram (7.1) holds.

UFD

≈ -UFD

≈ -FFD

FFD

BFD

≈ -CKD atomic domain

(7.1) For an integral domain R, we let ∼ be the associate relation on I (R).

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Proposition 7.6 ([2, Theorem 2.5]) Let R be an integral domain, and let ≈, ≈1 , and ≈2 be equivalence relations on I (R). Then the following statements hold. (1) If ≈1 ⊆ ≈2 and R is a ≈1 -FFD, then R is a ≈2 -FFD. In particular, if R is an FFD and ∼ ⊆ ≈, then R is a ≈-FFD. (2) If R is a ≈-CKD and a BFD, then R is a ≈-FFD. Proof (1) The first statement is a direct consequence of part (2) of Definition 7.1, while the second statement is a special case of the first statement. (2) Let r be a nonunit of R ∗ . Since R is a ≈-CKD, for every  ∈ N, the element r has only finitely many atomic factorizations that are non-equivalent with respect to ≈ and involve exactly  irreducibles. This, along with the fact that R is a BFD, implies that r has only finitely many atomic factorizations up to ≈-equivalence. Thus, R is a ≈-FFD.  In part (1) of Proposition 7.6, we observe that an FFD R can also be a ≈-FFD for an equivalence relation on I (R) satisfying ≈  ∼ (see, for instance, Example 7.2). On the other hand, none of the conditions in the hypothesis of part (2) of Proposition 7.6 is superfluous. In addition, although every ≈-FFD is a BFD (for any relation ≈ on the set of irreducibles), the reverse implication of part (2) of Proposition 7.6 does not hold. The following examples, which are part of [2, Example 2.6], illustrate these observations. Example 7.7 (1) Since every CKD is an FFD, it follows from part (1) of Example 7.5 that any CKD R containing at least one irreducible is a BFD that is not a ≈-CKD when ≈ is taken to be the diagonal relation on I (R). (2) Consider the additive submonoid M = 1/ p | p ∈ P of Q≥0 , and let R be the monoid domain Q[M]. We have seen in Example 4.8 that R satisfies ACCP but is not a BFD. In addition, we have seen in Example 7.4 that when ≈ is the full relation I (R)2 , the integral domain R is a BFD if and only if it is a ≈-FFD and also that R is atomic if and only if it is a ≈-CKD. As a result, R is a ≈-CKD that is not a ≈-FFD. (3) To see that the converse of part (2) of Proposition 7.6 does not hold, it suffices to take an FFD that is not a CKD, for instance, the ring of integers Z. The following theorem describes how the extended notion of a ≈-FFD behaves with respect to the D + M construction. Theorem 7.8 ([2, Theorem 2.10]) Let T be an integral domain, and let K and M be a subfield of T and a nonzero maximal ideal of T , respectively, such that T = K + M. For a proper subfield k of K , set R = k + M. Let ≈ be an equivalence relation on I (T ), and set ≈ = ≈ ∩I (R)2 . Then the following statements hold. (1) If T is quasilocal, then R is a ≈-FFD if and only if T is a ≈-FFD. (2) If T is not quasilocal, then R is a ≈ -FFD if T is a ≈-FFD.

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Proof (1) Since T is quasilocal, R is quasilocal and I (R) = I (T ) ⊆ M by Lemma 4.18, and then one can easily see that P(R) is empty. In addition, we have seen in the proof of Proposition 5.32 that R is atomic if and only if T is atomic. As a consequence, R is a ≈-FFD if and only if T is a ≈-FFD. This, together with the fact that I (R) = I (T ), guarantees that R is ≈ -FFD if and only if T is a ≈-FFD. (2) Suppose now that T is not quasilocal. In this case, R is not quasilocal. Once again, it follows from Lemma 4.18 that I (R) = I (T ) ∩ R, and one can check that P(R) = (P(T ) ∩ R) \ M (in this case, P(R) may be nonempty). Then R is a  ≈ -FFD if T is a ≈-FFD. There are integral domains R with a relation ≈ on I (R) such that R is a ≈-FFD, but R is neither a ≈-UFD nor an FFD. Example 7.9 Consider the monoid domain R = Q[M], where M is the additive monoid {0} ∪ Q≥1 . We have already seen in Example 4.7 that R is a BFD that is neither an FFD nor an HFD. Observe that the monoid domain R[Y ] is a BFD by Corollary 6.2 and that R[Y ] is not an FFD (resp., an HFD) because R is not an FFD (resp., an HFD). Finally, note that if T is the DVR we obtain by localizing R[Y ] at the maximal ideal Y R[Y ] and ≈ denotes the equivalence relation on R[Y ] defined by being associates in T , then R[Y ] is a ≈-FFD that is not a ≈-UFD. Lastly, we determine when the polynomial ring R[X ] is a ∼ K [X ] -FFD, where two elements of R[X ] are related with respect to ∼ K [X ] whenever they are associates in K [X ] (here K is the quotient field of R). Theorem 7.10 ([2, Theorem 2.14]) Let R be an atomic integral domain with quotient field K . Then R[X ] is a ∼ K [X ] -FFD if and only if R is a BFD. Proof Let ≈ denote ∼ K [X ] . For the direct implication, it suffices to note that if R[X ] is a ≈-FFD, then it is a BFD, and so R must be a BFD. Conversely, suppose that R is a BFD. It follows from Theorem 6.1 that R[X ] is also a BFD. Take a nonunit f ∈ R[X ], and take  ∈ N such that max L R[X ] ( f ) < . Observe that if two atomic factorizations of f are ≈-equivalent, then they must contain the same number of irreducibles in R and the same number of irreducibles in R[X ] \ R. For m, n ∈ N0 such that m + n ≤ , suppose that c1 . . . cm f 1 . . . f n and d1 . . . dm g1 . . . gn with ci , di ∈ R and f j , g j ∈ R[X ] \ R, are two atomic factorizations of f in R[X ]. If these factorizations are ≈-equivalent, then, after a possible reordering, f i K [X ] = gi K [X ]. Since both f i and gi divide f for every i ∈ [[1, n]] and the set {h K [X ] | f ∈ h R[X ]} is finite, we can conclude that f has only finitely many factorizations up to ≈-equivalence. Thus, R[X ] is a ≈-FFD.  Acknowledgements While working on this paper, the second author was supported by the NSF postdoctoral award DMS-1903069.

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Factorization and Irreducibility in Modules A. A. Altidor, H. E. Bruch, and J. R. Juett

Abstract Dan Anderson and his students laid the foundations for much of the definitions and theory concerning factorization in commutative rings with zero divisors. In this chapter we study module-theoretic generalizations of some notions they studied. Our focus is on formulating appropriate definitions of the various kinds of “irreducibility” and “atomicity.” We study the consequences of these definitions and investigate to what extent ring-theoretic results about these notions generalize to modules. Keywords Factorization · Module · Irreducibility · Atomicity 2010 Mathematics Subject Classification 13A05 · 13C99 · 13F15

1 Introduction In recent years, the study of non-unique factorization has received wide attention. The focus has primarily been on factorization in integral domains (e.g., [2]), where the theory and definitions are most straightforward, but this study has more recently been expanded to factorization in commutative rings and then to factorization in modules over commutative rings. An excellent introduction to this topic can be found in the ground breaking paper [7] by Dan Anderson and his former student Silvia ValdesA. A. Altidor Department of Mathematical Sciences, Bentley University, Waltham, MA 02452, USA e-mail: [email protected] H. E. Bruch Department of Mathematics, Austin Community College, Austin, TX 78745, USA e-mail: [email protected] J. R. Juett (B) Department of Computer Studies and Mathematics, University of Dubuque, Dubuque, IA 52001, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. Badawi and J. Coykendall (eds.), Rings, Monoids and Module Theory, Springer Proceedings in Mathematics & Statistics 382, https://doi.org/10.1007/978-981-16-8422-7_3

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Leon. This paper united previous authors’ work under one cohesive framework and has opened the floodgates to further investigation of the topic of factorization in commutative rings with zero divisors, being cited well over a hundred times. The various kinds of “irreducible” elements (ones that in some sense cannot be nontrivially factored) play a critical role in the theory of factorization in commutative rings, yet so far the existing literature has not even defined these notions for elements of modules. (At least not in a way consistent with the established ring-theoretic definitions.) Anderson and Valdes-Leon partially explored the topic of factorization in modules in [8], but their exploration was limited by the absence of a definition of “irreducibility” in this setting. In this chapter we will fill this gap, generalizing “irreducibility” and other notions from the theory of factorization in commutative rings to a module setting. We will show that many of the ring-theoretic factorization results derived by Anderson and his students generalize to modules, while also providing a few results that are new even in the commutative ring special case. Many concrete examples will be provided to illustrate the differences between the various notions we study and which results from the ring case do not generalize. Through the course of presenting our new results, we will also review some of the historical development of the ring-theoretic concepts by Anderson and his students. In Sect. 2 we will give module-theoretic generalizations of each of the four notions of “irreducibility” studied by Anderson and Valdes-Leon [7]. We will also study elements of modules that are “prime” in a sense closely related to the work of Behboodi and Koohi [10]. By extending this notion of “primeness,” we will arrive at a moduletheoretic analog of the “weakly prime” elements introduced by Galovich [16]. We will develop many characterizations and properties of the above six notions (such as how they behave with respect to direct products), determine all of the implications between them, and give several concrete examples to illustrate the differences. As a means of generating natural examples, we completely determine which elements of finitely generated or torsion abelian groups (i.e., Z-modules) satisfy each form of “irreducibility.” A commutative ring satisfies a form of “atomicity” if every nonzero nonunit can be written as a product of the appropriate kind of “irreducible” elements. Anderson and Valdes-Leon [8] previously gave definitions of various module-theoretic generalizations of “atomicity,” which by necessity referred to “irreducibility” in the base ring rather than the module. In Sect. 3 we will take a different approach, formulating generalizations of “atomicity” in terms of the notions of “irreducibility” we have defined for elements of modules. By analogy with Sect. 2, we will completely determine the implications between the six forms of atomicity we define, investigate how they behave with respect to direct products, and look at several concrete examples, including a complete characterization of which finitely generated or torsion abelian groups satisfy the various forms of atomicity. Before we begin, let us establish some conventions. Throughout, all rings will be commutative with 1 = 0 and all modules will be unital. Let R be a ring and M be an R-module. We respectively use Nil(R), J (R), and U (R) to denote the nilradical, Jacobson radical, and group of units of R. For submodules A and B of M, we set (A : B) := {x ∈ R | x B ⊆ A}. The annihilator of M is ann(M) := (0 : M) and

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 its set of zero divisors is Z(M) := 0=a∈M ann(a), where ann(a) := ann(Ra). We call R connected or indecomposable if it has no nontrivial idempotents, completely decomposable if it is a finite direct product of connected rings, quasilocal if it has a unique maximal ideal, semi-quasilocal if it has only finitely many maximal ideals, and (semi-)local if it is Noetherian and (semi-)quasilocal.

2 Notions of Irreducibility A major insight of Anderson and Valdes-Leon [7] was that different “associate” relations lead to different notions of “irreducibility” in commutative rings. In this section we will use a similar approach with our module-theoretic generalizations. Let M be an R-module. Following Anderson and Valdes-Leon [8], we say a, b ∈ M are R-associates (resp., strong R-associates, very strong R-associates), written a ∼ R b (resp., a ≈ R b, a ∼ = R b), if Ra = Rb (resp., a = ub for some unit u, (i) Ra = Rb and (ii) a = xb implies a = b = 0 or x ∈ U (R)). (These relations are symmetric [8, Sect. 2], so there is no grammatical ambiguity here.) For instance, in Z/5Z, the Z-associates of 1 are 1, 2, 3, and 4, the strong Z-associates of 1 are 1 and 4, and 1 has no very strong Z-associates. By contrast, in Z the notions of Z-associates, strong Z-associates, and very strong Z-associates coincide. When there is no danger of confusion, we drop the “R” from the preceding definitions and symbols, writing simply “associates,” “∼,” etc. Analogous conventions will apply to future phrases and symbols we define. We have ∼ ≤ ≈ ≤ ∼ = and the inequalities can be strict [8, Sect. 2]. Note that a ∈ M is (very) (strongly) associate to 0 if and only if a = 0. The following two simple results elaborate on when the various “associate” relations coincide for nonzero elements. In their statements, we say a is R-présimplifiable if a = xa implies x ∈ U (R). We call M R-présimplifiable [8] if each nonzero element is R-présimplifiable. (The notion of a présimplifiable ring is due to Bouvier [11– 13].) Examples of présimplifiable modules include torsion-free modules over integral domains and all modules over quasilocal rings [8, Sect. 2]. Proposition 2.1 Let M be a nontrivial R-module and 0 = a ∈ M. The following are equivalent. (1) (2) (3) (4) (5) (6) (7)

a is R-présimplifiable. a∼ = R a. a∼ = R b for some b ∈ M. Every R-associate of a is a very strong R-associate. ann(a) ⊆ J (R). a is R/ann(M)-présimplifiable and ann(M) ⊆ J (R). a = xb implies x and b are R-présimplifiable.

Proof Let R := R/ann(M). (6) ⇒ (4): Assume a ∼ R b, say a = xb and b = ya. So a = x ya and (6) implies R = (x y) + ann(M) ⊆ (x) + J (R). Therefore x ∈ U (R), showing that a ∼ = R b. (4) ⇒ (2) ⇔ (1) ⇒ (3): Clear. (3) ⇒ (5): Assume

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∼ b. For each x ∈ ann(a) = ann(b) and r ∈ R, we have Ra = R(1 − r x)b, hence a= 1 − r x ∈ U (R) by (3). Therefore ann(a) ⊆ J (R). (5) ⇒ (6): Assume ann(a) ⊆ J (R) and a = xa. Then (1 − x) + ann(M) ⊆ ann(a) ⊆ J (R), hence x ∈ U (R) and x ∈ U (R). (1) ⇒ (7): Assume a = xb is R-présimplifiable. If x = yx or b = yb, then a = x yb = ya, hence y ∈ U (R). So x and b are R-présimplifiable. (7) ⇒ (1): Clear.  Corollary 2.2 (cf. [8, Sect. 2]) The following are equivalent for a nontrivial Rmodule M. (1) M is R-présimplifiable. (2) The notions of R-associates, strong R-associates, and very strong R-associates coincide in M. (3) Z(M) ⊆ J (R). (4) M is R/ann(M)-présimplifiable and ann(M) ⊆ J (R). Proposition 2.1(6) and Corollary 2.2(4) illustrate a theme we will be pursuing throughout this chapter, examining the similarities and differences between factorization in M considered as an R-module and considered as an R/ann(M)-module. With many properties that we will study, we can freely pass back and forth between these two base rings, which often simplifies things. For the reader’s convenience, we record a couple of further very basic properties of the three “associate” relations in the following proposition. Here by a congruence relation on an R-module M, we mean an equivalence relation ≡ with the property that a ≡ b implies xa ≡ xb for all x ∈ R. Proposition 2.3 Let M be an R-module. (1) ∼ R and ≈ R are congruence relations on M. (2) ∼ = R is a symmetric and transitive relation on M, but is a reflexive (or equivalently congruence) relation on M if and only if M is R-présimplifiable. (3) If a ∼ = R c. = R b and b ∼ R c (a, b, c ∈ M), then a ∼ Proof Part (1) follows routinely from the definitions. (The fact that ∼ R and ≈ R are equivalence relations is noted in [8, Sect. 2].) Part (3) and the assertion about when ∼ = R is a reflexive/congruence relation follow from Proposition 2.1 and (1). So ∼ = R is transitive. It is symmetric by a trivial adjustment to the proof of [7, Theorem 2.2(1)].  By definition, units are excluded from being considered irreducible elements of rings. Thus, in order to define “irreducibility” in modules we will need to define the analog of units. Let M be an R-module. Following Anderson and Valdes-Leon [8], we say a ∈ M is (very) (strongly) R-primitive if Ra ⊆ Rb implies a is (very) (strongly) R-associate to b. They called a ∈ M R-superprimitive if xa = yb implies (x) ⊆ (y). (Other authors who have studied these notions, within the more limited context of torsion-free modules over integral domains, are Nicolas [21, 22], Costa [14], Lu [20], and Angermüller [9]. The terminology of Anderson and Valdes-Leon

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that we adopt is consistent with that of Costa, but Nicolas, Lu, and Angermüller used “irreducible” and “primitive” to mean what we have respectively called “primitive” and “superprimitive.” We made the choice we did because in the latter three authors’ terminology the “irreducible” elements of the R-module R would be the units, an inconsistency with the established ring-theoretic definitions.) Superprimitive ⇒ faithful very strongly primitive ⇒ very strongly primitive ⇒ strongly primitive ⇒ primitive [8, Proposition 2.3], but none of the implications reverse [8, Example 2.4 and Theorem 2.9]. Example 2.4 We give some examples to illustrate présimplifiability and the various forms of “primitivity.” (1) The (super)primitive elements of a ring are simply its units [8, Example 2.4(1)]. More generally, the primitive elements of a nonzero cyclic R-module R/I are precisely the units of the ring R/I . Also note that R/I is R-présimplifiable if and only if it is présimplifiable as a ring and I ⊆ J (R). (2) Let n ≥ 2 and consider the Z-module G := Z/nZ. An element m is never very strongly Z-primitive and it is (strongly) Z-primitive if and only if gcd(m, n) = 1 (and n = 2, 3, 4, or 6) [8, Example 2.4(2)]. No element of G is Z-présimplifiable, but G is a présimplifiable ring if and only if n is a prime power. (3) Let R be a ring. A polynomial f ∈ R[X ] is R-présimplifiable if and only if ann(c( f )) ⊆ J (R), where c( f ) denotes the ideal of R generated by the coefficients of f . So R[X ] is R-présimplifiable if and only if Z(R) ⊆ J (R) (or equivalently R is a présimplifiable ring). But R[X ] is présimplifiable as a ring if and only if Z(R) = Nil(R) [7, Sect. 6]. A polynomial f ∈ R[X ] is very strongly Rprimitive if and only if it is a “primitive” polynomial in the usual sense, i.e., c( f ) is not contained in any proper principal ideals (cf. [8, Example 2.4(3)]). If R is a domain, then 0 = f ∈ R[X ] is R-superprimitive if and only if (c( f )−1 )−1 = R [8, Example 2.4(3)], i.e., f is a “superprimitive” polynomial in the sense of Tang [23]. We collect some further basic facts about “primitivity” in the following proposition. Proposition 2.5 Let M be a nontrivial R-module and a ∈ M. (1) a is R-primitive if and only if it is R/ann(M)-primitive. (2) If ann(M) ⊆ J (R), then a is strongly R-primitive if and only if it is strongly R/ann(M)-primitive. (3) a is very strongly R-primitive (or equivalently R-primitive and R-présimplifiable) if and only if it is very strongly R/ann(M)-primitive and ann(M) ⊆ J (R). (4) a is R-superprimitive if and only if Rxa ⊆ Ryb implies (x) ⊆ (y). (5) Associates of superprimitive (resp., (very) (strongly) primitive) elements are superprimitive (resp., (very) (strongly) primitive). Proof Parts (1) and (4) follow easily from the definitions. Let R := R/ann(M).

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(2) If ann(M) ⊆ J (R), then U (R) = {x | x ∈ U (R)}, so the notions of strong Rassociates and strong R-associates coincide. The result immediately follows. (3) We may assume a = 0. By Proposition 2.1 and (1), we see that a is very strongly R-primitive ⇔ a is R-primitive and R-présimplifiable ⇔ a is R-primitive and R-présimplifiable and ann(M) ⊆ J (R) ⇔ a is very strongly R-primitive and ann(M) ⊆ J (R). (5) The superprimitive case follows immediately from (4). Now assume a is (very) (strongly) primitive and Ra = Rb ⊆ Rc. Then a is (very) (strongly) associate to b and c, hence b is (very) (strongly) associate to c by Proposition 2.3. Therefore b is (very) (strongly) primitive.  Anderson and Valdes-Leon [7] defined a nonunit a of a ring R to be (very) (strongly) irreducible if a = x y implies a is a (very) (strong) associate of x or y. A natural way to attempt to generalize the definition of irreducibility to a non-primitive element a of an R-module M is: a = xb implies Ra = x M or Rb. (This is similar to how a prime R-submodule of M is defined to be a proper R-submodule P such that P ⊇ Rxb implies P ⊇ x M or Rb.) However, this seems like a much stronger property than we really want. For example, consider (2, 0) in the Z-module Z × Z. If (2, 0) = x(a, b), then x is a unit or (a, b) is very strongly primitive, so (2, 0) ought to be “irreducible” by any reasonable definition. But (2, 0) = 2(1, 0) and Z(2, 0) = 2(Z × Z) or Z(1, 0). Our definitions of “irreducibility” take a different approach, using the following result as motivation. Proposition 2.6 A nonunit a of a ring is (very) (strongly) irreducible if and only if a = x yz implies a is a (very) (strong) associate of x z or yz. Proof (⇒): Assume a is (very) (strongly) irreducible, a = x yz, and a is not (very) (strongly) associate to yz. Then (a) = (x) = (x z). If a is strongly irreducible, then a ≈ x z by [7, Theorem 2.10]. If a is very strongly irreducible, then it is zero or présimplifiable, hence a ∼  = x z. (⇐): Take z := 1. Thus we say a non-primitive element a of an R-module M is (very) (strongly) Rirreducible if a = x yb implies a is a (very) (strong) associate of xb or yb. Anderson and Valdes-Leon [7] defined a nonunit x ∈ R to be m-irreducible if (x) ⊆ (y) implies (y) = (x) or R. By analogy, we say a non-primitive a ∈ M is m-R-irreducible if Ra ⊆ Rb ⊆ Rc implies Rb = Ra or Rc. For some simple examples of the preceding definitions, we note that the element 5 = 2 · 3 · 5 of Z/25Z is (m-)Z-irreducible but not strongly Z-irreducible, while the element 2 = 32 · 2 of Z/4Z is strongly Zirreducible but not very strongly Z-irreducible, and the element (0, 1) of Z × Z is strongly (Z × Z)-irreducible but not m-(Z × Z)-irreducible. (See Theorems 2.19 and 2.21 below for further details and generalizations.) Agargün, Anderson, and Valdes-Leon [1] called a nonunit p ∈ R weakly prime if ( p) ⊇ (x y) = (0) implies ( p) ⊇ (x) or (y). (These elements were introduced by Galovich [16], who simply called them “prime.”) By analogy, we call a non-R-primitive a ∈ M (weakly) F-Rprime if Ra ⊇ Rx yb (= 0) implies Ra ⊇ Rxb or Ryb. (So our F-prime elements correspond not to prime submodules but to a weaker notion introduced by Behboodi

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and Koohi [10]. They called a proper R-submodule P of M “weakly prime” if P ⊇ Rx yb implies P ⊇ Rxb or Ryb. We will refer to this notion as BK-prime to avoid confusion. An element a ∈ M is then defined to be (BK-)prime if Ra is a (BK-)prime R-submodule of M. So a is F-prime if and only if it is non-primitive and BK-prime. The notions of prime, BK-prime, and F-prime are equivalent for elements of rings, but they are distinct in modules. For instance, in the Z-module Z × Z, the element (1, 0) is prime but not weakly F-prime, while (2, 0) is F-prime but not prime.) Here we think of “F-prime” as an abbreviation for “factorially prime,” indicating that this is the sort of “primeness” connected to factorization. The following proposition gives an equivalent natural way to define (very) strong irreducibility. Proposition 2.7 An element a of a module is (very) strongly irreducible if and only if it is not (very) strongly primitive and a = x yb implies a is a (very) strong associate of xb or yb. Proof (⇒): Clear. (⇐): Assume a = x yb implies a is a (very) strong associate of xb or yb. We will finish the proof by showing that each associate b of a is a (very) strong associate. (So a is non-primitive if it is not (very) strongly primitive.) Write b = ya and a = xb = x ya. Then a is (very) strongly associate to xa or ya = b. If a ≈ xa, then a = y(xa) ≈ ya = b since ≈ is a congruence relation. If a ∼ = xa, then a is zero or présimplifiable, so associates of a are very strong associates.  Let R be a ring. Then 0 is prime (or equivalently (very strongly) irreducible) in R if and only if R is a domain, but 0 is m-irreducible in R if and only if R is a field [7, Theorem 2.13]. We generalize these facts. Theorem 2.8 Let M be an R-module. (1) 0 is always weakly F-prime in M but never présimplifiable or superprimitive. (2) 0 is (very strongly) primitive in M if and only if M is trivial. (3) The following are equivalent if M is nontrivial. (a) (b) (c) (d) (e) (f)

0 is (very strongly) irreducible in M. 0 is (very strongly) R/ann(M)-irreducible in M. 0 is F-prime in M. ann(a) is prime for each 0 = a ∈ M. {ann(a)}a∈M is a chain of radical ideals. {ann(N ) | N is a nonzero submodule of M} is a chain of prime ideals.

(4) 0 is m-irreducible in M if and only if ann(M) is maximal. Proof (Part (3) is partially given in [10, Proposition 2.1].) Parts (1)–(2) follow easily from the definitions, as does the equivalence of (3a)–(3c). We will later derive the equivalence of (3c)–(3f) as a special case of Theorem 2.18. (4) (⇒): Assume 0 is m-irreducible in M. Then M is nontrivial and every nonzero element is primitive. So every nonzero cyclic submodule Ra is simple, i.e.,

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Fig. 1 Implications Between Factorization Properties for Nonzero Elements of Modules

ann(a) is maximal. We show that ann(M) is maximal by demonstrating that ann(a) = ann(b) for all nonzero a, b ∈ M. Suppose not. Then x + y = 1 for some x ∈ ann(a) and y ∈ ann(b). Since x(a + b) = xb = (1 − y)b = b and y(a + b) = ya = (1 − x)a = a, we have 0  Ra  Ra + Rb = R(a + b), contradicting the m-irreducibility of 0. (⇐): If ann(M) is maximal, then every nonzero cyclic submodule of M is simple, hence 0 is m-irreducible in M.  For nonzero elements of rings, unit ⇒ présimplifiable ⇐ very strongly irreducible ⇒ m-irreducible ⇒ strongly irreducible ⇒ irreducible ⇐ weakly prime ⇐ prime, with no further nontrivial implications (cf. [7, Theorem 2.13]). As Fig. 1 illustrates below, the situation in modules is considerably more complex. We have intentionally omitted R/ann(M)-primitive, (m-)R/ann(M)-irreducible, and (weakly) FR/ann(M)-prime from Fig. 1, since clearly these properties are the same thing as R-primitive, (m-)R-irreducible, and (weakly) F-R-prime, respectively. Note that mirreducible does not imply strongly irreducible in modules. Theorem 2.9 The implications in Fig. 1 hold. Proof Let M be an R-module, 0 = a ∈ M, and R := R/ann(M). (R-présimplifiable) ⇔ (R-présimplifiable + ann(M) ⊆ J (R)): Proposition 2.1. (R-superprimitive) ⇒ (very strongly R-primitive): [8, Proposition 2.3(4)]. (Rsuperprimitive) ⇒ (R-superprimitive): Clear. (Very strongly R-primitive) ⇔ (Rprimitive + R-présimplifiable) ⇔ (very strongly R-primitive + ann(M) ⊆ J (R)): Proposition 2.5(3). (Very strongly R-primitive) ⇒ (strongly R-primitive) ⇒ (strongly R-primitive) ⇒ (R-primitive): Clear. (Strongly R-primitive + ann(M) ⊆ J (R)) ⇒ (strongly R-primitive): Proposition 2.5(2). (Very strongly R-irreducible) ⇒ (strongly R-irreducible) ⇒ (strongly R-irreducible) ⇒ (R-irreducible) ⇐ (weakly F-R-prime) ⇐ (F-R-prime): Clear. (Strongly R-irreducible + ann(M) ⊆ J (R)) ⇒

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(strongly R-irreducible): The proof of Proposition 2.5(2) applies. (Very strongly R-irreducible) ⇒ (R-présimplifiable): If a = 1 · 1a is very strongly R-irreducible, then a ∼ = R a, hence a is R-présimplifiable. (R-irreducible + R-présimplifiable) ⇒ (very strongly R-irreducible): If a is R-présimplifiable, then R-associates of a are very strong R-associates. (Very strongly R-irreducible) ⇔ (very strongly Rirreducible + ann(M) ⊆ J (R)): Note that a is very strongly R-irreducible ⇔ it is R-irreducible and R-présimplifiable ⇔ it is R-irreducible and R-présimplifiable and ann(M) ⊆ J (R) ⇔ it is very strongly R-irreducible and ann(M) ⊆ J (R). (Very strongly R-irreducible) ⇒ (m-R-irreducible): Since m-R-irreducible is equivalent to m-R-irreducible, we may assume a is very strongly R-irreducible. Write Ra  Rb ⊆ Rc, say a = xb and b = yc. Then a = x yc and Ra = Rb = Ryc, hence a ∼ = R xc by very strong R-irreducibility. So y ∈ U (R) and Rb = Rc, as desired. (m-Rirreducible) ⇒ (R-irreducible): Assume a = x yb is m-R-irreducible and Ra = Rxb. Then Rxb is R-primitive, so Rxb = Rb and Ra = Rx yb = Ryb.  Example 2.10 No further nontrivial implications can be added to Fig. 1. The element 1 (resp., 5) of Z/25Z is Z/25Z-superprimitive (resp., very strongly Z/25Zirreducible) but neither Z-présimplifiable, strongly Z-primitive, nor strongly Z¯ 2), ¯ (2, 2)) of irreducible. The element (2, 1) (resp., (0, 1), (0, 1), (0, 5), (0, 2, the faithful Z-module Z × Z/2Z (resp., Z × Z/2Z, Z × Z/5Z, Z × Z/25Z, Z × Z/4Z × Z/4Z, Z × Z/4Z) is very strongly Z-primitive (resp., strongly Z-primitive, Z-primitive, m-Z-irreducible and F-Z-prime, weakly F-Z-prime, very strongly Zirreducible) but not Z-superprimitive (resp., Z-présimplifiable, strongly Z-primitive, strongly Z-irreducible, F-Z-prime, weakly F-Z-prime). The element (0, 1) of the ring Z × Z is prime and strongly irreducible but neither m-irreducible nor very strongly irreducible. See Theorems 2.19 and 2.21 below for further details and generalizations related to these examples. We next prove several characterizations and basic properties of the various kinds of “irreducibility.” Lemma 2.11 Let M be an R-module. The following are equivalent for a ∈ M. (1) a = x yb with Ra  Rxb, Ryb implies Ra = R(x y)2 b. (2) Ra  Rxb implies Ra = Rxa. Therefore (2) holds if a is irreducible. Proof (For the special case M = R, cf. [7, Theorem 2.9].) (1) ⇒ (2): Assume Ra  Rxb, say a = x yb. If Ra = Ryb, then Rxa = Rx yb = Ra, while if Ra = Ryb, then (1) gives Ra = R(x y)2 b ⊆ Rx 2 yb = Rxa ⊆ Ra. (2) ⇒ (1): If a = x yb with  Ra  Rxb, Ryb, then (2) gives Ra = Rxa = Rx ya = R(x y)2 b. Lemma 2.12 Let M be an R-module. Pick b ∈ M and x ∈ R with Rxb = Rx 2 b. Then every associate of xb has the form uxb with (u) + ann(b) = R. Thus associates of idempotent elements of R are strong associates.

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Proof Say xb = sx 2 b and let e := sx. Then Rxb = Reb and e2 b = eb. It suffices to show that each associate a of eb has the form ueb with (u) + ann(b) = R. Say a = yeb and eb = za = yzeb. Then (ye + 1 − e)eb = a, where (ye + 1 − e) + ann(b) = R since (ye + 1 − e)(ze + 1 − e)b = (e + 1 − e)b = b.  Theorem 2.13 Let a be a non-primitive element of a nontrivial R-module M. The following are equivalent. (1) (2) (3) (4) (5) (6)

a is R-irreducible. Ra = Rx1 · · · xn b implies some Rx j b = Ra. Ra  Rxb, Ryb implies Ra  Rx yb. Ra  Rx1 b, . . . , Rxn b implies Ra  Rx1 · · · xn b. Ra  Rb  Rc implies ann(a)  ann(b) = ann(c). For each Rb  Ra, there is a prime ideal P ⊇ ann(b) such that a is R-primitive in Pb. (7) Every R-associate of a is R-irreducible. (8) a is R/ann(M)-irreducible. (9) a = xb with b = 0 implies x is irreducible or invertible in the ring R/ann(b). Proof (For the special case M = R of (1)–(7), cf. [4, Theorem 2.4].) (4) ⇒ (2) ⇒ (7) ⇒ (1) ⇔ (8): Clear. (1) ⇒ (3): The case a = 0 is trivial, so let us assume a is nonzero and R-irreducible with Ra  Rxb, Ryb. Lemma 2.11 gives Ra = Rxa = Rx ya ⊆ Rx yb. Suppose Ra = Rx yb. Then Ra = Rx ya = R(x y)2 b, so by Lemma 2.12 we have a = (ux)yb with (u) + ann(b) = R. Irreducibility implies Ra = Ruxb = Rxb or Ra = Ryb, a contradiction. (3) ⇒ (4): Use induction. (4) ⇒ (6): Assume (4) holds and Rb  Ra, say a = yb. Then S := {x ∈ R | Ra  Rxb} is a nonempty saturated multiplicative subset of R disjoint from (y) + ann(b), so we may enlarge (y) + ann(b) to a prime ideal P disjoint from S. Thus a is R-primitive in Pb. (6) ⇒ (2): Assume (6) holds and Ra = Rx1 · · · xn b. If Ra = Rb, then each Rxi b = Ra, so let us assume Ra  Rb. Let P be as in (6). Write x1 · · · xn b = yb with y ∈ P. Then x1 · · · xn ∈ (y) + ann(b) ⊆ P, so some x j ∈ P. Thus Ra = Rx j b since a is Rprimitive in Pb. (1) ⇒ (5): Assume a is R-irreducible and Ra  Rb  Rc, say b = yc and a = xb. Then Ra = Rya by Lemma 2.11, say a = sya. Then (1 − sy)a = 0 but (1 − sy)c = c − sb = 0 since Rb  Rc. Thus ann(a)  ann(c). It only remains to show that zb = 0 implies zc = 0. We need to show that (Rzc)M = 0M for each maximal ideal M. If y ∈ / M, then (Rzc)M = (Rzyc)M = (Rzb)M = 0M , so let us assume y ∈ M. Then (Ra)M = (Rya)M = 0M by Nakayama’s Lemma. Since a = x yc = (x + z)yc and Ra = Rb = Ryc, we have Ra = Rxc = R(x + z)c by R-irreducibility, hence (R(x + z)c)M = (Rxc)M = (Ra)M = 0M . Therefore (Rzc)M = 0M , as desired. (5) ⇒ (1): Assume a = x yb and Ra = Ryb. Then Ra  Ryb ⊆ Rb and Ra ⊆ Rxb  Rb. If z ∈ ann(a), then zx ∈ ann(yb) = ann(b) by (5), hence z ∈ ann(xb). So ann(a) = ann(xb) and hence Ra = Rxb by (5), as desired. (1) ⇒ (9): Assume a = xb is R-irreducible with b = 0. Then a is Rirreducible or R-primitive in Rb, hence R/ann(b)-irreducible or R/ann(b)-primitive in Rb by (8) and Proposition 2.5(1). So x is irreducible or invertible in the ring R/ann(b). (9) ⇒ (1): Write a = x yb, where we may assume b = 0. Then x y is

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irreducible or invertible in R/ann(b) by (9), hence (x y) = (x) or (y). So Ra = Rxb or Ryb, as desired.  Corollary 2.14 Let Ra1  · · ·  Ran with each ai an irreducible element of an R-module M. Then n ≤ 2. Proof (The special case M = R is [3, Corollary 2].) Suppose n ≥ 3. Then a3 is not primitive, say Ra3  Rb. Theorem 2.13 gives ann(a1 )  ann(a2 ) = ann(a3 ) and  ann(a2 )  ann(a3 ) = ann(b), a contradiction. Theorem 2.15 Let a be a non-primitive element of an R-module M. The following are equivalent. (1) (2) (3) (4)

a is strongly R-irreducible. Ra = Rx1 · · · xn b implies some x j b ≈ R a. Every R-associate of a is strongly R-irreducible. a is R-irreducible and every R-associate of a is a strong R-associate.

Proof (For the special case M = R, cf. [7, Theorem 2.10].) (1) ⇒ (4): We showed this in the proof of Proposition 2.7. (4) ⇒ (2): Follows from Theorem 2.13. (2) ⇒ (3): Assume Ra = Rc. Then c is not primitive. If c = x1 x2 b, then some x j b ≈ a ≈ c by (2). So c is strongly irreducible. (3) ⇒ (1): Clear.  Theorem 2.16 Let a be a non-primitive element of an R-module M. The following are equivalent. (1) a is m-R-irreducible. (2) If Ra = Rx1 · · · xn b, then for each i either xi b is R-primitive or a = u i xi b with (u i ) + ann(b) = R. (3) Every R-associate of a is m-R-irreducible. (4) a is m-R/ann(M)-irreducible. (5) Ra ⊆ Rb implies b is R-primitive or R-irreducible. (6) a = xb with b = 0 implies x is m-irreducible or invertible in the ring R/ann(b). Proof (For the special case M = R of (1)–(3), cf. [7, Theorem 2.12]. The equivalence of (1) and (5), which may be considered slightly counterintuitive, seems to be new even in this special case.) (1) ⇒ (2): Assume a is m-R-irreducible and Ra = Rx1 · · · xn b. Then each xi b is R-primitive or an R-associate of a. So consider an xi b with Ra = Rxi b, say a = yi xi b. If Ryi b = Rb, then (yi ) + ann(b) = R, so let us assume Ryi b  Rb. Then by m-R-irreducibility Ryi b = Ra = Rxi b. Therefore Rxi b = Ra = Ryi xi b = Rxi2 b, hence a = u i xi b with (u i ) + ann(b) = R by Lemma 2.12. (2) ⇒ (1) ⇔ (3) ⇔ (4) ⇒ (5): Clear. (5) ⇒ (1): Assume (5) holds and Ra  Rb ⊆ Rc, say b = xc. Then Ra  Rx 2 c by Theorem 2.13, so x 2 c is R-irreducible or R-primitive by (5), hence Rxb = Rx 2 c = Rxc = Rb. Say b = sxb. Then (1 − sx)b = 0, so c − sb = (1 − sx)c = 0 by Theorem 2.13. Thus Rb = Rc. (1) ⇒ (6): Adjust the proof of “(1) ⇒ (9)” in Theorem 2.13. (6) ⇒ (1): Write Ra  Rb ⊆ Rc. Then a is m-R/ann(c)-irreducible in Rc by (6), hence m-R-irreducible in Rc by (4). Therefore Rb = Rc. 

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Theorem 2.17 Let a be a nonzero non-primitive element of an R-module M. The following are equivalent. (1) (2) (3) (4) (5) (6) (7) (8)

a is very strongly R-irreducible. Ra = Rx1 · · · xn b implies some x j b ∼ = R a. Ra = Rx yb implies x ∈ U (R) or y ∈ U (R). If Ra = Rx1 · · · xn b, then either (i) each xi ∈ U (R) or (ii) exactly one x j is very strongly irreducible, xi ∈ U (R) for i = j, and b is very strongly R-primitive. Ra = Rxb implies x ∈ U (R) or b is very strongly R-primitive. Every R-associate of a is very strongly R-irreducible. a is R-présimplifiable and R-irreducible. a is very strongly R/ann(M)-irreducible and ann(M) ⊆ J (R).

Proof (For the special case M = R of (1)–(7), cf. [7, Theorem 2.5].) (1) ⇔ (7) ⇔ (8): Theorem 2.9. (7) ⇒ (2): Follows from Theorem 2.13. (2) ⇒ (6): Assume Ra = Rc. Then c is not primitive. If c = x1 x2 b, then some x j b ∼ =a∼ = c by (2). Thus c is very strongly irreducible. (6) ⇒ (1): Clear. (2) ⇒ (4): Assume Ra = / U (R). If Rx1 · · · xn b. By (2), some x j b ∼ = a, so xi ∈ U (R) for i = j. Assume x j ∈ x j = yz, then Ra = Rx j b = Ryzb and (2) implies y ∈ U (R) or z ∈ U (R). Therefore x j is very strongly irreducible. If b = wc, then Ra = Rx j wc and w ∈ U (R) by (2). Therefore b is very strongly primitive. (4) ⇒ (5): Clear. (5) ⇒ (3): If Ra = Rx yb and x ∈ / U (R), then (5) implies yb is very strongly R-primitive and y ∈ U (R). (3) ⇒ (7): Assume (3) holds. If a = xa, then a = x 2 a, hence x ∈ U (R). If a = x yb, then x ∈ U (R) or y ∈ U (R), hence Ra = Ryb or Rxb. So a is R-présimplifiable and R-irreducible.  Theorem 2.18 Let a be a non-primitive element of an R-module M. The following are equivalent. (1) a is F-R-prime. (2) Ra ⊇ I J N with I, J ideals of R and N a submodule of M implies Ra ⊇ I N or J N . (3) (Ra : Rb) is prime for each b ∈ M \ Ra. (4) {(Ra : Rb)}b∈M is a chain of radical ideals. (5) {(Ra : N ) | N is a submodule of M not contained in Ra} is a chain of prime ideals. Proof (The special case a = 0 is partially given, but without proof, in [10, Proposition 2.1].) (1) ⇒ (2): Assume a is F-prime, Ra ⊇ I J N , and Ra  J N . Then there are y ∈ J and c ∈ N with yc ∈ / Ra. We need to show that xb ∈ Ra for each x ∈ I and b ∈ N . If yb ∈ / Ra, then Rx yb ⊆ I J N ⊆ Ra implies xb ∈ Ra by F-primeness. So let us assume yb ∈ Ra. Then x yc, x y(b + c) ∈ Ra but yc, y(b + c) ∈ / Ra, so xc, x(b + c) ∈ Ra by F-primeness. Thus xb ∈ Ra, as desired. (2) ⇒ (5): The only nontrivial detail to check is that (2) implies (Ra : B) and (Ra : C) are comparable for all distinct submodules B and C of M. Without loss of generality, say x ∈ (Ra : B) \ (Ra : C). If yC ⊆ Ra, then x y(B + C) ⊆ Ra but x(B + C)  Ra, hence y B ⊆ y(B + C) ⊆ Ra by (2). Therefore (Ra : C) ⊆ (Ra : B), as desired.

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(4) ⇐ (5) ⇒ (3) ⇒ (1): Clear. (4) ⇒ (1): Assume (4) holds and Ra ⊇ Rx yb. Without loss of generality, say (Ra : Rxb) ⊆ (Ra : Ryb). Then Ra ⊇ Ry 2 b, hence Ra ⊇ Ryb by (4), as desired.  In Example 2.10 we referred several times to factorization in finitely generated abelian groups (considered as Z-modules). As a natural source of further examples, we will find all of the various kinds of “primitive” and “irreducible” elements in finitely generated or torsion abelian groups. The facts about strongly primitive and strongly irreducible elements are highly dependent on specific properties of Z, but we will derive most of our results in the more general context of modules over principal ideal domains (PIDs). Theorem 2.19 Let D be a PID but not a field, Max(D) be the set of maximal ideals of D, M be a nontrivial D-module, and T (M) be the torsion submodule of M. Assume M/T (M) is free. (This holds if M is finitely generated [17, Theorem 4.6.5] or torsion.) Then 3] and ∞  M = M(0) ⊕ T (M) for some submodule M(0) [18, Theorem where M( p) := {a ∈M| T (M) = ( p)∈Max(D) M( p) [17, Theorem 4.6.7], n=1  p n a = 0}. Let a ∈ M and write a = a(0) + ( p)∈Max(D) a( p) with a(0) ∈ M(0) and a( p) ∈ M( p). Let c(a) be the ideal generated by the coordinates of a(0), which does not depend on the choice of basis. (1) a is présimplifiable if and only if (i) a ∈ / T (M) or (ii) (D, ( p1 ), . . . , ( pn )) is semi-local and each a( pi ) = 0. (2) a√ is (very strongly) primitive if and only if either (i) c(a) = D, (ii) (0)  c(a) = ( p1 · · · pn )  D  with each pi prime and each a( pi ) ∈ / pi M( pi ), or n ( pi ) is a finite union of maximal ideals (and (iii) a ∈ T (M), Z(M) = i=1 / pi M( pi ). (D, ( p1 ), . . . , ( pn )) is semi-local), and each ai ( pi ) ∈ (3) a is superprimitive if and only if c(a) = D. (4) a is (m-)irreducible if and only if either √ (i) T (M) = 0 and c(a) is prime (and nonzero), (ii) T (M) = 0, (0)  c(a) = ( p1 · · · pn )  D with the pi ’s or a( p j ) ∈ / prime, a( p j ) ∈ p j M( p j ) for exactly one j, and either c(a)  ( p j )2 n ( pi ) is p 2j M( p j ), or (iii) a ∈ T (M) (and a = 0 if M = T (M)), Z(M) = i=1 a finite union of maximal ideals, a( p j ) ∈ p j M( p j ) for exactly one j, and either / p 2j M( p j ) or p j M( p j ) = 0. a( p j ) ∈ (5) a is very strongly irreducible if and only if either (i) a ∈ / T (M) and a is irreducible, (ii) a = 0 and either T (M) = 0 or ann(T (M)) is maximal, or n ( pi )), (iii) a ∈ T (M), (D, ( p1 ), . . . , ( pn )) is semi-local (and Z(M) = i=1 / p 2j M( p j ). a( p j ) ∈ p j M( p j ) for exactly one j, and a( p j ) ∈ (6) a is (weakly) F-prime if and only if either (i) T (M) = 0 and c(a) is prime, (ii) c(a) = ann(T (M)) = ( p) is maximal and a( p) = 0, (iii) a ∈ T (M), Z(M) =  n i=1 ( pi ) is a finite union of maximal ideals, some p j M( p j ) = Da( p j ), and M( pi ) = Da( pi ) for i = j (or (iv) a = 0, (v) Z(M) = ( p) is maximal, a ∈ pT (M) \ p 2 T (M), and pa = 0, or (vi) Z(M) = ( p) ∪ (q) with ( p) and (q) distinct maximal ideals, pM( p) = 0, a ∈ M(q) \ q M(q), and qa = 0). (7) If D = Z, then a is strongly primitive (resp., strongly irreducible) if and only if it is primitive (resp., irreducible) and either a ∈ / T (M), 4a = 0, or 6a = 0.

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Proof We begin with a few general observations that will facilitate our calculations throughout the proof. For each maximal ideal ( p) and s ∈ D \ ( p), we the D- and D( p) have (s) + ann(a( p)) = D, so Da( p) = D( p) a( p) and hence n submodules of M( p) coincide. Also note that, if a ∈ i=1 M( pi ), then for each k for large enough k. It follows that i we have D( p1 · · · pi−1 p i+1 · · · pn ) a = Da( pi ) if a ∈ T (M), then Da = ( p)∈Max(D) Da( p) = ( p)∈Max(D) D( p) a( p). (1) Recall from Proposition 2.1 a is présimplifiable if and only if ann(a) ⊆ that n M( pi ) is présimplifiable. Then ( p1 · · · pn ) ⊆ J (D). (⇒): Assume a ∈ i=1 √ ann(a) ⊆ J (D) ⊆ ( p1 · · · pn ), so (D, ( p1 ), . . . , ( pn )) is semi-local and each a( pi ) = 0. (⇐): If (i) or (ii) holds, then ann(a) ⊆ J (D), so a is présimplifiable. (2) In view of (1) and the fact that a is very strongly primitive if and only if it is primitive and présimplifiable, we only need to prove the characterization of primitivity. (⇒): √ Assume a is primitive. First consider the case where (0)  c(a)  D. Say c(a) n ) with each pi prime. By (very  strong) primitivity, we have n= ( p1 · · · p n pi M = i=1 ( pi M(0) ⊕ pi M( pi ) ⊕ (q)∈Max(D)\{( pi )} M(q)), hence a∈ / i=1 / pi M( pi ). Now assume a ∈ T (M). Suppose a( p) ∈ pM( p) for each a( pi ) ∈ some maximal ideal ( p) ⊆ Z(M), say a( p) = pb with 0 = b ∈ p). Then M( n ( pi ) is a Da  D(a − a( p) + b), a contradiction. Therefore Z(M) = i=1 / pi M( pi ). (⇐): Clearly a is finite union of maximal ideals and each a( pi ) ∈ very strongly primitive if (i) or (ii) holds. Now assume n (iii) holds. Then a is very ( pi ). So a is D-primitive strongly D S -primitive in T (M), where S := D \ i=1 in T (M) (or equivalently in M) since the D- and D S -submodules of T (M) coincide. (3) (⇒): Assume a is superprimitive, hence faithful. Thus Da(0) = c(a)b for some 0 = b ∈ M(0). Pick 0 = x ∈ D with x(a − a(0)) = 0, so Dxa = xc(a)b. Then (x) ⊆ xc(a) by superprimitivity, hence c(a) = D. (⇐): Assume c(a) = D. If Dxa ⊆ Dyb, then (x) = c(xa) ⊆ c(yb) ⊆ (y). So a is superprimitive. (4) (⇒): Assume a is irreducible. If T (M) = 0 (and a is m-irreducible), then a is very strongly irreducible, so c(a) is prime (and nonzero by Theorem 2.8(4) since ann(M) = (0) is not maximal). So let us assume T (M) = 0. First consider the √ case where a ∈ / T (M). Since a is not primitive, part (2) implies (0)  c(a) = ( p1 · · · pn )  D with each pi prime and a( p j ) ∈ p j M( p j ) for some j. Note / p 2j M( p j ), for otherwise a ∈ p 2j M(0) ⊕ that either c(a)  ( p j )2 or a( p j ) ∈  2 p j M( p j ) ⊕ (q)∈Max(D)\{( p j )} M(q) = p 2j M, violating (very strong) irreducibil/ pi M( ity. Similarly a( pi ) ∈  pi ) for each i = j, for otherwise a ∈ pi p j M(0) ⊕ pi M( pi ) ⊕ p j M( p j ) ⊕ (q)∈Max(D)\{( pi ),( p j )} M(q) = pi p j M. Now assume a ∈ T (M). If a is m-irreducible and M = T (M), then a = 0 by Theorem 2.8(4) since ann(M) = (0) is not maximal. Suppose there are distinct maximal ideals (q1 ), (q2 ) ⊆ Z(M) with each a(qi ) ∈ qi M(qi ), say a(qi ) = qi bi with 0 = bi ∈ M(qi ). Then Da = Dq1 q2 (a − a(q1 ) − a(q2 ) + b1 + b2 ) but Da = D(a − a (q2 ) + b2 ) = Dq1 (a − a(q1 ) − a(q2 ) + b1 + b2 ) and Da = D(a − a(q1 ) + b1 ) − a(q1 ) − a(q2 ) + b1 + b2 ), contradicting irreducibility. Therefore = Dq2 (a  n ( pi ) is a finite union of maximal ideals and a( p j ) ∈ p j M( p j ) Z(M) = i=1 for exactly one j. Assume a( p j ) = p nj b j for some n ≥ 2 and b j ∈ M( p j ). Then

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  Da = Dp nj (b j + i= j a( pi )) = Dp j (b j + i= j a( pi )) by the irreducibility of a. So D( p j ) a( p j ) = D( p j ) p nj b j = D( p j ) p j b j = 0 by Nakayama’s Lemma. Therefore a( p j ) = 0 is D( p j ) -irreducible in M( p j ), hence p j M( p j ) = 0 by Theorem 2.8(3). (⇐): Case (i) will follow from (6) and case (ii) is clear, so let us assume (iii) holds. Then a is non-primitive in T (M) by (2). Note that (if a = 0) and a is (m-)irreducible in T (M), then it is (m-)irreducible in M. So we may pass to T (M) and assume M is torsion. We need to show that Da  Db implies b is primitive. For i = j, we have Da( pi ) = Db( pi ) by (2), / p 2j M( p j ) or p j M( p j ) = 0, we hence Da( p j )  Db( p j ). Since either a( p j ) ∈ / p j M( p j ). So each b( pi ) ∈ / pi M( pi ) and b is primitive by (2), conclude b( p j ) ∈ as desired. (5) Since a is very strongly irreducible if and only if it is irreducible and either zero or présimplifiable, this follows by combining (1) and (4). (6) (⇒): The cases where a = 0 or T (M) = 0 are covered by (4), so let us assume a is nonzero and weakly F-prime and T (M) = 0. First consider the case where a ∈ / T (M). Since a is not primitive, we have a = pd for some 0 = d ∈ M and nonzero prime p ∈ D. We claim that ann(T (M)) = ( p). We need to show that pe = 0 for each maximal ideal (q) and e ∈ M(q). Say q k e = 0. Then 0 = q k a = q k p(d + e), hence Da ⊇ Dp(d + e) = D(a + pe) by weak F-primeness. (If (q) = ( p), then Da  Dq k d = Dq k (d + e) since c(a) is contained in a higher power of ( p) than c(q k d).) In fact a = a + pe since a(0) = (a + pe)(0) is faithful, hence pe = 0, as desired. Thus a( p) ∈ pM( p) =0 and c(a) = ( p) by (4). So let us assume a ∈ T (M). Part (4) implies n ( pi ) is a finite union of maximal ideals, a( p j ) ∈ p j M( p j ) for Z(M) = i=1 / p 2j M( p j ) or p j M( p j ) = 0. Say j = 1 and exactly one j, and either a( p j ) ∈ write a( p1 ) = p1 b with 0 = b ∈ M( p1 ). Then Da( p1 ) = Dp1 b = D( p1 ) p1 b  D( p1 ) b = Db. First assume n = 1 but p1 T (M) = Da. Then a ∈ p1 T (M) \ / Da. Let k be the least positive integer p12 T (M). Pick d ∈ T (M) with p1 d ∈ with p1k d = 0 and note that k ≥ 2. Suppose a is F-prime or p1 a = 0. Then p12 (b + p1k−2 d) = p1 a, so a + p1k−1 d = p1 (b + p1k−2 d) = xa for some x ∈ D. Thus (x − 1)a = p1k−1 d = 0, so p1 d ∈ Da by weak F-primeness, a contradick k tion. 1 a = Dp1 (d + n So we may assume n ≥ 2. For each d ∈ M( p1 ), we have Dp n i=2 a( pi ))  = 0 for k large enough, hence Da ⊇ Dp1 (d + i=2 a( pi )) (or Thus Da( p1 ) = equivalently Da( p1 ) ⊇ Dp1 d) since ais weakly F-prime. n n M( p )  = Da( p ). Reorder p1 M( p1 ). Now further assume that i i i=2 i=2 so that there is a d ∈ M( p2 ) \ Da( p2 ). Then Dp1 p2k (b + d) = Da( p1 ) ⊆ Da for k large enough but neither p1 (b + d) nor p2k (b + d) is in Da. Therefore a is not F-prime and p2k a( p1 ) + p1 p2k d = p1 p2k (b + d) = 0 for some k, hence p1 M( p1 ) = Da( p1 ) = Dp2k a( p1 ) ⊆ M( p1 ) ∩ M( p2 ) = 0. Suppose n ≥ 3. Use the Chinese Remainder Theorem to pick x, y ∈ D with x ≡ 1 (mod ann(b)), y ≡ 0 (mod ann(b)), x ≡ 0 (mod ann(d)), y ≡ 1 (mod ann(d)), and x ≡ y ≡ 1 (mod ann(a( pi ))) for i ≥ 3. Then 0 = x y(b + d + a − a( p1 ) − a( p2 )) = a − a( p1 ) − a( p2 ) ∈ Da but x(b + d + a − a( p1 ) − a( p2 )) = b + / Da and y(b + d + a − a( p1 ) − a( p2 )) = d + a − a( p1 ) a − a( p1 ) − a( p2 ) ∈ − a( p2 ) ∈ / Da, contradicting weak F-primeness. Therefore n = 2 and a ∈

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M( p2 ) \ p2 M( p2 ). Suppose p2 a = 0. Let e := d if p2 d = 0 and e := d + a if p2 d = 0. So e ∈ M( p2 ) \ Da and p2 e = 0. Pick the smallest positive integer k with p2k e = 0. Then Dp1 p2 (b + p2k−1 e + a) = Dp2 a = 0 and Dp2 (b + p2k−1 e + a) = D(b + p2 a)  Da, so Da ⊇ Dp1 (b + p2k−1 e + a)=D( p2k−1 e + a) by weak F-primeness. Therefore Da ⊇ Dp2k−1 e = 0, hence Da ⊇ Dp2 e since a is weakly F-prime. Thus Da contains Dp1 p2 (b + e) = Dp2 e = 0 but neither Dp2 (b + e) = D(b + p2 e) nor Dp1 (b + e) = De, contradicting weak F-primeness. (⇐): If T (M) = 0, then 0 is F-prime in M by Theorem 2.8(3). For the remainder of cases (i) and (ii), assume c(a) = ( p) is maximal, pT (M) = 0, and a( p) = 0. Then a ∈ pM, so a is not (very strongly) primitive. Write xa = yzb. Then yzb( p) = xa( p) = 0. Since 0 is F-prime in M by Theorem 2.8(3), we may assume xa = 0, hence x, y, z = 0. Then (x p) = c(xa) = c(yzb) = yzc(b), so ( p) ⊇ (y), (z), or c(b). If ( p) contains (y) (resp., (z), c(b) but neither (y) nor (z)), then z (resp., y, yz) divides x and yb( p) = 0 (resp., zb( p) = 0, b( p) = 0 since (yz) + ann(b) = D), hence Da contains Dyb (resp., Dzb, Db) since M(0) is torsion-free. Therefore a is F-prime. Now assume (iii) holds. Then a is not primitive by (2). Write Da ⊇ Dx1 x2 b, where we may assume each xi = 0, so b ∈ T (M). Then x1 x2 ∈ (Da( p j ) : Db( p j )) = ( p j M( p j ) : Db( p j )) = ( p j ) or D, so some xk ∈ (Da( p j ) : Db( p j )), while xk b( pi ) ∈ M( pi ) = Da( pi ) for i = j. Thus Da ⊇ Dxk b, showing that a is F-prime. Case (iv) is clear. Now assume either (v) or (vi) holds. Then a is irreducible by (4) and ann(a) is maximal. If xa = yzb = 0, then (x) + ann(a) = D, hence Da = Dxa = Dyzb and Da = Dyb or Dzb by irreducibility, as desired. (7) We may assume D = Z and 0 = a ∈ T (M). Because a is strongly primitive (resp., strongly irreducible) if and only if it is primitive (resp., irreducible) and every associate of a is a strong associate, it suffices to show that all associates of a are strong associates if and only if 4a = 0 or 6a = 0. (⇒): Assume associates of a are strong associates. We claim that the same holds for each a( pi ), where p1 , . . . , pn are the positive primes with a( pi ) = 0. Each associate of a( pi ) has the form ma( pi ) with m ∈ Z \ ( pi ). By the Chinese Remainder Theorem there is an integer y congruent to m modulo ann(a( pi )) and congruent to 1 modulo ann(a( p j )) for j = i. So Zya = Za, hence ya = ±a and ma( pi ) = ±a( pi ), as desired. Therefore the pi -primary cyclic group Za( pi ) has at most 2 generators, so pi = 2 or 3. Thus 5 is relatively prime to each pi , hence 5a = ±a, hence 4a = 0 or 6a = 0. (⇐): If 4a = 0 or 6a = 0, then Za ∼ = Z/2Z, Z/3Z, Z/4Z, or Z/6Z. Since each of these groups has no generators other than ±1, the result follows.  In the case of cyclic modules, the criteria of Theorem 2.19 can be simplified. Corollary 2.20 Let D be a PID, y ∈ D be a nonzero nonunit, and x ∈ D/(y). (1) x is D-présimplifiable if and only if (y) ⊆ xJ (D). (2) x is D-primitive if and only if (x, y) = D. (3) x is F-D-prime (or equivalently (m-)D-irreducible) if and only if (x, y) is prime.

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(4) x is very strongly D-irreducible if and only if (x, y) is prime and either (x) ⊆ (y) or (y) ⊆ xJ (D). (5) If D = Z, then: (a) x is strongly Z-primitive if and only if (x, y) = Z and y = ±2, ±3, ±4, or ±6 [8, Example 2.4(2)]. (b) x is strongly Z-irreducible if and only if (x, y) = ( p) is prime and y = ± p, ±2 p, ±3 p, ±4 p, or ±6 p. Proof (1) Say (x, y) = (d). Then x is D-présimplifiable ⇔ J (D) ⊇ ann(x) = ((y) : (x))=((y) : (d))=( dy ) ⇔ (y) ⊆ dJ (D) = (x, y)J (D) ⇔ (y) ⊆ xJ (D), where the first equivalence is by Proposition 2.1 and the last is by Nakayama’s Lemma. (2)–(3) In the Artinian principal ideal ring D/(y), the notions of prime and (m-)irreducible coincide. Thus the results follow from the fact that x is Dprimitive (resp., (m-)D-irreducible, F-D-prime) if and only if it is invertible (resp., (m-)D-irreducible, prime) in the ring D/(y). (4) Since x is very strongly D-irreducible if and only if it is D-irreducible and either zero or D-présimplifiable, this follows by combining (1) and (3). (5) By (2) (resp., (3)) and Theorem 2.19(7) we see that x is strongly Z-primitive (resp., strongly Z-irreducible) if and only if (x, y) = Z (resp., (x, y) is prime) and y divides 4x or 6x. The result follows.  We conclude the section with an examination of how the various factorization properties behave with respect to direct products. This provides us with another fertile source of examples. Theorem 2.21 For i = 1,. . . , n (2 ≤ n < ∞), let Mi be a nontrivial Ri -module. n n Ri , M := i=1 Mi , and a := (a1 , . . . , an ) ∈ M. Let R := i=1 (1) a is (strongly) R-associate to b := (b1 , . . . , bn ) if and only if each ai is (strongly) Ri -associate to bi . But a ∼ = R b if and only if a = b = 0 or each ai ∼ = Ri bi = 0. (2) a is R-présimplifiable (resp., R-(super)primitive, (very) strongly R-primitive) if and only if each ai is Ri -présimplifiable (resp., Ri -(super)primitive, (very) strongly Ri -primitive). (3) a is (m-)R-irreducible if and only if some a j is (m-)R j -irreducible and ai is Ri -primitive for i = j. (4) a is (very) strongly R-irreducible if and only if some (nonzero) a j is (very) strongly R j -irreducible and ai is (very) strongly Ri -primitive for i = j. (5) a is (weakly) F-R-prime if and only if (i) some a j is F-R j -prime and Mi = Ri ai for i = j (or (ii) a = 0 or (iii) n = 2 and after a suitable reordering a1 = 0 is R1 -irreducible and R2 a2 ∩ R2 b2 = 0 for each b2 ∈ M2 \ R2 a2 ). Proof (For the special case M = R of (1)–(4), cf. [7, Theorem 2.15].)

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n n (1) Since Ra = i=1 Ri ai and Rb = i=1 Ri bi , it follows that a ∼ R b if and only if each ai ∼ Ri bi . The (very) strongly n associate characterization follows U (Ri ). routinely from the fact that U (R) = i=1 (2) Part (1) implies (i) a is R-présimplifiable ⇔ a ∼ = R a = 0 ⇔ each ai ∼ = Ri ai = 0 ⇔ each ai is Ri -présimplifiable and (ii) a is (very) (strongly) Rprimitive ⇔ a is (very) (strongly) R-associate to each of its divisors in M ⇔ every ai is (very) (strongly) Ri -associate to each of its divisors in Mi ⇔ each (very) (strongly) is R-superprimitive ai is  nRi -primitive. Lastly, nnote that a  n n R x a ⊆ R y b implies R x ⊆ ⇔ i i i i i i i i i=1 i=1 i=1 i=1 Ri yi ⇔ each ai is Ri -superprimitive. (3)–(4) (⇒): We first prove the case where a is very strongly R-irreducible. Then a is not R-primitive, so by (2) we may assume a1 is not R1 -primitive. If a1 = x1 y1 b1 , then a = (x1 , 1, . . . , 1)(y1 , 1, . . . , 1)(b1 , a2 , . . . , an ), hence a ∼ = R (x1 b1 , a2 , . . . , an ) or (y1 b1 , a2 , . . . , an ) by very strong R-irreducibility, hence a1 ∼ = R1 x1 b1 or y1 b1 by (1). Therefore a1 is very strongly R1 -irreducible. Say a1 = z 1 c1 with R1 a1  R1 c1 . To show that ai is very strongly Ri -primitive for 2 ≤ i ≤ n, write Then a = (z 1 , 1, . . . , 1)(1, z 2 , . . . , z n )(c1 , . . . , cn ) and ai = z i ci . a  R (1, z 2 , . . . , z n )(c1 , . . . , cn ), so a ∼ = R (z 1 , 1, . . . , 1)(c1 , . . . , cn ). Therefore ai ∼ = Ri ci for 2 ≤ i ≤ n by (1), as desired. So a, being nonzero and very strongly R-irreducible, is R-présimplifiable, hence a1 = 0 by (2). The case where a is (strongly) R-irreducible is proven with a minor adaptation of the above argument. Finally, consider the case where a is m-R-irreducible. We can reorder if necessary so that a1 is R1 -irreducible and ai is Ri -primitive for 2 ≤ i ≤ n. If R1 a1  R1 b1 ⊆ R1 c1 , then Ra  R(b1 , a2 , . . . , an ) ⊆ R(c1 , a2 , . . . , an ), hence R(b1 , a2 , . . . , an ) = R(c1 , a2 , . . . , an ) and R1 b1 = R1 c1 by (1). Therefore a1 is m-R1 -irreducible. (⇐): We first prove the case where we can reorder so that a1 is nonzero and very strongly R1 -irreducible and ai is very strongly Ri -primitive for 2 ≤ i ≤ n. Then a is not R-primitive by (2). Write a = (x1 , . . . , xn )(y1 , . . . , yn )(b1 , . . . , bn ). Then ai = xi yi bi for 1 ≤ i ≤ n, so a1 ∼ = R1 x1 b1 or y1 b1 , while ai ∼ = Ri xi bi ∼ = Ri yi bi for 2 ≤ i ≤ n. Part ∼ (1) shows that a = R (x1 , . . . , xn )(b1 , . . . , bn ) or (y1 , . . . , yn )(b1 , . . . , bn ), as desired. (Here we used the fact that each ai = 0 when invoking (1).) A small adjustment to the above argument proves the (strongly) irreducible case. Now assume we can reorder so that a1 is m-R1 -irreducible and ai is Ri -primitive for 2 ≤ i ≤ n. Then a is not R-primitive by (2). Write Ra  R(b1 , . . . , bn ) ⊆ R(c1 , . . . , cn ). Then Ri ai = Ri bi = Ri ci for 2 ≤ i ≤ n since ai is Ri -primitive. Thus R1 a1  R1 b1 ⊆ R1 c1 , so R1 b1 = R1 c1 and R(b1 , . . . , bn ) = R(c1 , . . . , cn ). (5) (⇒): Assume a is weakly F-R-prime. If a = 0, then a is not F-R-prime by (3), so let us assume a = 0. Then a is R-irreducible, so by (3) we can reorder so that a1 is R1 -irreducible and ai is Ri -primitive for 2 ≤ i ≤ n. If R1 a1 ⊇ R1 x1 y1 b1 , then Ra ⊇ R(x1 , 1, . . . , 1)(y1 , 1, . . . , 1)(b1 , a2 , . . . , an ) = 0, hence Ra ⊇ R(x1 b1 , a2 , . . . , an ) or R(y1 b1 , a2 , . . . , an ), hence R1 a1 ⊇

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R1 x1 b1 or R1 y1 b1 . So a1 is F-R1 -prime. Without loss of generality, we may now assume there is a c2 ∈ M2 \ R2 a2 . Pick c1 ∈ M1 with R1 a1  R1 c1 , say a1 = x1 c1 . So (1, 0, 1, 1, . . . , 1)a = (x1 , 1, . . . , 1)(1, 0, 1, 1, . . . , 1)(c1 , c2 , a3 , . . . , an ) but a divides neither (x1 , 1, . . . , 1)(c1 , c2 , a3 , . . . , an ) nor (1, 0, 1, 1, . . . , 1)(c1 , c2 , a3 , . . . , an ). Therefore a is not F-R-prime and (1, 0, 1, 1, . . . , 1)a = 0, hence a1 = 0 and n = 2. All that remains is to show that R2 a2 ∩ R2 c2 = 0. Say y2 c2 ∈ R2 a2 . Then a divides (0, 1)(1, y2 )(c1 , c2 ) but neither (0, 1)(c1 , c2 ) nor (1, y2 )(c1 , c2 ), hence y2 c2 = 0 by weak Fprimeness, as desired. (⇐): First assume (without loss of generality) a1 is Mi = Ri ai for 2 ≤ i ≤ n. Then (Ra : R(b1 , . . . , bn )) = F-R1 -prime and  n Ri is a prime ideal of R for each (b1 , . . . , bn ) ∈ (R1 a1 : R1 b1 ) × i=2 M \ Ra, so a is F-R-prime by Theorem 2.18. Now assume n = 2, a1 = 0 is R1 -irreducible, and R2 a2 ∩ R2 b2 = 0 for each b2 ∈ M2 \ R2 a2 . Write (x1 , x2 )a = (y1 , y2 )(z 1 , z 2 )(c1 , c2 ) = 0. Then y1 z 1 c1 = 0 and x2 a2 = y2 z 2 c2 = 0, so R2 a2 ⊇ R2 c2 and either y1 c1 = 0 or z 1 c1 = 0. Thus Ra ⊇  R(y1 , y2 )(c1 , c2 ) or R(z 1 , z 2 )(c1 , c2 ), as desired. We note that nonzero weakly prime elements in non-connected rings are prime [6, Theorem 7] but this fact does not generalize to modules. For example, the element (0, (0, 1)) of the (Q × Q)-module Q × (Q × Q) is weakly F-prime but not F-prime.

3 Notions of Atomicity Anderson and Valdes-Leon [7] called a ring m-atomic (resp., (very) (strongly) atomic, p-atomic) if every nonzero nonunit is a product of m-irreducible (resp., (very) (strongly) irreducible, prime) elements. (So a p-atomic integral domain is simply a unique factorization domain (UFD).) They later generalized these definitions to modules as follows [8]. Let α ∈ {irreducible, strongly irreducible, very strongly irreducible, m-irreducible, weakly prime, prime} and β ∈ {primitive, strongly primitive, very strongly primitive, superprimitive}. Then an R-module M is (α, β)atomic if each nonzero non-β a ∈ M has an (α, β)-factorization, i.e., an expression a = x1 · · · xn b with b β and each xi α. (So if M is (α, β)-atomic, then every nonzero primitive element is an associate of a β element, hence β.) In this section we will generalize the ring-theoretic definitions in a different manner, referencing irreducibility in M rather than in R. We call M m-R-atomic (resp., (very) (strongly) R-atomic, (w)p-R-atomic) if each nonzero non-R-primitive element can be written in the form x1 · · · xn b with n ≥ 1 and each xi b m-R-irreducible (resp., (very) (strongly) R-irreducible, (weakly) F-R-prime). (It will follow from Proposition 3.1 below that these definitions are in fact consistent with the previously established ones in the case M = R.) For example, the Z-module Z/25Z is (w)p-Z-atomic and (m-)Zatomic but not strongly Z-atomic, while Z/4Z × Z/4Z is wp-Z-atomic but not p-Zatomic, while Z × Z/2Z is very strongly Z-atomic but not wp-Z-atomic, and Z × Z is strongly (Z × Z)-atomic but not m-(Z × Z)-atomic. (For further details and gen-

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eralizations, see Theorems 3.3 and 3.6 below.) Note that M is m-atomic (resp., (very) strongly atomic, (w)p-atomic) if and only if it is atomic and every nonzero irreducible element is m-irreducible (resp., (very) strongly irreducible, (weakly) F-prime). Also, every atomic module has an irreducible element since either 0 is m-irreducible or there are nonzero non-primitive elements. The following result gives an equivalent way that we could have phrased the definitions of “atomicity.” Proposition 3.1 An R-module M is m-R-atomic (resp., (very) (strongly) R-atomic, (w)p-R-atomic) if and only if each nonzero a ∈ M can be written in the form a = x1 · · · xn b with n ≥ 0, b primitive, and each xi b m-R-irreducible (resp., (very) (strongly) R-irreducible, (weakly) F-R-prime). Proof We will prove the atomic case. The other cases can be handled similarly. (⇒): Assume M is atomic and let a be a nonzero non-primitive element of M. Then a = x1 · · · xn b with n ≥ 1 and each xi b irreducible. If b is not divisible by some primitive element of M, then since M is atomic we have Rb  Rc1  Rc2  Rc3 with each ci ∈ M irreducible, contradicting Corollary 2.14. Thus b = yc with c ∈ M primitive. Reorder so that Rxi b = Rb for i ≤ m and Rxi b = Rb for i > m (where 0 ≤ m ≤ n). Then Rxm · · · xn yc = Rxm b, while irreducibility implies Rxi b = Rxi yc = Rxi c for i ≤ m. So x1 c, . . . , xm−1 c, (xm · · · xn y)c are irreducible  and a = x1 · · · xm−1 (xm · · · xn y)c, as desired. (⇐): Clear. Note that we cannot equivalently replace “primitive” with “(very) strongly primitive” in the characterization of (very) strongly atomic given in Proposition 3.1. (Contrast this remark with Proposition 2.7.) For example, the Z-module Z/5Z is very strongly atomic but its primitive elements are not strongly primitive. For rings, (i) very strongly atomic ⇒ m-atomic ⇒ strongly atomic ⇒ atomic and (ii) p-atomic ⇒ wp-atomic ⇒ strongly atomic. There are no further implications between these properties and none of these properties implies or is implied by présimplifiable. (For the facts in the preceding two sentences, cf. [7, Theorems 3.4, 3.7, and 4.9] and [1, Theorems 2.3 and 2.13].) The situation becomes considerably more complex for modules, as illustrated in Fig. 2 below. Note that (m-)R/ann(M)atomic and (w)p-R/ann(M)-atomic have been omitted from the diagram since these properties are obviously equivalent to (m-)R-atomic and (w)p-R-atomic, respectively. All of the implications in Fig. 2 follow immediately from Theorem 2.9. Note that neither p-atomic nor m-atomic implies strongly atomic for modules. We have proven that (w)p-atomic implies strongly R/ann(M)-atomic in various special cases (see Theorem 3.3(4) and Corollary 3.8(1)), but we have been unable to determine if the implication holds in general. Example 3.2 No further nontrivial implications can be added to Fig. 2, with the possible exception that (w)p-R-atomic might imply strongly R/ann(M)-atomic. The Prüfer 2-group Z(2∞ ) is Z(2) -présimplifiable but not Z(2) -atomic. The Z-module Z/25Z is p-Z-atomic, Z/25Z-présimplifiable, and very strongly Z/25Z-atomic but neither Z-présimplifiable nor strongly Z-atomic. The ring Z × Z is p-atomic

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Fig. 2 Implications between factorization properties of nontrivial modules

and strongly atomic but neither m-atomic nor présimplifiable. The Z(2) -module Z/4Z × Z/4Z is wp-Z(2) -atomic but not p-Z(2) -atomic. The faithful Z-module Z × Z/25Z (resp., Z × Z/2Z) is m-Z-atomic (resp., very strongly Z-atomic) but neither Z-présimplifiable nor strongly Z-atomic (resp., wp-Z-atomic). See Theorems 3.3 and 3.6 below for further details and generalizations related to these examples. As in the previous section, we find finitely generated or torsion abelian groups to be an abundant source of natural examples of the concepts we are studying. Again, with the exception of the “strong” properties, we can generalize our results to modules over PIDs. Theorem 3.3 With the hypotheses and notation of Theorem 2.19, the following results hold. (1) M is présimplifiable if and only if T (M) = 0 or D is local. n (2) M is (m-)atomic if and only if (i) T (M) = 0 or (ii) Z(M) = i=1 ( pi ) is a finite union of maximal ideals and for each i either pi M( pi ) = 0 or pi M( pi ) = pi2 M( pi ). In particular, the module M is m-atomic if T (M) is finitely generated. (3) M is very strongly atomic if and only if either (i) T (M) = 0, (ii) ann(T n (M)) ( pi )) is maximal, or (iii) (D, ( p1 ), . . . , ( pn )) is semi-local (and Z(M) = i=1 and each pi M( pi ) = pi2 M( pi ). (4) M is (w)p-atomic if and only if either (i) T (M) = 0, (ii) (D, ( p)) is local and ann(T (M)) = ( p), (iii) M is cyclic, or (iv) M = M( p) for some nonzero prime p and pM is cyclic (or (v) p 2 M = 0 for some nonzero prime p or (vi) ann(M) = ( pq) for distinct maximal ideals  ( p) and (q)). In case (ii) (resp.,  (iv), (v), (vi)), n D/( p) (resp., D/( p ) ⊕ we can write T (M) in the form λ∈ D/( p),  λ∈   2 D/( p) ⊕ D/( p ), D/( p) ⊕ D/(q)). In particular, λ∈ γ∈ λ∈ γ∈ the module M is strongly D/ann(M)-atomic if it is wp-D-atomic. (5) If D = Z, then M is strongly atomic if and only if T (M) is annihilated by 8,  Z/2Z 9, 12, or a positive prime. In this case T (M) has one of the forms λ∈     ⊕  Z/4Z ⊕ Z/8Z, Z/3Z ⊕ Z/9Z, Z/2Z ⊕ γ∈ δ∈ γ∈ λ∈ γ∈   λ∈ Z/4Z ⊕ δ∈ Z/3Z, or λ∈ Z/ pZ with p a positive prime.

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Proof (1) Recall from Corollary 2.2 that M is présimplifiable if and only if Z(M) ⊆ J (D). Since Z(M) is either (0) or a union of maximal ideals, the result follows. (2) First we show how the second sentence of (2) will follow n from the first. ( pi ) is a finite Assume T (M) is finitely generated, or equivalently Z(M) = i=1 union of maximal ideals and each M( pi ) is finitely generated. If pi M( pi ) = 0, then ann( pi M( pi )) ⊆ ( pi ), so pi M( pi ) = pi2 M( pi ) by Nakayama’s Lemma, as desired. Now we prove the first sentence of (2). (⇒): Assume M is atomic and T (M) =0. Then M has a primitive torsion element by Proposition 3.1, n ( pi ) is a finite union of maximal ideals by Theorem 2.19(2). so Z(M) = i=1 We need to show that p j M( p j ) = p 2j M( p j ) if p j M( p j ) = 0. Pick 0 = a ∈ p j M( p j ). Then a is not primitive in M by Theorem 2.19(2), so a = x1 · · · xm b with each xi b irreducible in M. By Theorem 2.19 parts (2) and (4), each xi b( p j ) is irreducible or primitive in M( p j ), but a = x1 · · · xm b( p j ) is not primitive in M( p j ), hence some xk b( p j ) is irreducible in M( p j ). Therefore xk b( p j ) ∈ p j M( p j ) \ p 2j M( p j ) by Theorem 2.19(4), as desired. (⇐): If a = x1 · · · xn b, then c(a) = x1 · · · xn c(b). Therefore, if a is not primitive and a∈ / T (M), there is a maximum-length expression of the form a = x1 · · · xn b / U (D) and each xi b is necessarily m-irreducible. Due to the with each xi ∈ preceding sentence and the fact that nonzero m-irreducible elements of T (M) are m-irreducible in M, we may pass to T (M) and assume M is torsion. First we prove the case where Z(M) = ( p) is maximal. If pM = 0, then M is matomic by Theorem 2.8(4) since every nonzero element is primitive. So assume there is an a ∈ M with pa ∈ / p 2 M. Say p m a = 0. Suppose M is not m-atomic. Carry out the following recursive construction. Because M is not m-atomic, there is a non-primitive element b1 of M that cannot be written in the form p n c with pc m-irreducible. Having constructed non-primitive elements b1 , . . . , bn of M that cannot be written in the above form with bi = pbi+1 for 1 ≤ i < n, we have bn ∈ p 2 M by Theorem 2.19 parts (2) and (4), say bn = pbn+1 with bn+1 ∈ pM. Then bn+1 is non-primitive by Theorem 2.19(2) and it cannot be written in the above form. Thus b1 = p m (a + bm+1 ), where p(a + bm+1 ) is mirreducible by Theorem 2.19 parts (2) and (4) since it is in pM \ p 2 M, a contradiction. Now we consider the general case where (ii) holds and the ( pi )’s are distinct. Then each M( pi ) is m-atomic. Pick a non-primitive a ∈ M. By Proposition 3.1, each a( pi ) = xi,1 · · · xi,m i bi with bi a primitive element of M( pi ), xi,k ∈ ( pi ), and each xi,k bi irreducible in M( pi ). (The special case a( pi ) = 0 is not technically part of the statement of Proposition 3.1, but it is easily resolved as follows. There is an m-irreducible element in M( pi ), which can be written in the form xi bi with xi ∈ ( pi ) and bi a primitive element of M( pi ), and 0 = xik bi for some k.) Since ann(b1 ), . . . , ann(bn ) are pairwise comaximal, we can use the Chinese Remainder Theorem to arrange for xi,k to be congruent n j = i. Therefore a = x1,1 · · · xn,m n i=1 bi , where each to 1  modulo ann(b j ) for  n xi,k i=1 bi = xi,k bi + j=i b j is m-irreducible in M by Theorem 2.19 parts (2) and (4).

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(3) (⇒): Assume n M is very strongly atomic and T (M) = 0. Part (2) implies ( pi ) is a finite union of distinct maximal ideals and for each i we Z(M) = i=1 pi M( pi ). Pick ai ∈ M( pi ) \ pi M( pi ). have either pi M( pi ) = 0 or pi2 M( pi ) = First assume some p j M( p j ) = 0. Then i= j ai is irreducible in M by Theorem 2.19(4), hence is very strongly irreducible since M is very strongly atomic, hence assume each pi M( pi ) = pi2 M( pi ). ann(T (M)) = p1 by Theorem 2.19(5). Now n 2 ai is irreducible in M by ThePick b1 ∈ p1 M( p1 ) \ p1 M( p1 ). Then b1 + i=2 orem 2.19(4), hence very strongly irreducible since M is very strongly atomic. Thus (D, ( p1 ), . . . , ( pn )) is semi-local by Theorem 2.19(5). (⇐): If (i), (ii), or (iii) holds, then M is atomic by (2) and every irreducible element is very strongly irreducible by Theorem 2.19 parts (4) and (5), hence M is very strongly atomic. (4) Once we have proven the equivalence in (4), the isomorphisms will follow from the fact that non-faithful modules over PIDs are direct sums of cyclic modules [19]. Note that the first two sentences of (4) imply M is cyclic or D/ann(M)présimplifiable if it is wp-atomic. Since wp-atomic implies strongly atomic for rings, the third sentence of (4) will also follow once we prove the equivalence. (To see that a wp-atomic ring R is strongly atomic, note that R is a finite direct product of présimplifiable rings by [1, Theorems 2.3 and 2.13]. So associates and strong associates coincide in R by Theorem 2.21(1) and hence R is strongly atomic.) (⇒): Assume M is wp-atomic and T (M) = 0. Part (2) implies Z(M) =  n i=1 ( pi ) is a finite union of distinct maximal ideals and for each i we have either pi M( pi ) = 0 or pi2 M( pi ) = pi M( pi ). Pick ai ∈ M( pi ) \ pi M( pi ). Let bi := 0 if pi M( pi ) = 0 and otherwise pick bi ∈ pi M( pi ) \ pi2 M( p i ). First assume M  = n ai is irreducible T (M). Pick d ∈ M(0) with c(d) = ( p1 ). Then d + b1 + i=2 in M by Theorem 2.19(4), hence weakly F-prime since M is wp-atomic. Thus ann(T (M)) = ( p1 ) by Theorem 2.19(6). Suppose D has a maximal ideal (q) = ( p1 ). Then there is an e ∈ M(0) with c(e) = (q) and e is irreducible but not weakly F-prime in M by Theorem 2.19 parts (4) and (6), contradicting the fact that M is wp-atomic. So let us assume M = T (M). First consider the case whereeither (a) n ≥ 3 or (b) n = 2 and M is p-atomic. For each i, the element bi + j=i a j is irreducible in M by Theorem 2.19(4), hence weakly F-prime since n So each M( pi ) = Dai by Theorem 2.19(6), hence M = n M is wp-atomic. Da = D i i=1 i=1 ai is cyclic. Now consider the case where n = 2 and M is not cyclic, say M( p2 ) is not cyclic. Again b1 + a2 is weakly F-prime in M, hence p1 M( p1 ) = 0 and p2 a2 = 0 by Theorem 2.19(6). But M( p2 ) is atomic by (2), so every element has a divisor not in p2 M( p2 )—see Theorem 2.19(2) and Proposition 3.1. So p2 M( p2 ) = 0 by the arbitrariness of a2 . Thus ann(M) = ( p1 p2 ). Finally, assume n = 1 and p1 M is not cyclic. Theorem 2.19(6) implies M has no F-prime elements, hence is not p-atomic, and that each irreducible element is annihilated by p1 . But every element of p1 M is divisible by an irreducible element of M—see Theorem 2.19(2). Thus p12 M = 0. (⇐): If any of (i)-(vi) hold, then M is atomic by (2). So we just need to show that (i)-(iv) (resp., (v) and (vi)) each imply that every irreducible element of M is F-prime (resp., weakly Fprime). Cases (i), (ii), (v), and (vi) immediately follow from Theorem 2.19 parts (4) and (6), while (iii) follows from Corollary 2.20(3). So assume (iv) holds, say

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pM = Da, where a is F-prime by Theorem 2.19(6). By Theorem 2.19(4), each irreducible b ∈ M is m-irreducible and Db ⊆ pM = Da, hence Db = Da and b is F-prime. (5) As in the proof of (4), the isomorphisms will follow once we prove the equivalences. Assume D = Z. Note that Theorem 2.19(7) implies that M is strongly atomic if and only if it is atomic and every torsion irreducible element is annihilated by 4 or 6. (⇒): Assume M is strongly atomic and T (M) is not annihilated by a positive prime. Because each torsion irreducible element is annihilated by 4 or 6, we have Z(M) ⊆ (2) ∪ (3) by part (2) and Theorem 2.19(4). For each g ∈ M(2) (resp., g ∈ M(3)), the element 2g (resp., 3g) is not primitive in M by Theorem 2.19(2), hence 8g = 0 (resp., 9g = 0) since M is atomic and torsion irreducible elements are annihilated by 4 or 6. So 72T (M) = 0, hence T (M) is a direct sum of copies of Z/2Z, Z/4Z, Z/8Z, Z/3Z, and Z/9Z [19]. So we may assume Z(M) = (2) ∪ (3) and it will suffice to show that 4g = 3h = 0 for 0 = g ∈ M(2) and 0 = h ∈ M(3). Since 2(g + h) is nonzero and not primitive (see Theorem 2.19(2)), it is a multiple of a torsion irreducible element, hence is annihilated by 4 or 6. But 8Z(g + h) = Zh = 0, so 12(g + h) = 0, hence 4g = 3h = 0, as desired. (⇐): If T (M) is annihilated by a nonzero prime, then M is very strongly atomic by (3), so let us assume T (M) is annihilated by 8, 9, or 12. Then Z(M) ⊆ (2) ∪ (3) and 8M(2) = 9M(3) = 0, so M is atomic by part (2) and we just need to show that each torsion irreducible g ∈ M is annihilated by 4 or 6. If 8T (M) = 0 (resp., 9T (M) = 0, 12T (M) = 0), then g ∈ 2T (M) (resp., g ∈ 3T (M), g ∈ 2T (M) ∪ 3T (M)) by Theorem 2.19(4), hence 4g ∈ 8T (M) = 0 (resp., 6g ∈ 18T (M) = 0, either 6g ∈ 12T (M) = 0 or 4g ∈ 12T (M) = 0), as desired.  Unsurprisingly, the criteria of Theorem 3.3 can be simplified for cyclic modules. Corollary 3.4 Let D be a PID and y be a nonzero nonunit of D. (1) (2) (3) (4)

D/(y) is D-présimplifiable if and only if D is local. D/(y) is m-D-atomic and p-D-atomic. D/(y) is very strongly D-atomic if and only if (i) y is prime or (ii) (y) ⊆ J (D)2 . If D = Z, then Z/yZ is strongly Z-atomic if and only if y is prime or y = ±4, ±6, ±8, ±9, or ±12.

Proof We prove (3). Parts (1), (2), and (4) follow immediately from Theorem 3.3. Let D := D/(y). By Theorem 3.3(3) we may assume y is not prime. Let ( p1 ), . . . , ( pn ) be the prime ideals containing n (y). Then D is very strongly D-atomic ⇔ each2pi is Dpi J (D) = p1 · · · pn J (D) ⇔ (y) ⊆ J (D) , where présimplifiable ⇔ (y) ⊆ i=1 the first equivalence is by (2), the second is by Corollary 2.20(1), and the third is by  the fact that J (D) = ( p1 · · · pn ) if (y) ⊆ J (D). We next examine how the various notions of atomicity behave with respect to direct products. To facilitate the statements of our results, we give the following definitions, some of which may be redundant. For each notion γ of “atomicity,” we

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define Fletcher γ by removing the “nonzero” from the definition of γ. (We choose this name in honor of Fletcher [15], whose definition of “unique factorization ring” also placed requirements on the factorizations of zero, contrary to most authors’ formulations of factorization definitions.) Since 0 is not a product of m-irreducible elements in a non-field domain, a ring is Fletcher m-atomic if and only if it is a field or an m-atomic ring with zero divisors. So Fletcher m-atomic is strictly stronger than m-atomic. However, since in a ring either 0 is prime (or equivalently (very strongly) irreducible) or a product of two nonzero nonunits, it follows that a Fletcher γ ring is the same thing as a γ ring for each γ = m-atomic. We do not know if this last conclusion generalizes to our module-theoretic definitions of atomicity. Lemma 3.5 Let D be a domain with quotient field K , M be a nontrivial torsion-free D-module, and 0 = a ∈ M. The following are equivalent. (1) (2) (3) (4)

a is superprimitive. K a ∩ M = Da. Da ⊇ Dxb implies Da ⊇ Db. Da ∩ Db = 0 for each b ∈ M \ Da.

Proof (1) ⇔ (2) ⇔ (3): [8, Proposition 2.5]. (3) ⇔ (4): [20, Proposition 1.1].



Theorem . . . , n (2 ≤ n < ∞), let Mi be a nontrivial Ri -module. Let n 3.6 For i = 1,  n Ri and M := i=1 Mi . R := i=1 (1) M is (m-)R-atomic if and only if each Mi is Fletcher (m-)Ri -atomic. (2) M is (very) strongly R-atomic if and only if each Mi is Fletcher (very) strongly Ri -atomic with every Ri -primitive element (very) strongly Ri -primitive (and 0 not Ri -irreducible in Mi ). (3) M is (w)p-R-atomic if and only if (either (i)) each Mi is a cyclic p-Ri -atomic module (or (ii) n = 2 and each ann(Mi ) is maximal or (iii) n = 2 and after a suitable reordering M1 is simple, R2 /ann(M2 ) is a UFD, and M2 is (prime, superprimitive)-atomic as an R2 /ann(M2 )-module). Throughout, if M satisfies some notion γ of atomicity, then it is in fact Fletcher γ. Proof (For the special case M = R, cf. [7, Theorem 3.4].) (1)–(2) We will prove the very strongly atomic case. The other cases are handled similarly. We use Proposition 3.1’s characterization of very strong atomicity. (⇒): Assume M is very strongly R-atomic and pick a non-primitive a1 ∈ M1 . Since M has an R-primitive element, each Mi has an Ri -primitive element bi by Theorem 2.21(2). Because M is very strongly R-atomic and (a1 , b2 , . . . , bn ) is nonzero and non-R-primitive by Theorem 2.21(2), we have (a1 , b2 , . . . , bn ) = (x1,1 , . . . , x1,n ) · · · (xm,1 , . . . , xm,n )(c1 , . . . , cn ) with m ≥ 1, (c1 , . . . , cn ) R-primitive, and each (xi,1 , . . . , xi,n )(c1 , . . . , cn ) very strongly R-irreducible. Each xi, j c j with 2 ≤ j ≤ n, being an associate of b j , is R j -primitive, so xi, j c j (and hence b j ) is very strongly R j -primitive by Theorem 2.21(4). Thus a1 = x1,1 x2,1 · · · xm,1 c1 , where each xi,1 c1 is very

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strongly R1 -irreducible by Theorem 2.21(4). By symmetry, we conclude that each Mi is Fletcher Ri -atomic with every Ri -primitive element very strongly Ri -primitive. Finally, since every R-irreducible element of M is very strongly R-irreducible, it follows from Theorem 2.21 parts (3) and (4) that 0 is not Ri -irreducible in any Mi . (⇐): Assume each Mi is Fletcher very strongly Ri -atomic with every Ri -primitive element very strongly Ri primitive and 0 not Ri -irreducible in Mi . Let (b1 , . . . , bn ) ∈ M. Then each bi = xi,1 · · · xi,m i ci with m i ≥ 0, ci very strongly Ri -primitive, and each xi, j ci very strongly Ri -irreducible (hence nonzero). Thus (b1 , . . . , bn ) = (x1,1 , 1, . . . , 1) · · · (x1,m 1 , 1, . . . , 1) · · · (1, . . . , 1, xn,1 ) · · · (1, . . . , 1, xn,m n ) (c1 , . . . , cn ), where each (1, . . . , 1, xi, j , 1, . . . , 1)(c1 , . . . , cn ) is very strongly R-irreducible by Theorem 2.21(4), as desired. n (3) By passing to R/ann(M) ∼ = i=1 Ri /ann(Mi ), we may assume each Mi is a faithful Ri -module. (⇒): Assume M is wp-R-atomic. Then each Mi is Fletcher Ri -atomic by (1). Moreover, every R-irreducible element of M is weakly F-R-prime, hence every Ri -irreducible element of each Mi is F-Ri prime by Theorem 2.21 parts (3) and (5). Therefore each Mi is Fletcher p-Ri atomic. Assume some M j , say M2 , is not cyclic. Then M has R-irreducible elements that are not F-R-prime by Theorem 2.21 parts (3) and (5). Thus M is wp-R-atomic but not p-R-atomic, hence n = 2 by Theorem 2.21(5). Since every R-irreducible element of M is weakly F-R-prime, it follows from Theorem 2.21 parts (3) and (5) that 0 is the only R1 -irreducible element of M1 , hence is m-R1 -irreducible since M1 is R1 -atomic. So R1 is a field by Theorem 2.8(4). Similarly, if M1 is not cyclic, then R2 is a field. So let us assume M1 is cyclic. For each R2 -primitive a2 ∈ M2 , the element (0, a2 ) ∈ M is R-irreducible by Theorem 2.21(3), hence weakly F-R-prime, hence R2 a2 ∩ R2 b2 = 0 for each b2 ∈ M2 \ R2 a2 by Theorem 2.21(5). So, if a2 and b2 are non-R2 -associate R2 -primitive elements of M2 and x2 b2 = 0, / R 2 a2 . then x2 a2 = x2 (a2 + b2 ) ∈ R2 a2 ∩ R2 (a2 + b2 ) = 0 since a2 + b2 ∈ Therefore all R2 -primitive elements of M2 have the same annihilator, namely ann(M2 ) = (0) by Proposition 3.1. Since M2 is not cyclic, Proposition 3.1 implies there are non-R2 -associate R2 -primitive c2 , d2 ∈ M2 . If x2 y2 = 0 for some x2 , y2 ∈ R2 with x2 = 0, then 0 = x2 c2 = x2 (c2 + y2 d2 ) ∈ R2 c2 ∩ R2 (c2 + y2 d2 ), so c2 + y2 d2 ∈ R2 c2 , hence y2 d2 ∈ R2 c2 ∩ R2 d2 = 0, hence y2 = 0. Therefore R2 is a domain and M2 is torsion-free since its primitive elements are faithful—see Proposition 3.1. So by Lemma 3.5 each R2 -primitive element of M2 is R2 -superprimitive. By Proposition 3.1, each element of M2 is a multiple of an R2 -superprimitive element, so it only remains to show that R2 is a UFD, or equivalently each nonzero nonunit x2 ∈ R2 is (associate to) a product of primes. Pick a superprimitive a2 ∈ M2 . Since M2 is Fletcher p-R2 -atomic, we have x2 a2 = p1 · · · pm b2 with b2 ∈ M superprimitive and each pi b2 F-R2 -prime. Then (x2 ) = ( p1 · · · pm ) by superprimitivity, so we just need to show that each pi is prime. Since pi b2 is F-R2 -prime in Rb2 , this follows by applying the canonical isomorphism Rb2 → R. (⇐): First assume each Mi is cyclic and p-Ri -atomic. Since

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∼ Ri is p-atomic and Fletcher p-atomic are equivalent for rings, each Mi = Fletcher p-Ri -atomic, hence M is Fletcher R-atomic by (1). In fact M is Fletcher p-R-atomic since its R-irreducible elements are F-R-prime by Theorem 2.21 parts (3) and (5). Next assume n = 2 and each Ri is a field. By Theorem 2.8(4), each Mi is Fletcher Ri -atomic with a unique Ri -irreducible element (namely 0), hence M is Fletcher R-atomic by (1). Using Theorem 2.21 parts (3) and (5), we see that R-irreducible elements of M are weakly FR-prime, so M is Fletcher wp-R-atomic. Lastly, assume n = 2, R1 is a field, M1 ∼ = R1 , R2 is a UFD, and M2 is (prime, superprimitive)-atomic. We claim that M2 is Fletcher p-R2 -atomic. It suffices to show that pa2 is F-R2 -prime for each prime p ∈ R2 and R2 -superprimitive a2 ∈ M2 . Write spa2 = x yb2 . Since M2 is (prime, superprimitive)-atomic, we can write b2 = zc2 with c2 R2 -superprimitive. If 0 = spa2 = x yzc2 , then by the faithfulness of c2 either x = 0, y = 0, or z = 0, hence xb2 = 0 or yb2 = 0. So let us assume 0 = spa2 = x yzc2 . Proposition 3.1 implies M2 is torsion-free, so we can apply Lemma 3.5 to see that R2 a2 = R2 c2 . Therefore (sp) = (x yz) by faithfulness, hence ( p) ⊇ (x z) or (yz) by primeness. Therefore R2 pa2 contains R2 x zc2 = R2 xb2 or R2 yzc2 = R2 yb2 , as desired. So each Mi is Fletcher p-Ri -atomic, hence M is Fletcher R-atomic by (1). By Theorem 2.21(3) and Lemma 3.5, every R-irreducible element of M has the form (a) (0, a2 ) with R2 a2 ∩ R2 b2 = 0 for each b2 ∈ M2 \ R2 a2 or (b) (a1 , a2 ) with M1 = R1 a1 and a2 F-R2 -prime. So R-irreducible elements of M are weakly F-R-prime by Theorem 2.21(5). Therefore M is Fletcher wp-R-atomic.  The condition on M2 in Theorem 3.6(3iii) is equivalent to M2 being a “factorial” R2 /ann(M2 )-module in the sense of Nicolas [21] (cf. [8, Theorem 4.4]). Such modules have been exhaustively studied by Nicolas [21, 22], Costa [14], Lu [20], Anderson and Valdes-Leon [8], and Angermüller [9]. Examples of factorial modules include projective modules over UFDs [9, Corollary 14]. While p-atomic and wp-atomic are the same for non-connected rings [6, Theorem 7], this fact does not generalize to modules. For example, the (Q × Q)-module Q × (Q × Q) is wp-atomic but not p-atomic. We close the chapter with two applications of Theorem 3.6. Corollary 3.7 Let M be a nontrivial atomic R-module. Then R/ann(M) is completely decomposable. Proof (The special case M = R is part of [7, Theorem 3.4].) By passing to R/ann(M), we reduce to the case where M is faithful. We may assume R is not connected, so M is Fletcher atomic by Theorem 3.6. With a trivial adjustment to the proof of Proposition 3.1, we obtain 0 = x1 · · · xn b with b ∈ M primitive and We show that R is completely each xi b irreducible. m decomposable by demonstrating m Ri implies m ≤ n. Write M = i=1 Mi with each Mi a (necessarily that R = i=1 faithful) Ri -module, xi = (xi,1 , . . . , xi,m ) with xi, j ∈ R j , and b = (b1 , . . . , bm ) with b j ∈ M j . Define yi := (yi,1 , . . . , yi,m ) ∈ R, where yi, j := 1 if xi, j b j is R j -primitive

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and otherwise yi, j := xi, j . Then each R j xi, j b j = R j yi, j b j , so Rxi b = Ryi b. Furthermore, because xi b is irreducible, Theorem 2.21(3) implies yi is 1 in all but one coordinate. Thus y1 · · · yn has at most n nonunit coordinates. By Theorem 2.21(2), every coordinate of b is primitive, so the fact that 0 = y1 · · · yn b implies all m coor dinates of y1 · · · yn are nonunits. Therefore m ≤ n, as desired. Corollary 3.8 Let M be a nontrivial R-module with R/ann(M) not connected. (1) If M is wp-R-atomic, then R/ann(M) is p-atomic and M is strongly R/ann(M)atomic. (2) M is p-R-atomic if and only if R/ann(M) is p-atomic and M is cyclic. Proof By passing to R/ann(M), we may assume R = R1 × R2 and M = M1 × M2 with each Mi a faithful Ri -module. Of course “⇐” in (2) is trivial. For the reverse implication, assume M is p-R-atomic. Then M ∼ = R by Theorem 3.6(3). So R, being a p-atomic ring, is a finite direct product of UFDs and Artinian local principal ideal rings [5, Theorem 2.4]. Therefore R is a finite direct product of strongly atomic rings, hence R (or equivalently M) is strongly R-atomic by Theorem 3.6(2). All that remains is to prove (1) in the case where M is wp-R-atomic but not p-R-atomic. Then each Ri is a UFD (hence p-atomic) and Mi is a torsion-free Ri -module by Theorem 3.6(3). Therefore R is p-atomic by Theorem 3.6(3). Furthermore, Theorem 2.21(1) implies that R-associates and strong R-associates coincide in M. So M is strongly R-atomic. 

References 1. A.G. Agargün, D.D. Anderson, S. Valdes-Leon, Unique factorization rings with zero divisors. Commun. Algebra 27, 1967–1974 (1999) 2. D.D. Anderson, D.F. Anderson, M. Zafrullah, Factorization in integral domains. J. Pure Appl. Algebra 69, 1–19 (1990) 3. D.D. Anderson, S. Chun, Irreducible elements in commutative rings with zero-divisors. Houston J. Math. 37, 741–744 (2011) 4. D.D. Anderson, S. Chun, Irreducible elements in commutative rings with zero-divisors. II. Houston J. Math. 39, 741–752 (2013) 5. D.D. Anderson, R. Markanda, Unique factorization rings with zero divisors. Houston J. Math. 11, 15–30 (1985) 6. D.D. Anderson, E. Smith, Weakly prime ideals. Houston J. Math. 29, 831–840 (2003) 7. D.D. Anderson, S. Valdes-Leon, Factorization in commutative rings with zero divisors. Rocky Mountain J. Math. 26, 439–473 (1996) 8. D.D. Anderson, S. Valdes-Leon, Factorization in commutative rings with zero divisors, II, in Factorization in integral domains ed. by D.D. Anderson (Marcel Dekker, 1997), pp. 197–219 9. G. Angermüller. Unique factorization in torsion-free modules, in Rings, Polynomials, and Modules ed. by M. Fontana, S. Frisch, S. Glaz, F. Tartarone, P. Zanardo (Springer, 2017), pp. 13–31 10. M. Behboodi, H. Koohy, Weakly prime modules. Vietnam J. Math. 32, 185–195 (2004) 11. A. Bouvier, Anneaux présimplifiables. C.R. Acad. Sci. Paris, 274, A1605–A1607, 1972

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12. A. Bouvier, Résultats nouveaux sur les anneaux présimplifiables. C.R. Acad. Sci. Paris 275, A955–A957 (1972) 13. A. Bouvier, Anneaux présimplifiables. Rev. Roumaine Math. Pures Appl. 713–724 (1974) 14. D.L. Costa, Unique factorization in modules and symmetric algebras. Trans. Am. Math. Soc. 224, 267–280 (1976) 15. C.R. Fletcher, Unique factorization rings. Proc. Camb. Philos. Soc. 65, 579–583 (1969) 16. S. Galovich, Unique factorization rings with zero divisors. Math. Mag. 51, 276–283 (1978) 17. T.W. Hungerford, Algebra, 8th edn. (Springer, New York, 2003) 18. I. Kaplansky, Modules over Dedekind rings and valuation rings. Trans. Am. Math. Soc. 72, 327–340 (1952) 19. G. Köthe, Verallgemeinerte abelsche gruppen mit hyperkomplexen operatorenring. Math. Z. 39, 31–44 (1935) 20. C. Lu, Factorial modules. Rocky Mountain J. Math. 7, 125–139 (1977) 21. A.-M. Nicolas, Modules factoriels. Bull. Sci. Math. 2(95), 33–52 (1971) 22. A.-M. Nicolas, Extensions factorielles et modules factorables. Bull. Sci. Math. 98, 117–143 (1974) 23. H.T. Tang, Gauss’ lemma. Proc. Am. Math. Soc. 35, 372–376 (1972)

On -potent Domains and -homogeneous Ideals Muhammad Zafrullah

Abstract Let  be a star operation of finite character. Call a -ideal I of finite type a -homogeneous ideal if I is contained in a unique maximal -ideal M = M(I ). A maximal ∗-ideal that contains a -homogeneous ideal is called potent and the same name bears a domain all of whose maximal -ideals are potent. One among the various aims of this article is to indicate what makes a -ideal of finite type a -homogeneous ideal, where and how we can find one, what they can do and to direct to sources that indicate how this notion came to be. Keywords Star operation · -homogeneous ideal -potent domain · -semi-homogeneous domain. 1991 Mathematics Subject Classification 13A15 · Secondary 13G05 · 06F20

1 Introduction Let  be a finite character star operation defined on an integral domain D throughout. (A working introduction to star operations, and the reason for using them, will follow.) Call a nonzero finitely generated ideal I a -homogeneous ideal, if I is contained in a unique maximal -ideal. The notion of a -homogeneous ideal has figured prominently in describing unique factorization of ideals and elements in [5] and it seems important to indicate some other properties and uses of this notion and notions related to it. Call a -maximal ideal M -potent if M contains a -homogeneous ideal and call a domain D -potent if each of the -maximal ideals of D is -potent. The aim of this article is to study some properties of -homogeneous ideals and of potent domains. We show for instance that while in a -potent domain every proper Dedicated to Dan Anderson. M. Zafrullah (B) Department of Mathematics, Idaho State University, Pocatello, ID 83209, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. Badawi and J. Coykendall (eds.), Rings, Monoids and Module Theory, Springer Proceedings in Mathematics & Statistics 382, https://doi.org/10.1007/978-981-16-8422-7_4

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-ideal of finite type is contained in some I  for I a -homogeneous ideal, the converse may not be true. We shall also indicate how these concepts can be put to use. Before we elaborate on that, it seems pertinent to give an idea of our main tool, the star operations. Indeed, the rest of what we plan to prove will be included in the plan of the paper after the introduction to star operations.

1.1 Introduction to Star Operations Let D be an integral domain with quotient field K , throughout. Let F(D) be the set of nonzero fractional ideals of D, and let f (D) = {A ∈ F(D)|A is finitely generated}. A star operation  on D is a closure operation on F(D) that satisfies D  = D and (x A) = x A for A ∈ F(D) and x ∈ K = K \{0}. With  we can associate a new staroperation s given by A → As = ∪{B  |B ⊆ A, B ∈ f (D)} for each A ∈ F(D). We say that  has finite character if  = s . Three important star-operations are the d-operation A → Ad = A, the v-operation A → Av = (A−1 )−1 = ∩{Dx|Dx ⊇ A, x ∈ K } where A−1 = {x ∈ K : x A ⊆ D} and the t-operation t = vs . Here d and t have finite character. A fractional ideal A is a -ideal if A = A and a -ideal A is of finite type if A = B  for some B ∈ f (D). If  has finite character and A is of finite type, then A = B  for some B ∈ f (D), B ⊆ A. A fractional ideal A ∈ F(D) is -invertible if there exists a B ∈ F(D) with (AB) = D; in this case we can take B = A−1 . For any -invertible A ∈ F(D), A = Av . If  has finite character and A is -invertible, then A is a finite type -ideal and A = At . Given two fractional ideals A, B ∈ F(D), (AB) denotes their -product. Note that (AB) = (A B) = (A B  ) . Given two star operations 1 and 2 on D, we write 1 ≤ 2 if A1 ⊆ A2 for all A ∈ F(D). So 1 ≤ 2 ⇔ (A1 )2 = A2 ⇔ (A2 )1 = A2 for all A ∈ F(D).Indeed for any finite character star-operation  on D we have d ≤  ≤ t. For a quick introduction to star-operations, the reader is referred to [20, Sections 32, 34] or [40], for a quick review. For a more detailed treatment see Jaffard [27]. These days star operations are being used to define analogues of various concepts. The trick is to take a concept, e.g., a PID and look for what the concept would be if we require that for every nonzero ideal I, I  is principal and voila! You have several concepts parallel to that of a PID. Of these t-PID turns out to be a UFD. Similarly a v-PID is a completely integrally closed GCD domain with the property that for each nonzero ideal we have Av principal. A t-Dedekind domain, on the other hand is a Krull domain and a v-Dedekind domain is a domain with the property that for each nonzero ideal A we have Av invertible see e.g. [18]. So when we prove a result about a general star operation  the result gets proved for all the different operations, d, t, v etc., in some form. Apart from the above, any terminology that is not mentioned above, is either standard or, will be introduced at the point of entry of a new concept. Suppose that  is a finite character star-operation on D. Then a proper -ideal is contained in a maximal -ideal and a maximal -ideal is prime. We denote the set of maximal -ideals of D by -Max(D). We have D = ∩D P where P ranges over

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-Max(D). From this point on we shall use  to denote a finite type star operation. Call D of finite -character if for each nonzero non unit x of D, x belongs to at most a finite number of maximal -ideals. It may be noted that while we talk in terms of general star operations, our main focus will be the d, t and v-operations. Apart from the introduction there are two sections in this paper. In Sect. 2 we talk about -homogeneous ideals, and -potent domains. We characterize -potent domains in this section, show that if D is of finite -character then D must be -potent, and characterize domains of finite -character, giving a new proof. In Sect. 3, we show how creating a suitable definition of a -homogeneous ideal will create a theory of unique factorization of ideals. Call an element r ∈ D -f-rigid (-factorial rigid), if r D is a -homogeneous ideal such that every proper -homogeneous ideal containing r is principal. Also call a -potent maximal -ideal M (resp., domain D) -f-potent if M (resp., every maximal -ideal of D) contains a -f-rigid element. We show that over a -f-potent domain a primitive polynomial f is super primitive i.e. if A f , the content of f, is such that the generators of A f have no non unit common factor, then (A f )v = D and indicate how to construct atomless domains. In this section we also offer a seamless patch to remove an error in the proof of a result in a paper by Kang [28] and show that D is t-super potent if and only if D[X ] S is t-f-potent, where X is an indeterminate and S = { f ∈ [X ]|(A f )v = D}. Here a -homogeneous ideal I is called -super homogeneous if every -homogeneous ideal J with J  ⊇ I is -invertible. We also show, by way of constructing more examples, in this section that if L is an extension of K the quotient field of D and X an indeterminate over D then D is t-f-potent if and only if D + X L[X ] is.

2 -homogeneous Ideals Work on this paper started in earnest with the somewhat simple observation that if D is -potent then every nonzero non unit x ∈ D is contained in I  for some homogeneous ideal I . The proof goes as follows: Because x is a nonzero non unit, x D is a proper -ideal and so must be contained in some maximal -ideal M. Now as D is -potent M = M(I ) for some -homogeneous ideal I. Consider J = (I, x) and note that (I, x) = D because x ∈ M and (I, x) is contained in a unique maximal -ideal and this makes J a -homogeneous ideal. This leads to the question: If D is a domain with a finite character star operation  defined on it in such a way that every nonzero non unit x of D is contained in I  for some -homogeneous ideal I of D, must D be -potent? This question came up in a different guise as: when is a certain type of domain ∗s -potent for a general star operation ∗ in [36] and sort of settled in a tentative fashion in Proposition 5.12 of [36] saying, in the general terms being used here, that: Suppose that D is a domain with a finite character star operation  defined on it. Then D is -potent provided (1) every nonzero non unit x of D is contained in I  for some -homogeneous ideal I of D and (2) for M, Mα ∈ -max(D), M ⊆ ∪Mα implies M = Mα for some α.

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The proof is something like: By (1) for every nonzero non unit x there is a homogeneous ideal Ix with (Ix ) containing x and so x ∈ M(Ix ). So M ⊆ ∪M(Ix ) and by (2) M must be equal to M(Ix ) for some x. Thus we have the following statement. Theorem 2.1 Let  be a finite character star operation defined on D. Then D is potent if D satisfies the following: (1)every nonzero non unit x of D is contained in I  for some -homogeneous ideal I of D and (2) For M, Mα ∈ -max(D), M ⊆ ∪Mα implies M = Mα for some α. Condition (2) in the statement of Theorem 2.1 has had to face a lot of doubt from me, in that, is it really necessary or perhaps can it be relaxed a little? The following example shows that condition (2) or some form of it is here to stay. Example 2.2 It is well known that the ring E of entire functions is a Bezout domain [20, Exercise 18, p 147]. It is also easy to check that a principal prime in a Bezout domain is maximal. Now we know that a zero of an entire function determines a principal prime in E and that the set of zeros of a nontrivial entire function is discrete, including multiplicities, the multiplicity of a zero of an entire function is a positive integer [22, Theorem 6]. Thus each nonzero non unit of E is expressible as a countable product of finite powers of distinct principal primes of height one. For the identity star operation d, certainly defined on E, only an ideal I generated by a power of a principal prime can be d-homogeneous. For if I is d-homogeneous, then I = (x1 , ..., xn )d = xE a principal ideal and hence a countable product of distinct primes. So I can only belong to a unique principal prime and has to be a finite prime power, to be d-homogeneous. To see that E falls foul of Theorem 2.1, let’s put S = { p| p a prime element in E}. Then for each non principal prime P of E we have P ⊆ ∪ p∈S pE because each element of P is divisible by some member(s) of S, but a non-principal prime cannot be contained in a principal prime. Condition (2) in Theorem 2.1 may remind some readers of the following result of Smith [34]: A ring D satisfies ∗ : (for P, Pα prime ideals of D, where α ∈ I, P ⊂ ∪Pα implies P ⊆ Pα for some α ) if and only if every prime ideal of D is the radical of a principal ideal in D. But of course the situations are different in that in Theorem 2.1 only maximal -ideals are considered whereas in Smith’s theorem any union of primes containing a prime P has to deliver the prime containing P. Once we know more about -homogeneous ideals we would know that rings do not behave in the same manner as groups do. To get an idea of how groups behave and what is the connection, the reader may look up [36]. Briefly, the notion of a -homogeneous ideal arose from the notion of a basic element of a lattice ordered group G (defined as b > 0 in G such that (0, b] is a chain). A basis of G, if it exists, is a maximal set of mutually disjoint strictly positive basic elements of G. According to [12] an l.o. group G has a basis if and only if every strictly positive element of G exceeds a basic element. So if we were to take D being potent as having a basis (every proper -ideal of finite type being contained in I  for a -homogeneous ideal

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I ) then every proper -ideal of finite type being contained in I  for a -homogeneous ideal does not imply that D is potent, as we have seen in Example 2.2. We next tackle the question of where -homogeneous ideals can be found. Call D of finite -character if every nonzero non unit of D is contained in at most a finite number of maximal -ideals. Again, a domain of finite -character could be a domain of finite character (every nonzero non unit belongs to at most a finite number of maximal ideals) such as an h-local domain or a semilocal domain or a PID or a domain of finite t-character such as a Krull domain. Proposition 1 A domain D of finite -character is -potent. Proof Let M be a maximal -ideal of D and let x be a nonzero element of M. If x belongs to no other maximal -ideal then x D is -homogeneous and M is potent. So let us assume that M, M1 , M2 , ..., Mn is the set of all maximal -ideals containing x. Now consider the ideal A = (x, y) where y ∈ M\(∪Mi ) for i = 1, ..., n. Obviously A ⊆ M but A  Mi because of y. Note that A cannot be contained in any maximal -ideal other than M, for if N were any maximal -ideal containing A then N would belong to {M, M1 , M2 , ..., Mn } because of x. And N cannot be any of the Mi , because of y. Thus A is a -homogeneous ideal contained in M and M is -potent. Since M was arbitrary we have the conclusion.  The above proof is essentially taken from the proof for part (2) of Theorem 1.1 of [2]. Now how do we get a domain of finite -character? The answer is somewhat longish and interesting. Bazzoni conjectured in [7] and [8] that a Prufer domain D is of finite character if every locally principal ideal of D is invertible. Reference [23] were the first to verify the conjecture using partially ordered groups. Almost simultaneously [25] proved the conjecture for r -Prufer monoids, using Clifford semigroups of ideals and soon after I chimed in with a very short paper [41]. The ring-theoretic techniques used in this paper verified the Bazzoni conjecture for Prufer domains, verified it for a generalization of Prufer domains called Prufer v-multiplication domains (PVMDs) and helped prove Bazzoni-like statements for other, suitable, domains that were not necessarily Prufer/PVMD. (Recall that D is a PVMD if every t-ideal A of finite type of D is t-invertible i.e. (A A−1 )t = D.) In the course of verification of the conjecture I mentioned a result due to Griffin from [21] that says: Theorem 2.3 A PVMD D is of finite t-character if and only if each t-invertible tideal of D is contained in at most a finite number of mutually t-comaximal t-invertible t-ideals of D. As indicated in the introduction of [41] the set of t-invertible t-ideals of a PVMD is a lattice ordered group under t-multiplication and the order defined by reverse containment of the ideals involved and that the above result for PVMDs came from the use of Conrad’s F-condition. Stated for lattice ordered groups Conrad’s F-condition says: Every strictly positive element exceeds at most a finite number of mutually disjoint elements. This and Theorem 2.3, eventually led the authors of [16], to the following statement.

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Theorem 2.4 (cf. Theorem 1 of [16]) Let D be an integral domain,  a finite character star operation on D and let  be a set of proper, nonzero, -ideals of finite type of D such that every proper nonzero -finite -ideal of D is contained in some member of  . Let I be a nonzero finitely generated ideal of D with I  = D. Then I is contained in an infinite number of maximal -ideals if and only if there exists an infinite family of mutually -comaximal ideals in  containing I . This theorem catapulted the consideration of finiteness of character from Pruferlike domains to consideration of finiteness of -character in general domains. But, there was an error in the proof. There was no reason for the error as the technique, Conrad’s F-condition, involved in the proof of Theorem 2.4 had been used at other places such as [15, 33] and, later, [17] but there it was. The sad story has been included in [42] and I see no point in dwelling on it, especially because the statement of the theorem was correct, [10]. However, for the record I include below a proof of 2.4, that shows how such results should actually be proved. Lemma 2.5 A nonzero finitely generated ideal I is -homogeneous if and only if for each pair X, Y of proper -ideals of finite type containing I we have that (X + Y ) is proper. Proof See [16].



Remark 2.6 (1)Note that if A and B are proper -ideals such that (A + B) = D and if C is any proper -ideal containing B then (A + C) = D, since (A + C) = (A + B + C) , (2) note also the change of definition of a -homogeneous ideal. In [16] an I was called -homogeneous if I is a -ideal of finite type that is contained in a unique maximal -ideal and in this paper, following [5] and [26], I call a finitely generated nonzero ideal I -homogeneous if I is contained in a unique -maximal ideal. As it is explained in [43], the two definitions are equivalent. Theorem 2.7 Let  be a finite type star operation defined on an integral domain D and let  be a set of nonzero -ideals of finite type such that every nonzero -ideal is contained in at least one member of . Then D is of finite -character if and only if every -ideal of finite type of D is contained in at most a finite number of mutually -comaximal -ideals of finite type from . Proof Let A be a -ideal of finite type of D. Lemma 2.5 ensures that if there are no two mutually -coprime -ideals of finite type(from ) containing A, then A itself contains a -homogeneous ideal I such that A = (I ) , see [43]. Next we show (I) that every -ideal of finite type of D is contained in at least one I  ∈  for a -homogeneous ideal I of D. For suppose that there is a -ideal A of finite type of D that is not contained in any I  for a -homogeneous ideal I of D. Then obviously A is not -homogeneous. So there are at least two proper -ideals A1 , B1 of finite type, such that (A1 + B1 ) = D and A ⊆ A1 , B1 . Obviously, neither of A1 , B1 is homogeneous. As B1 is not -homogeneous there are at least two -comaximal proper -ideals B11 , B12 of finite type containing B1 . Now by Remark 2.6 A1 , B11 , B12 are mutually -comaximal proper -ideals containing A and by assumption none of these

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is -homogeneous. Let B123 and B22 be two -comaximal proper -ideals containing B12 . Then by Remark 2.6 and by assumption, A1 , B11 , B22 , B123 are proper mutually -comaximal -ideals containing A and none of these ideals is homogeneous, and so on. Thus at stage n we have a collection: A1 , B11 , B22 , ..., Bnn , B12...n,n+1 that are proper mutually -comaximal -ideals containing A and none of these ideals is homogeneous. Also as each of the ideals in the above list is contained in a member of , which must be -comaximal with the other members of the list, we can conclude that the list belongs to . Now the process is never ending and has the potential of delivering an infinite number of mutually -comaximal proper -ideals of finite type containing A, contrary to the finiteness condition. Whence the conclusion. Let S be the set of all the -homogeneous ideals containing A and define a function ϕ : S → -Max(D) by ϕ(H ) = M(H ) the unique maximal -ideal containing H ∈ S. Obviously ϕ is a well-defined function. Let {Mα } = ϕ(S) and note that |ϕ(S)| < ∞. For if we were to choose exactly one -homogeneous ideal Hα from each Mα then {Hα } is the set of mutually disjoint -homogeneous ideals containing A, which must be finite by the condition. Next claim that ϕ(S) = {M1 , M2 , ...Mn } is the set of all the maximal -ideals containing A. For if P ∈ -Max(D)\ϕ(S) were a maximal -ideal n Mi . As (A, y) ⊆ P we have (A, y) = D, containing A then there is y ∈ P\ ∪i=1 forcing (A, y) and hence A, in a homogeneous ideal H ∈ / S, a contradiction. For the converse let A be a nonzero finitely generated ideal in a domain D of finite -character and let, by way of contradiction, {Ai |(Ai ) = D} be infinitely many mutually -comaximal finitely generated nonzero ideals containing A. Since Ai are mutually -co-maximal the sets of maximal -ideals ∪{Mαi } containing {Ai } would be infinite forcing A in an infinite set of distinct maximal -ideals. (Dan Anderson’s help with the proof is gratefully acknowledged.)  So, if we must construct a -homogeneous ideal we know where to go. Otherwise there are plenty of -potent domains, with one kind studied in [26] under the name -super potent domains.

3 What -homogeneous Ideals Can Do This much about -homogeneous ideals and potent domains leads to the questions: What else can -homogeneous ideals do? These ideals arose and figure prominently in the study of finite -character of integral domains, as we have seen above. The domains of -finite character where the -homogeneous ideals show their full force are the -Semi Homogeneous (-SH) Domains. Indeed as I indicate in Sect. 4 of [42], these ideals have been with me ever since I wrote Chapter One of my doctoral thesis. It turns out, and it is easy to see, that if I and J are two -homogeneous ideals that are similar, i.e. that belong to the same unique maximal -ideal (i.e. M(I ) = M(J ) in the notation and terminology of [5]) then I J is -homogeneous belonging to the same maximal -ideal. With the help of this and some auxiliary results it can then be shown that if an ideal K is a -product of finitely many -homogeneous ideals then K

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can be uniquely expressed as a -product of mutually -comaximal -homogeneous ideals. Based on this a domain D is called a -semi homogeneous (-SH) domain, if every proper nonzero principal ideal of D is expressible as a -product of finitely many -homogeneous ideals. It was shown in [5, Theorem 4] that D is a -SH domain if and only if D is a -h-local domain (D is a locally finite intersection of localizations at its maximal -ideals and no two maximal -ideals of D contain a common nonzero prime ideal.) Now if we redefine a -homogeneous ideal so that the -product of two similar, newly defined, -homogeneous ideals is a -homogeneous ideal meeting the requirements of the new definition, we have a new theory. To explain the process of getting a new theory of factorization merely by producing a suitable definition of a -homogeneous ideal we give below one such theory. Definition 3.1 Call a -homogeneous ideal I -axial homogeneous ( -a-homogeneous), if for every finitely generated ideal J with J  ⊇ I there is a positive integer n such that (J n ) is contained in a proper principal ideal. Also call a domain D a -ASH (-axial-SH) domain if for every nonzero non unit x of D, x D is a -product of finitely many -axial homogeneous ideals and call D a -axial potent domain if every maximal -ideal of D contains a -axial homogeneous ideal. Note that if I is a -axial homogeneous ideal and A is a -homogeneous ideal containing I, the proper principal ideal (d), that contains (An ) , must belong to M(I ). For if d belongs to any other maximal ideal N , then A will be contained in N too. Let I and J be two similar -axial homogeneous ideals. Since M(I ) = M(J ) we conclude that (I J ) is a -homogeneous ideal. Next let F be a -homogeneous ideal such that F ∗ contains I J. Then since F + I contains I we have ((F + I )n ) contained in a proper principal ideal (d), forcing (F n ) ⊆ (d). Now that we have shown that the -product of two similar -axial homogeneous ideals is a -axial homogeneous ideal similar to them, we can develop the theory of -ASH domains exactly along the lines of -SH domains discussed in [5]. To establish that the theory of -ASH domains is not an empty theory we have the following result on offer. First let’s recall that an integral domain D is a -AGCD domain if for every finite set of nonzero elements x1 , ..., xr ∈ D there is a positive integer n = n(x1 , ..., xr ) such that (x1n , ..., xrn ) is principal. Also, D is a pre-Schreier domain if x|yz implies x = r s where r |y and s|z, for all x, y, z ∈ D\{0}. Proposition 2 (a) In a -AGCD domain D every -homogeneous ideal is -axial and thus a -AGCD -SH domain is a -ASH domain, (b) Every t-homogeneous ideal in a pre-Schreier domain D is a t-axial ideal and so a pre-Schreier t-SH domain is a t-ASH domain and (c) A weakly Krull domain D such that every height one prime is the radical of a principal ideal is a t-ASH domain. Proof (a) Recall from Lemma 2.3 of [35] that for any integral domain D and for any finite set x1 , ..., xr ∈ D\{0}, (x1 , ..., xr )nr ⊆ (x1n , ..., xrn ) ⊆ (x1 , ..., xr )n and if  is defined on D, then  can be applied throughout. Now in a -AGCD D, (x1n , ..., xrn ) = (d), for some n = n(x1 , ..., xr ) and so for some m(= nr ) we have ((x1 , ..., xr )m ) contained in a proper principal ideal. Next if in a -AGCD domain

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D, I is a -homogeneous ideal then I  = D and depending upon the choice of x1 , ..., xr ∈ D\{0}, such that I = (x1 , ..., xr ) and on n = n(x1 , ..., xr ), we can find a positive integer m such that (I m ) is contained in a proper principal ideal, (x1n , ..., xrn ) = (d), that is contained in (I n ) and hence in M(I ). The same can be said about any -homogeneous ideal J with J  containing I, forcing I to be -axial. Consequently a -AGCD domain that is a -SH domain is a -ASH domain. (b) In Lemma 2.1 of [39] it was shown that if D is a Schreier domain and A a finitely generated ideal of D with Av = D then A must be contained in a proper principal ideal. As a Schreier domain is an integrally closed pre-Schreier domain and as the property of being integrally closed was not used in the proof of [39, Lemma 2.1] we conclude that the result holds for pre-Schreier domains. So in a pre-Schreier domain a t-homogeneous ideal I is such that I is contained in a proper principal ideal and so is the case of any t-homogeneous ideal containing I, making I a t-axial homogeneous ideal. Consequently a pre-Schreier domain that is a t-SH domain is a t-ASH domain. (c) Let us note that if D is a one dimensional quasi-local domain and if a, x1 , ..., xr are nonzero non units in D then a|xin for some positive integer n. Next let P be a height one prime ideal of the weakly Krull domain described √ in the statement of part (c) and let a be an element of P such that P = (a). Also let (x1 , ..., xr )t be a t-homogeneous ideal contained in P. (Such ideals can be arranged as in the proof of Proposition 1.) Now a|xin for some n and all i = 1, ..., r in D P because D P is one dimensional and a|xin in D Q for every height one prime Q = P, because a ∈ / Q. But then a|xin in D and, by [35, Lemma 2.3], we have n m ((x1 , ..., xr ) ) ⊆ (x1 , ..., xrn ) ⊆ (a), which gives ((x1 , ..., xr )m )t ⊆ (x1n , ..., xrn )t ⊆ (a). But then every t-homogeneous ideal contained in a particular height one prime is a t-axial ideal. Now it is well known that for each nonzero non unit x in a weakly Krull domain D, x D = (x D P1 ∩ D) ∩ ... ∩ (x D Pn ∩ D) = ((x D P1 ∩ D)...(x D Pn ∩ D))t where each of x D Pi ∩ D is a t-ideal of finite type contained only in Pi and hence is a t-axial homogeneous ideal.  Next, each of the definitions of -homogeneous ideals can actually give rise to -potent domains in the same manner as the -super potent domains of [26]. In [26], for a star operation  of finite character, a -homogeneous ideal is called -rigid. The -maximal ideal containing a -homogeneous ideal I may be called a -potent maximal -ideal, as we have already done. Next we may call the -homogeneous ideal I -super-homogeneous if each -homogeneous ideal J with J  containing I is -invertible and we may call a -potent domain D -super potent if every maximal -ideal I of D contains a -super homogeneous ideal. But then one can study A-potent domains where A refers to a -homogeneous ideal that corresponds to a particular √ definition. For example a -homogeneous ideal J is said to be of type 1 in [5] if J = M(J ). So we can talk about -type 1 potent domains as domains each of whose maximal -ideals contains a -homogeneous ideal of type 1. The point is, to each suitable definition say A of a -homogeneous ideal we can study the -Apotent domains as we studied the -super potent domains in [26]. Of course the theory corresponding to definition A would be different from that of other -potent domains. For example each of the maximal -ideal of the -type 1 potent domain would be

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the radical of a -homogeneous ideal etc.. Now as it is usual we present some of the concepts that have some direct and obvious applications, stemming from the use of -homogeneous ideals. For this we select the -f-potent domains for a study.

3.1 -f-potent Domains Let  be a finite type star operation defined on an integral domain D which is not a field unless specifically stated, throughout this section. Call a nonzero non unit element r of D -factorial rigid ( -f-rigid) if r belongs to a unique maximal ideal and every finite type -homogeneous ideal containing r is principal. Indeed if r is a -f-rigid element then r D is a -f- homogeneous ideal and hence a -super homogeneous ideal, we may also call r D a -f-rigid ideal. So the terminology and the theory developed in [5] applies. Note here that every non unit factor s of a -frigid element r is again -f-rigid because of the definition. Note also that if r, s are similar -f-rigid elements (i.e. r D, s D are similar -f-homogeneous ideals) then r s is a -f-rigid element similar to r and s and so r |s or s|r. Also if r is -f-rigid then r n is -f-rigid for any positive integer n. Following definition 10 of [5] we say that D is a t-f-SH domain if every nonzero non-unit of D is expressible as a product of finitely many t-f-rigid (i.e. t-f-homogeneous) elements. By Theorem 17 of [5] a t-f-SH domain is a GCD domain. Example 3.2 Every power of a principal prime is a t-f-rigid element. Call a maximal -ideal M -f-potent if M contains a -f-rigid element and call a domain D -f-potent if every maximal -ideal of D is -f-potent. Example 3.3 UFDs PIDs, Semirigid GCD domains, prime potent domains are all t-f-potent. (Domains in which every maximal t-ideal contains a prime element may be called prime potent. Indeed a prime element generates a maximal t-ideal [24, 13.5]. (So a domain in which every maximal t-ideal contains a prime element is simply a domain in which every maximal t-ideal is principal.) The definition suggests right away that if r is -f-rigid and x any element of D then (r, x) = s D for some s ∈ D and applying the v-operation to both sides we conclude that GC D(r, x) = (r, x)v of r exists with every nonzero element x of D and that for each pair of nonzero factors u, v of r we have u|v or v|u; that is r is a rigid element of D, in Cohn’s terminology [11]. Indeed it is easy to see, if necessary with help from [5], that a finite product of -f-rigid elements is uniquely expressible as a product of mutually -comaximal -f-rigid elements, up to order and associates and that if every nonzero non unit of D is expressible as a product of -f-rigid elements then D is, at least, a semirigid GCD domain of [37]. Also, as we shall show below, a t-f-potent domain of t-dimension one (i.e. every maximal t-ideal is of height one) is a GCD domain of finite t-character, but generally a t-f-potent domain is far from

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being a GCD domain. Before we delve into examples, let’s prove a necessary result, by adjusting Theorem 4.12 of [13] and its proof to our situation. Proposition 3 Let D be an integral domain and let L be an extension field of the field of fractions K of D. Then each nonzero ideal F of R = D + X L[X ] is of the form f (X )J R = f (X )(J + X L[X ]) , where J is a D-submodule of L and f (X ) ∈ R such that f (0)J ⊆ D. If F is finitely generated, J is a finitely generated D-submodule of L. Proof Indeed if F = f (X )J R such that f (0)J ⊆ D, then F is an ideal of R. Next let F = ({ f α }α∈ ) and let F be such that F ∩ D = 0. According to [14, Lemma 1.1], the following are equivalent for an ideal F of R: (1) F is such that F ∩ D = 0, (2) F ⊇ X L[X ] and (3) F L[X ] = L[X ]. Further if any of these hold, then F = F ∩ D + X L[X ] = (F ∩ D)R and taking f (X ) = 1, J = F ∩ D we have the stated form. Let’s now consider the case when I L[X ] = L[X ]. In this case I L[X ] = f (X )L[X ] where f (X ) is a variable polynomial of L[X ]. Two cases arise: (i) f (0) = 0 and (ii) f (0) = 0. In case of (i) f (X ) = l + Xg(X ) = l(1 + (X/l)g(X )) where l ∈ L\{0}. Let f  = f (X )/l so that f  (0) = 1 and proceed as follows. Since f  (X ) ∈ R, F/ f  (X ) is a fractional ideal of R and hence an R-submodule. Because in L[X ], every element of F is divisible by f  (X ) F = f  (X )({ f α / f  }α∈ ) = f  (X )({rα0 + X sα (X )}α∈ ). Also because f  (0) = 1 we have rα0 ∈ D. So F/ f  (X ) = ({rα0 + X sα (X )}α∈ ) is an ideal of R. But since F/ f  (X )L[X ] = L[X ] we have X L[X ] ⊆ F/ f  (X ) = ({rα0 + X sα (X )}α∈ ). Forcing F/ f  (X ) = ({rα0 }α∈ ) + X L[X ], where J = ({rα0 }α∈ ) is an ideal of D that can be termed as a D-submodule of L , [14, Lemma 1.1]. But then F = f  J R, as desired. Finally in case (ii) we have F L[X ] = f (X )L[X ] where f (0) = 0 and we can take f (X ) = X r g(X ), where we can assume that g(0) = 1. Next every element and hence every generator of F is divisible by f (X ), in L[X ] and so we can write F = f (X )({ f α / f }α∈ ) = f (X )({lα h α (X )X sα }), where lα ∈ L\{0}, h α (X ) ∈ L[X ] with h α (0) = 1 and sα ≥ 0. Now F/ f (X ) = ({lα h α (X )X sα }) has the now familiar property that F/ f (X )L[X ] = L[X ]. But this time it means that (a) ({lα h α (X )X sα }) ∩ L = (0) and (b) there is a non-empty subset  of  such that sδ = 0 for all δ ∈ . By (a) we have X L[X ] ⊆ F/ f (X ) and by (a) and (b) we have the D-submodule ({lδ }δ∈ ) ⊆ F/ f (X ) of L . Thus F/ f (X ) ⊇ ({lδ }δ∈ ) + X L[X ]. For the reverse containment let h(X ) ∈ F/ f (X ). If h(0) = 0, then h(X ) ∈ X L[X ]. If, on the other hand, h(0) = g = 0, then h(X ) = g + X k(X ) and as f (X )h(X ) ∈ F we have f (X )(g + X k(X )) ∈ F = ({ f α }α∈ ). This gives f (X )(g + X k(X )) = rα1 (X ) f α1 (X ) + ... + (X ) ∈ R. Since, ) = lαi h αi (X )X sαi we rαn (X ) f αn (X ), where rαi   in L[X ], sf αi (X )/ f (X n n αi + X sαi (X )) have g + Xk(X ) = i=1 rαi (X ) lαi h αi (X )X ) = i=1 (r αi0   n sαi ) . Next since the left hand side of g + X k(X ) = lαi h αi (X )X i=1 (r αi0 +   sαi X sαi (X )) lαi h αi (X )X ) has a nonzero constant term, some of the αi are in . Without loss of generality we can assume that the first r of the sαi are zero and thoseαi are δi . But then comparing the constants (by setting X = 0) we get g = ri=1 (rδi0 ) lδi ∈ ({lδ }δ∈ ). Whence g + X k(X ) ∈ ({lδ }δ∈ ) + X L[X ] and F/ f (X ) ⊆ ({lδ }δ∈ ) + X L[X ]. Thus F/ f (X ) = ({lδ }δ∈ ) + X L[X ] = (({lδ }δ∈ ))R or F = f (X )J R where J is a D-submodule of L such that f (0)J ⊆ D.

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For the finitely generated case let’s note that if F = ( f 1 , f 2 , ..., f n )(D + X L[X ]) is an ideal of R = D + X L[X ] with I = F ∩ D = (0) then F = ( f 10 , f 20 , ..., f n0 + X L[X ])(D + X L[X ]); where f i0 are the constant terms of f i . This is because, if F ∩ D = (0) then X L[X ] ⊆ F. Thus the constant terms are all contained in F. This 10 , f 20 , ..., f r 0 + X L[X )). But for f (X ) ∈ F we have f (X ) =  gives F ⊇ ( f ai (X ) f i (X ) = ai0 f i0 + X h(X ) ⊆ f 10 , f 20 , ..., f n0 + X L[X ]. Thus we have F = ( f 10 , f 20 , ..., f n0 )(D + X L[X ]) = f (X )J (D + X L[X ]) where f (X ) = 1 and ( f 10 , f 20 , ..., f n0 ) = J, an ideal of D. If on the other hand I = F ∩ D = (0), F L[X ] = f (X )L[X ] where f (X ) ∈ L[X ] and f i (X ) = f (X )h i (X ) with h i (X ) ∈ L[X ]. Now suppose that f (0) = 0. Then we can assume that f (0) = 1 and so h i (X ) = (h i0 + X ki (X )) where h i0 ∈ D and so F = f (X )(h 10 + X k1 (X ), ..., h n0 + X kn (X ))R. Since (F/ f (X ))L[X ] = L[X ], (F/ f (X )) ∩ D = (0). Resulting in X L[X ] ⊆ F/ f (X ) = (h 10 + X k1 (X ), ..., h n0 + X kn (X ))R and thus making F/ f (X ) = (h 10 , ..., h n0 ) + X L[X ])R or F = f (X )(h 10 , ..., h n0 )R, where (h 10 , ..., h n0 ) = J. Finally if f (0) = 0, then f (X )L[X ] ⊆ X L[X ] and so F = f (X )J (D + X L[X ]) where J is a D-submodule of L as determined in the following. Since f (0) = 0 we can write f (X ) = X r g(X ), where g(0) = 1. This gives in return F = ( f 1 , f 2 , ..., f n ) (D + X L[X ], f i = g(X )li X r +si h i (X ) where si ≥ 0. Thus we have F/ f (X ) = {l1 h 1 (X )X s1 , ..., ln h n (X )X sn ). But as (F/ f (X ))L[X ] = L[X ], F/ f (X ) = (l1 h 1 (X ) X s1 , ..., ln h n (X )X sn ) must contain some l ∈ L\{0} and consequently some si = 0. This results in X L[X ] ⊆ (l1 h 1 (X )X s1 , ..., ln h n (X )X sn ) making F/ f (X ) = ((l1 , l2 , ..., lr ) + X L[X ]), where J = (l1 , l2 , ..., lr ) is a D-submodule of L because F/ f (X ) is an R-module. But then F = f (X )J R, as claimed.  Before we proceed further, it seems pertinent to give an idea of the prime and maximal t-ideals of R = D + X L[X ]. As Proposition 3 indicates a general nonzero ideal I of R is of the form I = f (X )F R where f ∈ R and F is a D-submodule of L such that f (0)F ⊆ D. Moreover I can be of two types: (i) I such that I ∩ D = (0), which is equivalent to I L[X ] = L[X ], by [14, Lemma 1.1] and (ii) I such that I ∩ D = (0). (And this, by [14, Lemma 1.1], is equivalent to I L[X ] = L[X ] which indeed is the same as I L[X ] = f (X )L[X ].) So there are two kinds of prime ideals. The prime and maximal ideals of type (i) are given as ideals P of R such that P = (P ∩ R) + X L[X ] where P is maximal if and only if P ∩ R is, by [14, Theorem 1.3]. A prime ideal of type (ii) is either X L[X ], or a contraction from a prime of L[X ] of the form f (X )L[X ], via [14, Theorem 1.3, Lemmas 1.2, 1.5]. If indeed f is such that f (0) = 0 then by [14, Theorem 1.3, Lemma 1.5] f (X )L[X ] contracts to f (X )R where f (X ) is irreducible and f (0) = 1 and f (X )R is a maximal height one prime. So maximal ideals of type (ii) are precisely of height one and are principal. This means that if D is not a field, maximal ideals of type (ii) are principal of height one and so (maximal) t-ideals. This leaves maximal t-ideals of type (i). For that note that D + X L[X ] has the D + M form. Thus if I is a nonzero ideal of D then (I + X L[X ])v = Iv + X L[X ] = Iv (D + X L[X ]), by [6, Proposition 2.4] and using this we can also conclude that (I + X L[X ])t = It + X L[X ] = It (D + X L[X ]). Thus if P is a maximal t-ideal of R with P ∩ D = (0), then P = (P ∩ D) + X L[X ]

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where (P ∩ D) is a maximal t-ideal of D. On the other hand if ℘ is a maximal t-ideal of D then so is P = ℘ + X L[X ]. Indeed if D is a field, nonzero prime ideals of R = D + X L[X ] are either principal of the form (1 + Xg(X ))R or the prime X L[X ] and all are maximal of height one. Lemma 3.4 Let D be an integral domain (that is not a field) and let L be an extension field of the field of fractions K of D. Then (a) d ∈ D\(0) is a t-f-rigid element of D if and only if d is a t-f-rigid element of R = D + X L[X ] and (b) An element α of X L[X ] is t-f-rigid if and only if α = l X r , D is a valuation domain and K = L. (In part (b) “t-f-rigid” can be replaced by “d-f-rigid” or just “rigid".) Proof Let d be a t-f-rigid element of D then d D is a t-f-rigid ideal, so any t-ideal of finite type, of D, containing d D is principal. Next consider d ∈ D + X L[X ]. Any tideal of finite type F of R containing d intersects D and so has the form (F ∩ D) + X L[X ], according to [14, Lemma 1.1]. Consequently F contains d D + X L[X ]. We show that F is principal. For this let F = (a1 + X f 1 (X ), ..., an + X f n (X )t = ((a1 , ..., an ) + X L[X ])v = ((a1 , ..., an )v + X L[X ]). But ((a1 , ..., an )v + X L[X ]) = F ⊇ d D + X L[X ] forces (a1 , ..., an )v ⊇ d D. Also d D being t-f-rigid, (a1 , ..., an )v = d  must be principal, whence F = d  D + X L[X ] = d  (D + X L[X ]) = d  R is principal. Now note that according to [14], every prime ideal M of R that intersects D is of the form M ∩ D + X L[X ] and using [6, Proposition 2.4] we can show that every maximal t-ideal M of D + X L[X ] that intersects D is of the form M ∩ D + X L[X ] where M ∩ D is a maximal t-ideal of D and that, conversely, if m is a maximal ideal of D then m + X L[X ] is a maximal ideal of R. Thus, finally, if m is the unique maximal t-ideal of D containing d then m + X L[X ] is a maximal t-ideal of R containing d. Note that if N were another maximal t-ideal containing d then N ∩ D would be another maximal t-ideal of D containing d, a contradiction. Thus d is a t-f-rigid ideal of R. For (b), let α ∈ X L[X ]. Then α = l X r g(X ) where g(0) = 1 and l ∈ L\{0}. So l X and X are t-f-rigid for every l ∈ L\{0} because X, l X |α 2 , at least. Thus l X |X or X |l X, being two similar rigid elements. This forces l ∈ D or l −1 ∈ D for each l ∈ L\{0} and this makes L = q f (D) and D a valuation domain. Also if g(X ) = 1 we do not have g(X )|X , nor do we have X |g(X ). Thus α = l X r as claimed. Conversely if D is a valuation domain and L = q f (D) are as given, then R = D + X K [X ] is a Bezout domain and X belongs to a unique maximal ideal M + X L[X ], where M is the maximal ideal of D. Since R is Bezout, every finitely  generated ideal containing X is principal, same with l X r . Proposition 4 Let D be an integral domain that is not a field and let L be an extension field of the field of fractions K of D. Then D is t-potent if and only if R = D + X L[X ] is. If D is a field, then R = D + X L[X ] is t-potent. Proof Note that, according to [14, Lemma 1.2], every prime ideal P of R that is not comparable to X L[X ] contains an element of the form 1 + Xg(X ), so must contain a prime element of the form 1 + Xg(X ) and so must be a principal prime. Note also that D is not a field, so no homogeneous ideal of R is contained in X L[X ]. We next show

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that a finitely generated ideal F  X L[X ] of R is t-homogeneous if and only if F is of the form A + X L[X ], where A is a t-homogeneous ideal of D or F is generated by a prime power of the form (1 + X h(X ))n . For this note that if F = A + X L[X ] and A is contained in a unique maximal t-ideal P of D then A + X L[X ] is contained in the maximal t-ideal P + X L[X ] and any maximal t-ideal of R that contains A + X L[X ] also contains P + X L[X ]. Also if A + X L[X ] is contained in a unique maximal t-ideal then the maximal t-ideal that contains A + X L[X ] would have to be of the form P + X L[X ] where P contains A. Thus A is a t-homogeneous ideal of D if and only if A + X L[X ] is of R. Next, an ideal generated by a prime power is t-homogeneous anyway. Conversely let F be a finitely generated nonzero ideal of R. Then by Proposition 3, F = f (X )(J + X L[X ]) where f (X ) ∈ R. As F is not contained in X L[X ] there are two possibilities (i) f (X ) = 1 forcing J to be a finitely generated ideal of D or (ii) f (0) = 1. If in addition F has to be thomogeneous, then in case (i) F is contained in a prime ideal of the form P + X L[X ] and in case (ii) in a prime ideal incomparable with X L[X ], forcing J = D. In the first case F = J + X L[X ] where J is a t-homogeneous ideal belonging to P and in the second case F = f (X )R, where f (X )R, being homogeneous, is contained in a principal prime that contains an element of the type 1 + Xg(X ) [14, Lemmas 1.2, 1.5] and so is a prime power of the desired type. If D is a field then R = D + X L[X ] is one dimensional. So all maximal ideals are t-ideals, of which one is X L[X ] and all the others are principal of the form (1 + Xg(X ))R, where 1 + Xg(X ) is irreducible in L[X ]. Indeed every finitely generated ideal of R contained entirely in X L[X ] is t-homogeneous belonging to X L[X ], in this case. (Ideals such as X R, (X, l X ) etc.)  Corollary 1 Let D be an integral domain that is not a field and let L be an extension field of the field of fractions K of D. Then D is t-f-potent if and only if R = D + X L[X ] is. In case D is a field R = D + X L[X ] is t-f-potent if and only if D = L . Proof Suppose that D is t-f-potent. As in the proof of Proposition 4 every maximal t-ideal P of R that is not comparable to X L[X ] contains an element of the form 1 + Xg(X ), so must contain a prime element of the form 1 + Xg(X ) and so must be a principal prime. Next the maximal t-ideals comparable to X L[X ] are of the form P + X L[X ] where P is a maximal t-ideal of D. Since D is t-f-potent P contains a tf-rigid element, which is also a t-f-rigid element of R, by Lemma 3.4. So P + X L[X ] contains a t-f-rigid element of R. In sum, every maximal t-ideal of R contains a tf-rigid element of R and so R is t-f-potent. Conversely suppose that R is t-f-potent. Then, as for each maximal t-ideal M of D, M =M + X L[X ] is a maximal t-ideal of R, so must contain a t-f-rigid element r of R. We claim that there is a t-f-rigid r ∈ M. For if not and r = g + X h(X ) ∈ M, then two cases arise (i) g = 0 and (ii) g = 0. Case (i) is possible only if X h(X ) = 0 and this forces r ∈ M. Next, by (b) of Lemma 3.4, case (ii) is possible only if r = l X r , D is a valuation domain and L = K . But then every nonzero non-unit of D, is t-f-rigid and in M = M + X K [X ]. Thus each maximal t-ideal of D contains a t-f-rigid element of D. In case D is a field we proceed as follows. If D = L, then R = L[X ] which is a PID and so t-f-potent (and d-f-potent). If on the other hand R = D + X L[X ] is t-f-potent, with D a field, then

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X, l X must be t-f-rigid, for each l ∈ L\{0}. But then X |l X or l X |X and in either case l ∈ D.  Recall, from [1], that a GCD domain of finite t-character that is also of t-dimension 1 is called a generalized UFD (GUFD). Example 3.5 If D is a UFD (GUFD, Semirigid GCD domain) that is not a field and L an extension of the quotient field K of D, then the ring D + X L[X ] is a t-f-potent domain. The t-f-potent domains and their examples are nice but we must show that they have some useful properties. We have here the most striking property. Let X be an indeterminate over D. A polynomial f = ai X i is called primitive if its content A f = (a0 , a1 , ..., an ) is a primitive ideal, i.e., (a0 , a1 , ..., an ) ⊆ a D implies a is a unit and super primitive if (A f )v = D. It is known that while a super primitive polynomial is primitive, a primitive polynomial may not be super primitive, see e.g. Example 3.1 of [4]. A domain D is called a PSP domain if each primitive polynomial f over D is superprimitive, i.e. if f being primitive implies (A f )v = D. Proposition 5 A t-f-potent domain D has the PSP property.  Proof Let f = ai X i be primitive i.e. (a0 , a1 , ..., an ) ⊆ a D implies a is a unit and consider the finitely generated ideal (a0 , a1 , ..., an ) in a t-f- potent domain D. Then (a0 , a1 , ..., an ) is contained in a maximal t-ideal M associated with a t-f-rigid element r (of course M = M(r D)) if and only if (a0 , a1 , ..., an , r )t = s D = D. Since every maximal t-ideal of a t-f-potent domain is associated with a t-f-rigid element, we  conclude that in a t-f-potent domain D, f = ai X i primitive implies that A f is contained in no maximal t-ideal of D; giving (A f )v = D which means that each primitive polynomial f in a t-f-potent domain D is actually super primitive.  Now PSP implies AP i.e. every atom is prime, see e.g. [4]. So, in a t-f-potent domain every atom is a prime. If it so happens that a t-f-potent domain has no prime elements then the t-f-potent domain in question is atomless. Recently atomless domains have been in demand. The atomless domains are also known as antimatter domains. Most of the examples of atomless domains that were constructed were the so-called pre-Schreier domains, i.e. domains in which every nonzero non unit a is primal (is such that (a|x y implies a = r s where r |x and s|y). One example (Example 2.11 [4]) was laboriously constructed in [4] and this example was atomless and not pre-Schreier. As we indicate below, it is easy to establish a method of telling whether a t-f-potent domain is pre-Schreier or not. Cohn in [11] called an element c in an integral domain D primal if (in D) c|a1 a2 implies c = c1 c2 where ci |ai . Cohn [11] assumes that 0 is primal. We deviate slightly from this definition and call a nonzero element c of an integral domain D primal if for all a1 , a2 ∈ D\{0}, c|a1 a2 implies c = c1 c2 such that ci |ai . He called an integral domain D a Schreier domain if (a) every (nonzero) element of D is primal and (b) D is integrally closed. We have included nonzero in brackets because while he meant to include zero as a primal element, he mentioned that the group of divisibility of

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a Schreier domain is a Riesz group. Now the definition of the group of divisibility G(D)(= { ab D : a, b ∈ D\{0}} ordered by reverse containment) of an integral domain D involves fractions of only nonzero elements of D, so it’s permissible to restrict primal elements to be nonzero and to study domains whose nonzero elements are all primal. This is what McAdam and Rush did in [31]. In [38] integral domains whose nonzero elements are primal were called pre-Schreier. It turned out that pre-Schreier domains possess all the multiplicative properties of Schreier domains. So let’s concentrate on the terminology introduced by Cohn as if it were actually introduced for pre-Schreier domains. Cohn called an element c of a domain D completely primal if every factor of c is primal and proved, in Lemma 2.5 of [11] that the product of two completely primal elements is completely primal and stated in Theorem 2.6 a Nagata type result that can be rephrased as: Let D be integrally closed and let S be a multiplicative set generated by completely primal elements of D. If D S is a Schreier domain then so is D. This result was analyzed in [4] and it was decided that the following version ([4, Theorem 4.4] of Cohn’s Nagata type theorem works for pre-Schreier domains. Theorem 3.6 (Cohn’s Theorem for pre-Schreier domains) Let D be an integral domain and S a multiplicative set of D. (i) If D is pre-Schreier, then so is D S . (ii) (Nagata type theorem) If D S is a pre-Schreier domain and S is generated by a set of completely primal elements of D, then D is a pre-Schreier domain. Now we have already established above that if r is a t-f-rigid element then (r, x)v is principal for each x ∈ D\{0}. But then (r, x)v is principal for each x ∈ D\{0} if and only if (r ) ∩ (x) is principal for each x ∈ D\{0}. So, r is what was called in [3] an extractor. Indeed it was shown in [3] that an extractor is completely primal. Thus we have the following statement. Corollary 2 Let D be a t-f-potent domain. Then D is pre-Schreier if and only if D S is pre-Schreier for some multiplicative set S that is the saturation of a set generated by some t-f rigid elements. (Proof. If D is pre-Schreier then D S is pre-Schreier anyway. If on the other hand D S is pre-Schreier and S is (the saturation of a set) multiplicatively generated by some t-f- rigid elements. Then by Theorem 3.6, D is pre-Schreier.) We may note here that if D S is not pre-Schreier for some multiplicative set S, then D is not pre-Schreier. So the decision making result of Cohn comes in demand only if D S is pre-Schreier. Of course in the Corollary 2 situation, the saturation S of the multiplicative set generated by all the t-f-rigid elements of D, leads to "if D S is not pre-Schreier then D is not pre-Schreier for sure and if D S is pre-Schreier then D cannot escape being a pre-Schreier domain". Example 3.7 Let D = ∩i=n i=1 Vi be a finite intersection of distinct non-discrete rank one valuation domains with quotient field K = q f (D), X an indeterminate over D and let L be a proper field extension of K . Then (a) D + X L[X ] is a non-preSchreier, t-f-potent domain and (b) D + X L[X ](X ) is an atomless non-pre-Schreier, t-f-potent domain.

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Illustration: (a) It is well known that D is a Bezout domain with exactly n maximal ideals, Mi [29], with Vi = D Mi . Thus D= ∩D Mi and each of Mi being a t-ideal must, each, contain a t-homogeneous ideal by Proposition 1. R = D + X L[X ] is t-f-potent by Corollary 1. Now D + X L[X ] is not pre-Schreier for the following reason. The set D ∗ = D\(0) is a multiplicative set of R and R D∗ = (D + X L[X ]) D∗ = K + X L[X ] which is an atomic domain and not a UFD if K = L and an atomic pre-Schreier domain must be a UFD. For part (b) note that since R = D + X L[X ] is t-f-potent, so is D and all the maximal (t-)ideals of D + X L[X ](X ) are Mi + X L[X ](X ) where Mi are the maximal ideals of D and each Mi contains a t-f-rigid element. Finally as (D + X L[X ](X ) ) D∗ = K + X L[X ](X ) which cannot be Schreier, being atomic and non-UFD. One more result that can be added needs introduction to a neat construction called the Nagata ring construction these days. This is how the construction goes. Let  be a star operation on a domain D, let X be an indeterminate over D and let S = { f ∈ D[X ]| (A f ) = D}. Then the ring D[X ] S is called the Nagata construction from D with reference to  and is denoted by N a(D,  ). Indeed N a(D,  ) = N a (D,  f ) Proposition 6 ([28, Proposition 2.1]) Let  be a star operation on R. Let  f be the finite type star operation induced by . Let S = { f ∈ D[X ]|(A f ) = D}. Then (1) S = D[X ]\ ∪ M∈ M[X ] where  is the set of all maximal  f -ideals of D. (Hence S is a saturated multiplicatively closed subset of D[X ].), (2) {M[X ] S } is the set of all maximal ideals of [D X ] S . As pointed out in [19], proof of Part (1) of the following proposition has a minor flaw, in that for a general domain it uses a result ([20, 38.4]) that is stated for integrally closed domains. The fix offered in [19] is a new result and steeped in semistar operations. We offer, in the following, a simple change in the proof of [28, (1) Proposition 2.2.] to correct the flaw indicated above. Proposition 7 ([28, Proposition 2.2]) Let T be a multiplicatively closed subset of D[X ] contained in Sv = { f ∈ D[X ]|(A f )v = D}. Let I be a nonzero fractional ideal of D. Then (1) (I [X ]T )−1 = I −1 [X ]T , (2) (I [X ]T )v = Iv [X ]T and (3) (I [X ]T )t = It [X ]T . (1) It is clear that I −1 [X ]T ⊆ (I [X ]T )−1 . Let u ∈ (I [X ]T )−1 . Since for any a ∈ I \{0} we have (I [X ]T )−1 ⊆ a −1 D[X ]T ⊆ K [X ]T we may assume that u = f / h with f ∈ K [X ] and h ∈ T . Then f ∈ (I [X ]T )−1 . Hence f I [X ]T ⊆ D[X ]T . Hence b f ∈ D[X ]T for any b ∈ I . Now b f g ∈ D[X ] for some g ∈ Sv . So (Ab f g )v ⊆ D. By [32, Proposition 2.2.], (Ab f g )v = (Ab f A g )v = (Ab f )v , since (A g )v = D and hence vinvertible. Therefore b A f ⊆ (b A f )v = (Ab f )v ⊆ D for any b ∈ I. Hence A f ⊆ I −1 . Hence f ∈ I −1 [X ] and f / h ∈ I −1 [X ]T . Therefore (I [X ]T )−1 = I −1 [X ]T .

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Theorem 3.8 ([28, Theorem 2.4]) Let  be a finite type star operation on D. Let I be a nonzero ideal of D. Then I is -invertible if and only if I [X ] S is invertible. Theorem 3.9 ([28, Proposition 2.14]) Let  be a star operation on D. Then any invertible ideal of D[X ] S is principal. Thus we have the following corollary. Corollary 3 Let I be a t-ideal of finite type of D. Then I is t-invertible if and only if I [X ] Sv is principal. Proof If I [X ] Sv is principal, then I [X ] Sv is invertible and so I is t-invertible, by Theorem 3.8. Conversely let F be a finitely generated ideal such that Ft = I. Then F is t-invertible and so, by Theorem 3.8, is F[X ] Sv invertible and hence principal  by Theorem 3.9. But then F[X ] Sv = (F[X ] Sv )t = I [X ] Sv . Lemma 3.10 Let I be nonzero finitely generated ideal of D. Then It [X ] Sv is dhomogeneous if and only if I is t-homogeneous. Consequently It [X ] Sv is t-f-rigid if and only if I is t-super homogeneous. Proof Let I be a t-homogeneous ideal of D. Then I [X ] Sv is finitely generated and that It [X ] Sv is a t-ideal of finite type is an immediate consequence of Proposition 7. If M is the unique maximal t-ideal containing I, then at least M[X ] Sv ⊇ I [X ] Sv . Suppose that N is another maximal ideal of D[X ] Sv containing I [X ] Sv . But by Proposition 6, N = N [X ] Sv for some maximal t-ideal N of D. That is N = D ∩ N [X ] Sv ⊇ D ∩ I [X ] Sv ⊇ I . This forces N = M and consequently N [X ] Sv = M[X ] Sv making I [X ] Sv d-homogeneous. Conversely if I [X ] Sv is d-homogeneous contained in a unique M[X ] Sv ,suppose that N is another maximal t-ideal of D containing I. Then again N [X ] Sv ⊇ I D[X ] Sv which is d-homogeneous, a contradiction unless N = M. The consequently part follows from Corollary 3.  Let’s call a domain -f-r-potent if every maximal -ideal of D contains a -f-rigid element. Proposition 8 Let D be an integral domain with quotient field K , X an indeterminate over D and let Sv = { f ∈ D[X ]|(A f )v = D}. Then (a) D is t-potent if and only if D[X ] Sv is d-potent and (b) D is t-super potent if and only if D[X ] Sv is d-f-r-potent Proof (a) Suppose that D is t-potent. Let M[X ] Sv be a maximal ideal of D[X ] Sv and let I be a t-homogeneous ideal contained in M. By Lemma 3.10, It [X ] Sv is dhomogeneous, making M[X ] Sv d-potent. Conversely suppose that D[X ] Sv is d-potent and let M be a maximal t-ideal of D. Then M[X ] Sv is a maximal ideal of D[X ] Sv and so contains a d-homogeneous ideal I = ( f 1 , f 2 , ..., f n )D[X ] Sv . Now let I = ( f 1 , f 2 , ..., f n ). Then I = I D[X ] Sv and I ⊆ (A I )t [X ] Sv ⊆ M[X ] Sv , since M[X ] Sv is a t-ideal and f i ∈ M[X ] Sv ∩ D[X ]. This gives I = I D[X ] Sv ⊆ (A I )t [X ] Sv ⊆ M[X ] Sv making (A I )[X ] Sv another homogeneous ideal, contained in M[X ] Sv and containing I. But then (A I ) ⊆ M is a t-homogeneous ideal, by Lemma 3.10. For part (b) use part (a) and Corollary 3. 

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Next, another property that can be mentioned “off hand” is given in the following statement. Theorem 3.11 A t-f-potent domain of t-dimension one is a GCD domain of finite t-character. A domain of t-dimension one that is of finite t-character is called a weakly Krull domain. (D is weakly Krull if D = ∩D P where P ranges over a family F of height one prime ideals of D and each nonzero non unit of D belongs to at most a finite number of members of F.) A weakly Krull domain D is dubbed in [5] as -weakly Krull domain or as a √ type 1 -SH domain. Here a -homogeneous ideal I is said to be of type 1 if M(I ) = I  and D is a type 1 -SH domain if every nonzero non unit of D is a -product of finitely many -homogeneous ideals of type 1. In the following lemma we set  = t. Lemma 3.12 A t-f-potent weakly Krull domain is a type 1 t-f-SH domain. Proof A weakly Krull domain is a type 1 t-SH domain. But then for every pair I, J of similar homogeneous ideals I n ⊆ Jt and J m ⊆ It for some positive integers m, n. So J is a t-f-rigid ideal if I is and vice versa. Thus in a t-f-potent weakly Krull domain the t-image of every t-homogeneous ideal is principal. Whence every nonzero non unit of D is expressible as a product of t-f-rigid elements which makes D a t-f-SH domain and hence a GCD domain.  Proof of Theorem 3.11 Use Theorem 5.3 of [26] for  = t to decide that D is of finite t-character and of t-dimension one. Indeed, that makes D a weakly Krull domain that is t-f-potent. The proof would be complete once we apply Lemma 3.12 and note that a t-f-SH domain is a GCD domain and of course of finite t-character.  Generally a domain that is t-f-potent and with t-dimension > 1, is not necessarily GCD nor of finite t-character. Example 3.13 Let D = Z + X L[[X ]] where Z is the ring of integers and L is a proper extension of Q the ring of rational numbers. Indeed D is prime potent and two dimensional but neither of finite t-character nor a GCD domain. There are some special cases, in which a t-f-potent domain is GCD of finite t-character. i) If every nonzero prime ideal contains a t-f-rigid ideal. (Use (4) of Theorem 5 of [5]) along with the fact that D is a t-f-SH domain if and only if D is a t-SH domain with every t-homogeneous ideal t-f-rigid. Thus a t-f-potent domain of t-dim 1 is of finite character. ii) If D is a t-f-potent PVMD of finite t-character that contains a set S multiplicatively generated by t-f-rigid elements of D and if D S is a GCD domain then so is D. (This involves a straightforward use of Theorem 3.6 and the fact that a pre-Schreier PVMD is a GCD domain.) I’d be doing a grave injustice if I don’t mention the fact that before there was any modern day multiplicative ideal theory there were prime potent domains as Z the

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ring of integers and the rings of polynomials over them. It is also worth mentioning that there are three dimensional prime potent Prufer domains of finite character that are not Bezout. The examples that I have in mind are due to Loper [30]. These are non-Bezout Prufer domains whose maximal ideals are generated by principal primes. While those examples are so important that it’s hard not to mention them, they are so intricate that one can’t do justice to them in a few lines. Acknowledgements I thank the referee for suggesting some important corrections. Any remaining errors or mistakes are all mine.

References 1. D.D. Anderson, D.F. Anderson and M. Zafrullah, A generalization of unique factorization, Boll. Un. Mat. Ital. A(7)9(1995), 401-413 2. D.D. Anderson, G.W. Chang, M. Zafrullah, Integral domains of finite t-character. J. Algebra 396, 169–183 (2013) 3. D.D. Anderson and M. Zafrullah, P.M. Cohn’s completely primal elements, Zero-dimensional Commutative Rings, 115—123, Lecture Notes in Pure and Appl. Math., 171, Dekker, New York, 1995 4. D.D. Anderson, M. Zafrullah, The Schreier property and Gauss’ lemma. Bollettino U. MI 8, 43–62 (2007) 5. D.D. Anderson and M. Zafrullah, On -semi homogeneous integral domains, in Advances in Commutative Algebra, Editors: A. Badawi and J. Coykendall, pp. 7-31 6. D.F. Anderson, A. Rykaert, The class group of D + M. J. Pure Appl. Algebra 52, 199–212 (1988) 7. S. Bazzoni, Groups in the class semigroups of Prüfer domains of finite character, Comm. Algebra 28 (11) (2000) 5157 5167 8. S. Bazzoni, Clifford regular domains, J. Algebra 238 (2001) 703 722 9. G. Brookfield, D. Rush, Convex polytopes and factorization properties in generalized power series domains" in Rocky Mountain. J. Math. 38(6), 1909–1919 (2008) 10. G.W. Chang, H. Hamdi, Bazzoni’s conjecture and almost Prüfer domains. Comm. Algebra (2019). https://doi.org/10.1080/00927872.2018.1543426 11. P.M. Cohn, Bezout rings and their subrings. Proc. Cambridge Philos. Soc. 64, 251–264 (1968) 12. P. Conrad, Some structure theorems for lattice ordered groups. Trans. Amer. Math. Soc. 99, 212–240 (1961) 13. D.L. Costa, J.L. Mott, M. Zafrullah, The construction D + X D S [X ]. J. Algebra 53, 423–439 (1978) 14. D.L. Costa, J.L. Mott, M. Zafrullah, Overrings and dimensions of general D + M constructions. J. of Nat. Sci. and Math. 26, 7–14 (1986) 15. T. Dumitrescu, Y. Lequain, J. Mott, M. Zafrullah, Almost GCD domains of finite t-character. J. Algebra 245, 161–181 (2001) 16. T. Dumitrescu, M. Zafrullah, Characterizing domains of finite -character. J. Pure Appl. Algebra 214, 2087–2091 (2010) 17. T. Dumitrescu, M. Zafrullah, t -Schreier domains. Comm. Algebra 39, 808–818 (2011) 18. J. Elliott, Rings, modules and closure operations, Springer Monographs in Mathematics. Springer, Cham 19. M. Fontana, E. Houston and M.H. Park, Idempotence and divisoriality in Prufer-like domains, https://arxiv.org/pdf/1811.09210.pdf 20. R. Gilmer, Multiplicative Ideal Theory, Marcel-Dekker 1972 21. M. Griffin, Some results on v-multiplication rings. Canad. J. Math. 19, 710–722 (1967)

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22. O. Helmer, Divisibility properties of integral functions. Duke J. Math. 6, 345–356 (1940) 23. W.C. Holland, J. Martinez, W.Wm. McGovern, M. Tesemma, Bazzoni’s conjecture, J. Algebra 320 (4) (2008) 1764 1768 24. F. Halter-Koch, Ideal Systems - An Introduction to Multiplicative Ideal Theory (Marcel Dekker, New York, 1998) 25. F. Halter-Koch, Clifford semigroups of ideals in monoids and domains. Forum Math. 21, 1001– 1020 (2009) 26. E. Houston and M. Zafrullah, -Super potent domains, J. Commut. Algebra (to appear) 27. P. Jaffard, Les Syst‘emes d’ Ide ’aux (Dunod, Paris, 1962), p. 17 28. B.G. Kang, Prüfer v-Multiplication domains and the ring R[X ] Nv . J. Algebra 123, 151–170 (1989) 29. I. Kaplansky, Commutative Rings (Allyn and Bacon, Boston, 1970) 30. K. Alan Loper, Two Prufer domain counterexamples, J. Algebra 221 (1999), 630–643 31. S. McAdam and D.E. Rush, Schreier rings, Bull. London Math.SOC. 10 (1978), 77-80 32. J. Mott, Budh Nashier and M. Zafrullah, Contents of polynomials and invertibility 18(5), 1569–1583 (1990) 33. J. Mott, M. Rashid, M. Zafrullah, Factoriality in Riesz groups. J. Group Theory 11(1), 23–41 (2008) 34. W. Smith, A covering condition for prime ideals. Proc. Amer. Math. Soc. 30, 451–452 (1971) 35. S.Q. Xing, D.D. Anderson and Muhammad Zafrullah, Two generalizations of Krull domains, (pre-print) 36. Y. Yang, M. Zafrullah, Bases of pre-Riesz groups and Conrad’s F-condition. Arab. J. Sci. Eng. 36, 1047–1061 (2011) 37. M. Zafrullah, Semirigid GCD domains. Manuscripta Math. 17, 55–66 (1975) 38. M. Zafrullah, On a property of pre-Schreier domains. Comm. Algebra 15, 1895–1920 (1987) 39. M. Zafrullah, Well behaved prime t-ideals. J. Pure Appl. Algebra 65, 199–207 (1990) 40. M. Zafrullah, Putting t-invertibility to use, Non-Noetherian Commutative Ring Theory, in: Math. Appl., vol. 520, Kluwer Acad. Publ., Dordrecht, 2000, pp. 429-457 41. M. Zafrullah, t-invertibility and Bazzoni-like statements. J. Pure Appl. Algebra 214, 654–657 (2010) 42. M. Zafrullah, On -homogeneous ideals, https://arxiv.org/pdf/1907.04384.pdf 43. M. Zafrullah, Help Desk, https://lohar.com/mithelpdesk/hd1902.pdf

On the Set of Molecules of Numerical and Puiseux Monoids Marly Gotti and Marcos M. Tirador

Abstract Additive submonoids of Q≥0 , also known as Puiseux monoids, are not unique factorization monoids (UFMs) in general. Indeed, the only unique factorization Puiseux monoids are those generated by one element. However, even if a Puiseux monoid is not a UFM, it may contain nonzero elements having exactly one factorization. We call such elements molecules. Molecules were first investigated by W. Narkiewicz in the context of algebraic number theory. More recently, F. Gotti and the first author studied molecules in the context of Puiseux monoids. Here we address some aspects related to the size of the sets of molecules of various subclasses of Puiseux monoids with different atomic behaviors. In particular, we positively answer the following recent realization conjecture: for each m ∈ N≥2 there exists a numerical monoid whose set of molecules that are not atoms has cardinality m. Keywords Puiseux monoid · Numerical monoid · Atomic monoid · Atomicity · Factorization · Molecule · Atom · BFM · FFM · UFM 2010 Mathematics Subject Classification Primary: 20M13 · Secondary: 06F05, 20M14

1 Introduction Let M be a cancellative and commutative monoid. A factorization of a non-invertible element x ∈ M is a formal product a1 · · · a of atoms (i.e., irreducible elements), up to permutations and associates, such that x = a1 · · · a in M; in this case,  is called the length of the factorization. Following P. M. Cohn, we call M atomic if every M. Gotti (B) Department of Research and Development, Biogen, Cambridge, MA 02142, USA e-mail: [email protected] M. M. Tirador Facultad de Matemática y Computación, Universidad de La Habana, San Lázaro y L, Vedado Habana 4, Havana 10400, Cuba © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. Badawi and J. Coykendall (eds.), Rings, Monoids and Module Theory, Springer Proceedings in Mathematics & Statistics 382, https://doi.org/10.1007/978-981-16-8422-7_5

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non-invertible element of M has a factorization. In addition, M is called a unique factorization monoid (or a UFM) if every non-invertible element of M has a unique factorization. Although each UFM is clearly atomic, an element of an atomic monoid may have more than one factorization (even infinitely many). For instance, this is the case √ of the element 6 in the multiplicative monoid of the ring of algebraic integers Z[ −5]; notice that 6 = 2 · 3 = (1 −

√ √ −5)(1 + −5).

√ A didactic exposition of the factorization-theoretical aspects of Z[ −5] can be found in [9]. Following [21], we say that a non-invertible element x ∈ M is a molecule if x has exactly one factorization in M, and we let M (M) denote the set consisting of all molecules of M. Perhaps, the first systematic study of molecules was carried out by W. Narkiewicz back in the 1960s in multiplicative monoids of rings of algebraic integers of quadratic number fields [24, 25] and later in rings of integers of general number fields [23]. This is hardly a surprise given that factorization theory has its origin in algebraic number theory, one of the pioneering works being [7]. More recently, the molecules of additive monoids such as numerical monoids and some generalizations of them have been studied in [21, Sects. 3–4]. Let A (M) denote the set of atoms of a cancellative and commutative monoid M. Clearly, A (M) is contained in M (M). In this paper, we study the sizes of the sets of molecules that are not atoms in additive submonoids of Q≥0 . One of the initial motivations of this project was the following realizability question posed by F. Gotti and the first author in [21]. Conjecture 1.1 For every n ∈ N≥2 there exists a numerical monoid N such that |M (N ) \ A (N )| = n. With the statement of Conjecture 1.1 in mind, we say that a class C of cancellative and commutative monoids is molecular if for every n ∈ N≥2 there exists a monoid M in C such that |M (M) \ A (M)| = n. Clearly, a molecular class must contain infinitely many non-isomorphic monoids. In the first part of this paper, we provide a positive answer to Conjecture 1.1, i.e., we prove that the class consisting of all numerical monoids is molecular. Following D. D. Anderson, D. F. Anderson, and M. Zafrullah [2], we say that an atomic monoid M is a finite factorization monoid (or an FFM) if every element of M has only finitely many factorizations, and we say that M is a bounded factorization monoid (or a BFM) if for every element in M there is a bound for the set of lengths of its factorizations. It is clear that U F M =⇒ F F M =⇒ B F M =⇒ AT M,

(1.1)

where ATM stands for atomic monoid. The chain of implications (1.1) is a fragment of a larger diagram of atomic classes that first appeared in [2], where it was illustrated that none of the implications in (1.1) is reversible in the class consisting of integral

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domains. The original larger diagram of atomic classes of integral domains was further investigated in the sequel [1, 3, 4]. It was recently proved in [16, Theorem 4.3] that none of the implications in (1.1) is reversible in the class consisting of semigroup rings F[X ; S], where F is a field and M is an additive monoid consisting of rationals. A Puiseux monoid is an additive submonoid of Q≥0 . It is well known that none of the implications in (1.1) is reversible in the class of Puiseux monoids (see examples in Sect. 4.1). Unlike numerical monoids, Puiseux monoids are not, in general, atomic. The second part of this paper is devoted to construct, for each implication in (1.1), a molecular subclass of Puiseux monoids whose members witness the failure of the corresponding reverse implication. For instance, in Theorem 5.1 we construct a subclass of Puiseux monoids that is molecular, whose monoids are BFMs but not FFMs. We also construct a molecular class consisting of non-atomic Puiseux monoids.

2 Preliminary We let N and N0 := N ∪ {0} denote the set of positive and nonnegative integers, respectively, and we let P denote the set of primes. In addition, for X ⊆ R and r ∈ R, we set X ≥r := {x ∈ X | x ≥ r }; in a similar way, we use the notations X >r , X ≤r , and X 0 , the unique n, d ∈ N such that q = n/d and gcd(n, d) = 1 are denoted by n(q) and d(q), respectively. We call n(q) and d(q) the numerator and denominator of q, respectively. Throughout this paper, the term monoid refers to a cancellative and commutative semigroup with identity. Since all monoids here are assumed to be commutative, we shall write them additively unless otherwise is specified. In addition, we shall tacitly assume that all monoids in this paper are reduced, in the sense that the only invertible element they contain is the identity element. Let M be a monoid. For a subset S of M, we let S denote the smallest (under inclusion) submonoid of M containing S. We say that M is generated by S if M = S. In addition, M is called finitely generated if it can be generated by one of its finite subsets. An element a ∈ M \ {0} is called an atom if whenever a = u + v for some u, v ∈ M either u = 0 or v = 0. The set of atoms of M is denoted by A (M), and M is called atomic if M = A (M). The free commutative monoid on A (M) is denoted by Z(M), and the elements of Z(M) are called factorizations of M. If z := a1 + · · · + a ∈ Z(M) for some a1 , . . . , a ∈ A (M), then  is called the length of the factorization z and is denoted by |z|. Since Z(M) is free, there exists a unique monoid homomorphism π : Z(M) → M satisfying that π(a) = a for all a ∈ A (M). For x ∈ M, the set Z(x) := π −1 (x) is called the set of factorizations of x. Since M need not be atomic, Z(x) may be empty for some x ∈ M.

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Definition 2.1 Let M be a monoid. An element x ∈ M \ {0} is a molecule if |Z(x)| = 1. We let M (M) denote the set consisting of all molecules of M. For each x ∈ M, the set L(x) := {|z| | z ∈ Z(x)} is called the set of lengths of x. Clearly, the set of lengths of a molecule is a singleton. Suppose now that M is atomic. We say that M is a UFM (or a unique factorization monoid) if every noninvertible element of M is a molecule. In addition, M is called an FFM (or a finite factorization monoid) if Z(x) is finite for all x ∈ M while M is called a BFM (or a bounded factorization monoid) if L(x) is finite for all x ∈ M. It is clear that every UFM is an FFM, every FFM is a BFM, and every BFM is atomic. A submonoid N of (N0 , +) is said to be a numerical monoid 1 if N0 \ N is a finite set. If N0 \ N is not empty, then N is said to be a proper numerical monoid; in this case, the maximum of N0 \ N is known as the Frobenius number of N and is denoted by F(N ). It is not hard to verify that a numerical monoid is always finitely generated and has a unique minimal set of generators, which is precisely its set of atoms. The embedding dimension of N is the cardinality of its generating set. As numerical monoids are finitely generated, they are FFMs [15, Proposition 2.7.8], and so BFMs. Numerical monoids have been actively investigated (see [13] and references therein) and have many connections to several areas of mathematics (see [5] for some applications). On the other hand, a submonoid M of (Q≥0 , +) is called a Puiseux monoid. Unlike numerical monoids, Puiseux monoids may not be finitely generated or atomic: for instance, M = 1/2n | n ∈ N0  is clearly non-finitely generated and A (M) is empty. Further contrasts with numerical monoids are given by the existence of atomic Puiseux monoids that are not BFMs as it is the case of 1/ p | p ∈ P (see Example 4.4) and the existence of Puiseux monoids that are BFMs but not FFMs as it is the case of N0 ∪ Q≥n , where n is a positive integer (see Example 4.3). Puiseux monoids have only been systematically studied recently in connection to factorization theory (see [8, 10] and references therein). In addition, Puiseux monoids have appeared in the literature in connection to commutative ring theory (see [12, 22]) and, more recently, in the non-commutative context of monoids of matrices [6].

3 Molecules of Interval Numerical Monoids In this first section we describe the set of molecules of numerical monoids generated by discrete intervals. To begin with, let us provide a formal definition. Definition 3.1 We call a numerical monoid N an interval numerical monoid provided that A (N ) consists of consecutive integers. For a ∈ N and n ∈ 0, a − 1, we let Na,n denote the interval numerical monoid generated by the set {a + j | j ∈ 0, n}. 1

Numerical monoids have been widely investigated under the term numerical semigroups.

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Interval numerical monoids were first investigated by García-Sánchez and Rosales in [14] where, among other results, they found a formula for the Frobenius number. They proved that for a ∈ N and n ∈ 0, a −1, the Frobenius number of the interval a − 1. numerical monoid Na,n is F(Na,n ) = a−1 n It follows immediately that A (Na,n ) = {a + j | j ∈ 0, n}. However, the set of molecules of Na,n is not that easy to determine, and providing a full description of M (Na,n ) is our primary purpose in this section. Example 3.2 For every a ∈ N≥2 , the embedding-dimension-two numerical monoid Na,1 = a, a + 1 is an interval numerical monoid. Clearly, A (Na,1 ) = {a, a + 1}, and it is not hard to verify that M (Na,1 ) = {(m + n)a + n | m ∈ 0, a, n ∈ 0, a − 1, and (m, n) = (0, 0)}; for more information, see [21, Proof of Theorem 3.7]. In the next theorem we determine the set of molecules of Na,n for a ∈ N≥3 and n ∈ 1, a − 1. First, we argue the following lemma. Lemma 3.3 Let a, n ∈ N such that a ≥ 3 and n ∈ 1, a − 1, and let n 1 and n 2 be the smallest positive integers such that n 1 a ∈ a + j | j ∈ 1, n and n 2 (a + n) ∈

a + j | j ∈ 0, n − 1. Then n1 = n2 + 1 =

a  n

+ 1.

Proof Set M = a + j | j ∈ 0, n − 1. If x := n 2 (a + n) ∈ M, then x has a factorization in M with length greater than n 2 . Now the inequality (n 2 + 1)a ≤ n 2 (a + n) follows from the fact that (n 2 + 1)a is the smallest element of M that has a factorization of length greater than n 2 . Therefore a ≤ n 2 n, which yields n 2 ≥ a/n. It remains to prove that a/n(a + n) ∈ M. From a, a + n − 1 ⊆ M, one can easily see that (a/n + 1)a, (a/n + 1)(a + n − 1) ⊆ M. Now since a/n ≤ a/n and a/n + 1 ≤ a + n, we have that (a/n + 1)a ≤ a/n(a + n) ≤ (a/n + 1)(a + n − 1). Thus, a/n(a + n) ∈ M. We can prove that n 1 = a/n + 1 using similar arguments.  We are now in a position to establish the main result of this section. Theorem 3.4 Let a, n ∈ N such that a ≥ 3 and n ∈ 1, a − 1, and let n 1 and n 2 be the smallest positive integers such that n 1 a ∈ a + j | j ∈ 1, n and n 2 (a + n) ∈ a + j | j ∈ 0, n − 1. Then M (Na,n ) \ A (Na,n ) = M1 ∪ M2 ∪ M2 ∪ M3 ∪ M4 ∪ M4 , where

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M1 M2 M2 M3 M4 M4

= { ja | j ∈ 2, n 1 − 1}, = { ja + (a + 1) | j ∈ 1, n 1 − 2}, = {(n 1 − 1)a + (a + 1)} if a/n ∈ Z, and M2 = ∅ if a/n ∈ / Z, = { j (a + n) | j ∈ 2, n 2 − 1}, = { j (a + n) + (a + n − 1) | j ∈ 1, n 2 − 2}, and = {(n 2 − 1)(a + n) + (a + n − 1)} if a/n ∈ Z, and M4 = ∅ if a/n ∈ / Z.

Proof The element 2a is clearly a molecule. In addition, if ja is a molecule for some j < n 1 − 1, then ( j + 1)a ∈ / a + j | j ∈ 1, n, and so any factorization of ( j + 1)a yields a factorization of ja (after canceling one copy of a), whence ( j + 1)a is also a molecule. We have proved inductively that M1 consists of molecules of Na,n . To verify that each element of M2 is also a molecule, fix k ∈ 1, n 1 − 2. Let n ci (a + i) be a factorization of ka + (a + 1), where c0 , . . . , cn ∈ N0 . Then z := i=0 c j > 0 for some j ∈ 1, n, and so z  := (c j−1 + 1)(a + ( j − 1)) + (c j − 1)(a + j) +



ci (a + i)

i∈0,n\{ j−1, j}

is a factorization of (k + 1)a. Since (k + 1)a is a molecule (as proved in the previous paragraph), z  = (k + 1)a in Z(Na,n ), and so the equalities a + ( j − 1) = a and c j−1 + 1 = k + 1 hold. Thus, j = 1 and c0 = c j−1 = k. As c1 ≥ 1, the equality c0 = k forces the equalities c1 = 1 and ci = 0 for every i ≥ 2. Then z = ka + (a + 1) and, therefore, ka + (a + 1) must be a molecule. So M2 also consists of molecules. Let us check that the singleton M2 contains a molecule when n divides a. Write a = kn for some k ∈ N≥2 . It follows from Lemma 3.3 that k = a/n = n 1 − 1. Notice, on the other hand, that if the element (n 1 − 1)a + (a + 1) has a factorization z different from the obvious one, such a factorization must have length at most k. As a result, k(a + n) = (k + 1)a < π(z) ≤ k(a + n), which is not possible. Thus, (n 1 − 1)a + (a + 1) must be a molecule. Verifying that M3 (resp., M4 and M4 ) consists of molecules can be done following the same lines we just used to argue that M1 (resp., M2 and M2 ) consists of molecules. Therefore M1 ∪ M2 ∪ M2 ∪ M3 ∪ M4 ∪ M4 is a subset of M (Na,n ) \ A (Na,n ). To prove that the reverse inclusion holds, take m ∈ M (Na,n ) \ A (Na,n ) and let n ci (a  + i) be the only factorization of m, where c0 , . . . , cn ∈ N0 . Since m z := i=0 n ci ≥ 2. is not an atom, i=0 / {0, 1, n − 1, n}. Suppose for the sake of a contradiction that c j ≥ 1 for some j ∈ If c j ≥ 2, then one could replace 2(a + j) in z by (a + ( j − 1)) + (a + ( j + 1)) to obtain a factorization of m different from z. Hence c j = 1, and so there exists ck > 0 for some k = j. We first assume that k < j. If k = 0, then as a + 1 = a + j one could replace a + (a + j) by (a + 1) + (a + ( j − 1)) to obtain a factorization of m different from z. On the other hand, if k > 0, then after replacing (a + k) + (a + j) by (a + (k − 1)) + (a + ( j + 1)) we would obtain again a factorization of m different from z. As a result, k < j generates contradictions. The case of k > j

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can be handled mutatis mutandis to generate contradictions. Thus, c j = 0 when j∈ / {0, 1, n − 1, n}. Let us split the rest of the proof into the following two cases. CASE 1: n ≥ 3. Write z = c0 a + c1 (a + 1) + cn−1 (a + (n − 1)) + cn (a + n). Reasoning as in the previous paragraph, one finds that either c0 = c1 = 0 or cn−1 = cn = 0. Suppose first that cn−1 = cn = 0, and so z = c0 a + c1 (a + 1). In this case, c0 ≤ n 1 − 1 and c1 ∈ {0, 1}, as otherwise one could replace 2(a + 1) by a + (a + 2) to obtain a factorization of m different from z. If c1 = 0, it is / a + j | j ∈ 1, n, and so m ∈ M1 . Then suppose that c1 = 1. clear that c0 a ∈ If c0 < n 1 − 1, then m ∈ M2 . Accordingly, suppose that c0 = n 1 − 1 and so that m = n 1 a + 1. Because n 1 a ∈ a + j | j ∈ 1, n we can replace n 1 a by a sum of atoms in {a + 1, . . . , a + n} in m = n 1 a + 1. Provided that a + n does not divide n 1 a, this will yield a factorization of m different from z. Therefore n 1 a = n  (a + n) for some n  ∈ N. By Lemma 3.3 and the minimality of n 2 , one sees that n  ≥ n 2 = n 1 − 1. This, along with n 1 a = n  (a + n), guarantees that n  = n 1 − 1. Then n 1 a = (n 1 − 1)(a + n) can be rewritten as a = (n 1 − 1)n. Hence n divides a, and so m ∈ M2 . The case of c0 = c1 = 0 follows analogously. CASE 2: n = 2. Write now z = c0 a + c1 (a + 1) + c2 (a + 2). As m is a molecule, c0 c2 = 0 and c1 ≤ 1. Assume first that c2 = 0. Then z = c0 a + c1 (a + 1), and one can check that m ∈ M1 ∪ M2 ∪ M2 as we did in CASE 1. If c2 = 0, then c0 = 0 and so z = c1 (a + 1) + c2 (a + 2). This case can be handled similarly to the case of  z = c0 a + c1 (a + 1) to conclude that m ∈ M3 ∪ M4 ∪ M4 . Corollary 3.5 Take a, n ∈ N such that a ≥ 3 and n ∈ 1, a − 1. Then  |M (Na,n )| =

4a/n + n − 5 if n  a . 4a/n + n − 3 if n | a

4 Molecularity of the Class of Numerical Monoids In this section, we formally introduce the fundamental classes of Puiseux monoids we shall be concerned with, and then we prove Conjecture 1.1.

4.1 Atomic Classes of Puiseux Monoids There are three classes of atomic Puiseux monoids that we will present in this subsection with the intention of later investigating the sets of molecules of their members. Let C1 denote the class of all Puiseux monoids that are FFMs but not UFMs. Example 4.1 Let M be a finitely generated Puiseux monoid. It follows from [15, Proposition 2.7.8] that M is an FFM, and it follows from [18, Proposition 4.3.1] that

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M is a UFM if and only if M ∼ = (N0 , +). In particular, each nontrivial numerical monoid belongs to C1 . The class C1 also contains non-finitely generated monoids. Example 4.2 Consider the Puiseux monoid M = (3/2)n | n ∈ N0 . Since M can be generated by an increasing sequence, namely the increasing powers of 3/2, it follows from [17, Theorem 5.6] that M is an FFM. In addition, [18, Proposition 4.3.1] guarantees that M is not a UFM. Since A (M) = {(3/2)n | n ∈ N0 } by [10, Proposition 4.3], the monoid M is a non-finitely generated monoid that belongs to C1 . Let C2 denote the class of all Puiseux monoids that are BFMs but not FFMs. It is clear that the classes C1 and C2 are disjoint. Example 4.3 For every n ∈ N it is clear that M = N0 ∪ Q≥n is a Puiseux monoid. 4.5] that M is a BFM. As 0 is not a limit point of M • , it follows from [17, Proposition In addition, one can easily check that A (M) = {1} ∪ Q ∩ (n, n + 1) . Now notice that for every m ∈ N≥2 the equalities 2n + 1 = (n + 1/m) + (n + 1 − 1/m) determine infinitely many distinct factorizations in Z M (2n + 1). Hence M is not an FFM. Therefore the class C2 contains infinitely many Puiseux monoids. Finally, we denote by C3 the class of all atomic Puiseux monoids that are not BFMs. Clearly, the class C3 is disjoint from C1 ∪ C2 . Example 4.4 Let P be a set containing infinitely many prime numbers, and consider the Puiseux monoid M = 1/ p | p ∈ P. It is not hard to verify that M is atomic with A (M) = {1/ p | p ∈ P} (see [10, Theorem 4.5]). On the other hand, M is not a BFM because the fact that p 1p ∈ Z(1) for every p ∈ P implies that P ⊆ L(1). So the class C3 contains infinitely many members. The fundamental questions we are interested in are related to the size of the set M (M) \ A (M), where M is a Puiseux monoid. In particular, we would like to know what are the possible sizes of the set M (M) \ A (M) in the distinct classes of Puiseux monoids determined by the chain of implications 1.1.

4.2 A Conjecture on Molecularity We say that a class C of monoids is molecular if for every n ∈ N≥2 , there exists a monoid M in C such that |M (M) \ A (M)| = n. Clearly, Conjecture 1.1 can be rephrased by saying that the class consisting of all numerical monoids is molecular. We now offer a proof of this conjecture.

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Theorem 4.5 The class of numerical monoids is molecular. Proof Let N denote the class consisting of all proper numerical monoids, and for N in N set m(N ) := |M (N ) \ A (N )|. Let us prove that S := {m(N ) | N ∈ N } = N≥2 . It is easy to see that S ⊆ N≥2 . For the reverse inclusion, fix s ∈ N≥2 . Observe that if we set n = 2 in Corollary 3.5, then we obtain that |M (Na,2 ) \ A (Na,2 )| = 2a − 4. Therefore if s is even with s ≥ 6 we have that |M (Ns/2,2 ) \ A (Ns/2,2 )| = s − 4. As a result, S contains all even numbers in N≥2 . To prove that S contains all odd numbers in N≥2 , we will describe the sets of molecules of the numerical monoids Na := a, a + 1, a + 3, a + 4 for every a ∈ N≥6 . Fix a ∈ N≥6 . It is clear that A (Na ) = {a, a + 1, a + 3, a + 4}. For each n ∈ N, we set In := {x ∈ Na | n ∈ L(x)}. Notice that I1 = A (Na ), and I j =  ja, j (a + 4) for every j ∈ N≥2 . Out of the nine elements in I2 , only 2a + 4 has different factorizations of length 2. We shall prove inductively that for every n ∈ N≥3 each element in the discrete interval na + 3, n(a + 4) − 3 ⊆ In has at least two factorizations of length n (in particular, they will fail to be molecules). The elements 3a + 4, 3a + 5, 3a + 7, 3a + 8 ∈ I3 can be written as the addition of 2a + 4 and the atoms a, a + 1, a + 3, a + 4, respectively. Therefore each of them has at least two different factorizations of length 3. In addition, one can readily verify that each of the elements 3a + 3, 3a + 6, 3a + 9 ∈ I3 has at least two different factorizations of length 3. Suppose that for n ≥ 4, each of the elements (n − 1)a + 3, (n − 1)a + 4, . . . , (n − 1)(a + 4) − 3 ∈ In−1 has at least two different factorizations of length n − 1. Adding a to each of these elements, we find that each of the elements na + 3, na + 4, . . . , n(a + 4) − 7 ∈ In has at least two different factorizations of length n. Furthermore, n(a + 4) − 6, n(a + 4) − 5 ∈ In can be written as the addition of the element (n − 1)(a + 4) − 5 and the atoms a + 3 and a + 4, respectively. Similarly, n(a + 4) − 4, n(a + 4) − 3 can be written as the addition of the element (n − 1)(a + 4) − 3 and the atoms a + 3 and a + 4, respectively. Hence each of the elements n(a + 4) − 6, n(a + 4) − 5, n(a + 4) − 4, n(a + 4) − 3 ∈ In also has at least two different factorizations of length n. This concludes our inductive argument. Lastly, it is clear that for every n ∈ N≥3 the smallest three and the largest three elements of the discrete interval In each has exactly one factorization of length n. Now we just need to determine the elements of Na that belong to more than one I j . Notice that if x ∈ I j ∩ Ik for some j < k, then x ∈ I j ∩ I j+1 . When this is the case, one finds that j (a + 4) = max I j ≥ min I j+1 = ( j + 1)a, and so 4 j ≥ a. Set m := min{ j ∈ N | I j ∩ I j+1 = ∅}. Since max I j − min I j = 4 j and min I j+1 − min I j = a for every j ∈ N, the three largest elements of I j will be contained in I j+1 for every j ≥ m + 1. Similarly, the smallest three elements of I j will be contained in I j−1 for every j ≥ m + 2.

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Next we find the size m(Na ) of M (Na ) \ A (Na ) when a ≡ 1 (mod 4). First, we assume that m > 2. In this case, there are 8 molecules in I 2 , there are 6(m − 3) molecules in I3 ∪ · · · ∪ Im−1 , there are 3 + 3 − (4m − a + 1) molecules in Im , and there are 3 − (4m − a + 1) molecules in Im+1 . Hence m(Na ) = 8 + 6(m − 3) + 3 + 2(2 − (4m − a)) = 2a − 2m − 3 = 2a − 2

a  4

− 3.

In the case of m = 2, we have that a ∈ {6, 7, 8} and it can  be readily seen for each of such values of a, that the equality m(Na ) = 2a − 2 a4 − 3 holds. Finally, we are ready to verify that every odd number in N≥3 belongs to S. To do so, take a = 4k − i for 0 ≤ i ≤ 2 (and therefore, k ≥ 2), and observe that 1 + 2N≥2 = {(6k + i) − 8 | k ∈ N≥2 , i ∈ {1, 3, 5}} = {m(N4k−i ) | k ∈ N≥2 , i ∈ {0, 1, 2}} ⊆ {m(Na ) | a ∈ N≥6 } ⊆ S. To verify that 3 also belongs to S, it suffices to observe that M ( 2, 3) = {4, 5, 7}.  We conclude that N≥2 ⊆ S, and so {|M (N ) \ A (N )| : N ∈ N } = N≥2 . As C1 contains every numerical monoid, one obtains the following corollary. Corollary 4.6 The class C1 is molecular.

5 Molecularity of Further Classes of Puiseux Monoids In this section, we turn our attention to classes of non-finitely generated Puiseux monoids, and study them in the same direction we studied the class of numerical monoids in Sect. 4.

5.1 Molecularity of C2 We have seen in Corollary 4.6 that the class C1 is molecular. We proceed to provide a similar result for the class C2 . Theorem 5.1 The class C2 is molecular. Proof Fix n ∈ N≥2 . By Theorem 4.5, there exists a numerical monoid N satisfying |M (N ) \ A (N )| = n. Set a0 := min M (N ) and K = max M (N ). Therefore δ :=  a0 /K satisfies 0 < δ < 1. Now consider the Puiseux monoid M = A, where A = a∈A (N ) [a, a + δ) ∩ Q. As 0 is not a limit point of M • , it follows from [17,

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Proposition 4.5] that M is a BFM and, in particular, an atomic monoid. We proceed to verify that A (M) = A. To do this, take x ∈ M ∩ N with x ≤ K , and write x = a1 + · · · + a for  ∈ N and a1 , . . . , a ∈ A. Then suppose, by way of contradiction, that m := max{ai − ai  | i ∈ 1, } > 0. In this case, the fact that x ∈ N guarantees the last inequality of   (ai − ai ) ≥ 1. (5.1) δ > m ≥ i=1

Using (5.1) we obtain that x ≥ a0 = δ K > K , which is a contradiction. As a consequence, a1 , . . . , a ∈ N . Now take  a ∈ A and write a = a1 + · · · + a for  ∈ N and a1 , . . . , a ∈ A. Because i=1 (ai − ai ) ≥ a − a, we can write a = b1 + · · · + b , where bi ∈ [ai , ai ] ∩ Q for every i ∈ 1, . The inclusion a ∈ M ∩ N≤K , along with our argument in the previous paragraph, now implies that b1 , . . . , b ∈ N . As a ∈ A (N ), it follows that  = 1, and so a ∈ A (M). Hence A (M) = A. Since M is a BFM, proving that M belongs to C2 amounts to verifying that M is not an FFM. To do so, take r ∈ (0, δ) ∩ Q and set x := 2a0 + r . It is clear that x ∈ M. On the other hand, it can be readily seen that a0 + r/2 ± 1/n ∈ A (M) for every n ∈ N



with n ≥ 2/r . As a result, the equalities x = a0 + r2 − n1 + a0 + r2 + n1 (for every n ∈ N≥2/r ) yield infinitely many factorizations of x. As a consequence, M is not an FFM. Next we show that no element x ∈ M \ (A (M) ∪ N ) belongs to M (M). Since x∈ / A (M) ∪ N , there exists z ∈ Z(x) having a length-2 subfactorization a1 + a2 such that r := a1 − a1  > 0. Take n ∈ N such that 1/n < min{r, a2  − a2 + δ}, a2 + 1/n ∈ A (M). Then, after replacing a1 + a2 in z by and

that a1 − 1/n,

note a1 − n1 + a2 + n1 , one obtains a factorization of x different from z. Hence x is not a molecule. The inclusion A (N ) ⊆ A (M), together with our argument in the previous paragraph, ensures that M (M) \ A (M) ⊆ M (N ). On the other hand, if x ∈ M (N ), then x ∈ M ∩ N≤K and so each factorization of x in M is also a factorization of x in N . Hence x must belong to M (M). As a result, M (M) \ A (M) = M (N ) \ A (N ), from which we conclude that |M (M) \ A (M)| = n.  Unlike the case of numerical monoids, there are monoids M in C2 satisfying |M (M) \ A (M)| = 1. For instance, consider for every n ∈ N the monoid {0} ∪ Q≥n . It is easy to see that the only molecule of {0} ∪ Q≥n that is not an atom is 2n. In the direction of Corollary 4.6 and Theorem 5.1, we have the following conjecture. Conjecture 5.2 The class C3 is molecular.

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5.2 Molecularity of Non-Atomic Puiseux Monoids Each of the monoids we have treated so far is atomic. However, there are plenty of non-atomic Puiseux monoids. Let C4 consist of all non-atomic Puiseux monoids. In the same direction of our previous results, we have the following proposition. Proposition 5.3 The class C4 is molecular. Proof Fix n ∈ N≥2 . By Theorem 4.5 there exists a numerical monoid N such that |M (N ) \ A (N )| = n. Take p ∈ P such that p > max M (N ), and consider the Puiseux monoid   p M = N ∪ n

n ∈ N . 2 Clearly, none of the elements of M of the form p/2n belongs to A (M). Therefore A (M) ⊆ A (N ). Furthermore, since p > max M (N ) ≥ max A (N ), one can readily verify that A (M) = A (N ). Because A (M) is a finite set and 0 is a limit point of M • , the Puiseux monoid M cannot be atomic, that is, M belongs to C4 . Moreover, A (M) = A (N ) implies that Z N (x) = Z M (x) for all x ∈ N . Hence a molecule of N remains a molecule in M, i.e., M (N ) ⊆ M (M). On the other hand, if x ∈ M (M), then Z M (x) is nonempty and, therefore, A (M) = A (N ) ensures that x ∈ N . As Z N (x) = Z M (x), it follows that x ∈ M (N ). Thus, M (M) ⊆ M (N ), and so we obtain that M (M) = M (N ). This, together with the fact that A (M) = A (N ), ensures that |M (M) \ A (M)| = n, which concludes our proof because n was taken  arbitrarily in the set N≥2 .

6 Infinite Molecularity This last section is devoted to exploring the extreme case when the set M (M) \ A (M) has infinite cardinality (as before, M is taken to be a Puiseux monoid). This motivates the question as to whether one can find Puiseux monoids satisfying this property in each of the Ci classes introduced in previous sections. In order to address this question, the following definition is pertinent. Definition 6.1 We say that a Puiseux monoid M is infinitely molecular provided that |M (M) \ A (M)| = ∞. As we shall reveal in the next propositions, each of the Ci classes contains an infinite subclass consisting of non-isomorphic Puiseux monoids that are infinitely molecular. Proposition 6.2 There exists an infinite subclass of C1 consisting of infinitely molecular Puiseux monoids.

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Proof Consider for every r ∈ Q>1 \ N the Puiseux monoid Mr := r n | n ∈ N0 . It follows from [10, Proposition 4.3] that Mr is atomic with A (Mr ) = {r n | n ∈ N0 }. Indeed, since Mr is generated by the increasing sequence (r n )n∈N0 , it follows from [17, Theorem 5.6] that Mr is an FFM. Notice that Mr is not a UFM; for instance, n(r )1 and d(r )r are two distinct factorizations in Z(n(r )). Then Mr belongs to C1 . Now it follows from [8, Lemma 3.2] that, for every n ∈ N, the element 1 + r n is a molecule of Mr and, therefore, 1 + r n ∈ M (Mr ) \ A (Mr ). Hence |M (Mr ) \ A (Mr )| = ∞. Finally, take t ∈ Q>1 \ N such that Mr and Mt are isomorphic monoids. It follows from [19, Proposition 3.2] that Mt = q Mr for some q ∈ Q>0 . Since multiplication by q is an increasing function, it must take 1 = min Mr• to 1 = min Mt• . Then q = 1, and so t = r . As a result, C1 contains infinitely many non-isomorphic infinitely molecular Puiseux monoids.  We proceed to show that the class C2 also contains plenty of infinitely molecular Puiseux monoids. Proposition 6.3 There exists an infinite subclass of C2 consisting of infinitely molecular Puiseux monoids. Proof Let P be an infinite set of primes, and let ( pn )n∈N be a strictly increasing sequence with underlying set P. For each n ∈ N, let Rn denote the localization of . , pn } and let Mn be the ring Z at the multiplicative monoid generated by { p1 , . . the Puiseux monoid {0} ∪ (Rn ∩ Q≥n+1/ pn ). Now set M P = n∈N0 Mn , where M0 is taken to be N0 . We can readily verify that M P is closed under addition, whence it is a Puiseux monoid. As min M P• = 1, it follows that 0 is not a limit point of M P• , and so M P is a BFM by [17, Proposition 4.5]. To argue that M P is not an FFM, first notice that every element in Rn ∩ [n + 1/ pn , n + 1 + 1/ pn ) whose denominator is divisible by pn is an atom of M P . Since       1 1 2 1 1 n+ , + k + n+1+ − k ∈ Z M P 2n + 1 + pn pn pn pn pn for every k ∈ N≥2 , it follows that 2n + 1 + p2n is an element of M P with infinitely many factorizations. Therefore M P is not an FFM and, as a result, M P belongs to the class C2 . To prove that M P is infinitely molecular, consider the set S := { npn +pnpn +1 | n ∈

N}. Since npn +pnpn +1 = 1 + n + p1n , we find that S is a subset of M P consisting of elements that are not atoms. Observe that none of the elements of M P that is strictly less than n + p1n has a denominator divisible by pn . This, along with the fact that whenever a1 + · · · + ak is a factorization of

npn + pn +1 pn

the prime pn must

divide d(ai ) for some i ∈ 1, k, implies that the only factorization of npn +pnpn +1 must

be 1 + n + p1n . As a result, S is an infinite set of molecules that are not atoms, which implies that M P is an infinitely molecular Puiseux monoid. Finally, write P as the disjoint union of countably many disjoint infinite sets, namely, P = n∈N Pn . It

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follows immediately from [19, Proposition 3.2] that the monoid M Pn and M Pm are not isomorphic when m = n. Hence C2 contains an infinite subclass of non-isomorphic infinitely molecular Puiseux monoids.  The class C3 also contains infinitely many non-isomorphic Puiseux monoids that are infinitely molecular. To argue this, we use a subfamily of the monoids introduced in Example 4.4. Proposition 6.4 There exists an infinite subclass of C3 consisting of infinitely molecular Puiseux monoids. Proof Let P be an infinite set of odd primes, and consider the Puiseux monoid M P = 1/ p | p ∈ P introduced in Example 4.4. We have already seen that M P belongs to the class C3 . In addition, it follows from [21, Proposition 4.10] that 2/ p is a molecule of M P for every p ∈ P and, therefore, {2/ p | p ∈ P} is an infinite set of molecules of M P that are not atoms. Hence M P is infinitely molecular. Mimicking our argument in the proof of Proposition 6.3, we can argue that the construction used in this proof yields infinitely many non-isomorphic infinitely molecular Puiseux  monoids in C3 . For the sake of completeness let us show that the class C4 also contains infinitely many Puiseux monoids that are infinitely molecular. Proposition 6.5 There exists an infinite subclass of C4 consisting of infinitely molecular Puiseux monoids. Proof Fix  a prime psuch that p ≥ 5, and then consider the Puiseux monoid M p :=  M2/ p ∪ 23n n ∈ N , where M2/ p is the Puiseux monoid (2/ p)n | n ∈ N0 . Clearly, M p is not atomic; indeed, one can readily check that 3/2 cannot be written as a sum of atoms. Hence M p belongs to the class C4 . It is not hard to argue that A (M p ) = A (M2/ p ). In addition, for every n ∈ N, [8, Lemma 3.1] guarantees that the element 1 + (2/ p)n is a molecule of M2/ p that is not an atom. This, together with the fact that A (M p ) = A (M2/ p ), implies that 1 + (2/ p)n is a molecule of M p that is not an atom for every n ∈ N. As a consequence, |M (M p ) \ A (M p )| = ∞. Finally, suppose that M p is isomorphic to Mq for some q ∈ P≥5 . By [19, Proposition 3.2], there exists r ∈ Q>0 such that Mq = r M p . Since multiplication by r is increasing, it must send 1 = max A (M p ) to 1 = max A (Mq ) and it must send 2/ p to 2/q. Thus, r = 1, which implies that q = p. Hence C4 contains infinitely many non-isomorphic infinitely molecular Puiseux monoids.  Acknowledgements The authors would like to thank Felix Gotti for many valuable suggestions during the preparation of this paper. The authors are also grateful to an anonymous referee whose feedback helped improve the final version of this paper.

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Where Some Inert Minimal Ring Extensions of a Commutative Ring Come from, II David E. Dobbs

Abstract For commutative quasi-local rings A ⊂ B, the prequel developed examples illustrating the diverse behavior that can be exhibited (as A ⊂ B varies) by the set of rings D such that A ⊆ D ⊂ B and the extension D ⊂ B is an inert minimal ring extension. This note gives two foundational results that could be used to reorganize the examples from the prequel in a more efficient and transparent way. We also give a result pointing the way to further work on analogous problems in case the base ring A is not quasi-local. Following a survey of some recent work on minimal ring extensions, we also give a list of open questions concerning the minimal ring extensions of a finite local ring that is not a field. Keywords Commutative ring · Ring extension · Minimal ring extension · Integrality · Inert extension · Maximal ideal · Minimal field extension 2010 Mathematics Subject Classification Primary 13B99 · Secondary 13B21

1 Introduction This note is a sequel to [9]. To appreciate our recent focus on inert (minimal ring) extensions, we recommend reading the Introduction of [9]. Although a variety of constructions figured in building the examples in [9], one can sensibly ask if, in retrospect, one can state and prove some foundational results that underlie a significant number of those constructions. The main purpose of this note is to give an affirmative answer to that question by producing two such foundational results: see Theorem 2.1 and Proposition 2.2. Section 2 closes with an example showing how some of the foundations given in this note can be generalized to begin a study of the To Dan Anderson, in celebration of his distinguished career. D. E. Dobbs (B) Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1320, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. Badawi and J. Coykendall (eds.), Rings, Monoids and Module Theory, Springer Proceedings in Mathematics & Statistics 382, https://doi.org/10.1007/978-981-16-8422-7_6

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analogous problems where the base ring A is not assumed to be quasi-local. Then, after surveying some pertinent recent research on minimal ring extensions, Sect. 3 collects six open questions concerning the competitions for prevalence among the various kinds of minimal ring extensions of a finite local ring which is not a field. All rings and algebras considered below are commutative and unital; all inclusions of rings, ring extensions and algebra/ring homomorphisms are unital. If A is a ring, then Spec(A) (resp., Max(A)) denotes the set of all prime (resp., maximal) ideals of A and dim(A) denotes the Krull dimension of A. As usual, ⊂ denotes proper inclusion and X, Y denote algebraically independent indeterminates over the ambient ring(s). For any ring extension A ⊆ B, we let [A, B] denote the set of intermediate rings of the given ring extension and we say that A ⊆ B is a minimal ring extension if |[A, B]| = 2. (For any set S, we let |S| denote the cardinal number of S.) The reader may find it helpful to have a copy of [9] at hand. In addition, we will assume familiarity with the following four indispensable pieces of background material: if A ⊆ B are finite rings, then B is integral over A (cf. [18, Theorem XIII.1]); if A is a finite ring and A ⊂ B is a minimal ring extension, then B is a finite ring [3, Proposition 7]; if A ⊂ B is an integral minimal ring extension, then the conductor (A : B) is the crucial maximal ideal of A ⊂ B, in the sense of [13, Théorème 2.2 (i)]; and for any ring A, the integral minimal ring extensions A ⊂ B can be partitioned via what is known as the “ramified-decomposed-inert” trichotomy, by combining the Ferrand-Olivier classification of the minimal ring extensions of a field [13, Lemme 1.2] with a standard homomorphism theorem (cf. [13, Proposition 4.1], [10, Lemma II.3]). Any unexplained material is standard, as in [1, 15, 17].

2 Results We begin with two foundational results that can be used repeatedly to reorganize the presentation of examples in [9]. Given quasi-local rings (A, M) ⊂ (B, N ), Theorem 2.1 reduces the study of the set of rings D ∈ [A, B] such that D ⊂ B is an inert extension to the study of the set of certain fields F ∈ [R, K ], where R is a certain associated quasi-local domain and K is a certain associated field. (In detail, R := A/(N ∩ A), K := B/N , and the relevant fields F ∈ [R, K ] are such that F ⊂ K is a minimal field extension.) On the other hand, Proposition 2.2 shows, given a quasi-local domain R which is a subset of a field K , a classical way to construct quasi-local rings (A, M) ⊂ (B, N ) such that one has canonical identifications R = A/(N ∩ A) and K = B/N . Both Theorem 2.1 and Proposition 2.2 go beyond the context of [6, Theorem 2.5 (b)], as we do not assume that B is integral over A (in particular, we do not assume that A and B are finite rings) or that M ⊆ N (or, in case M ⊆ N , that the field extension A/M ⊆ B/N is algebraic). Moreover, Theorem 2.1 shows that in such studies (given (A, M) ⊂ (B, N )), one can replace A with A + N , which is a quasi-local ring, in effect thus reducing considerations to only two cases, namely, where M ⊆ N and where M ⊃ N .

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Theorem 2.1 Let (A, M) ⊂ (B, N ) be quasi-local rings. Put P := N ∩ A and A∗ := A∗ (B) := A + N . Use the composition of the A-algebra isomorphism A/P → A∗ /N and the inclusion map A∗ /N → B/N to view the domain A/P as a subring of the field B/N . Let S := {D ∈ [A, B] | D ⊂ B is an iner t (minimal ring) extension} and T := {F ∈ [A/P, B/N ] | F is a minimal subfield of B/N }. Then: (a) Let D ∈ S. Then A∗ ⊆ D, D/N ⊂ B/N B = B/N is a minimal field extension, and B is integral over D. Moreover, D is integral over A∗ if and only if D/N is integral over A/P. Furthermore, D is integral over A ⇔ B is integral over A ⇔ D/N is integral over A/P and each element of N is integral over A. (b) A∗ is a quasi-local ring, say with maximal ideal Q. Moreover, M ⊆ N if and only if Q = N ; and M  N if and only if Q ⊃ N . (c) S = {D ∈ [A∗ , B] | D ⊂ B is an inert (minimal ring) extension}. (d) The assignment D → D/N establishes a bijection S → T that is, in fact, an order-isomorphism of posets (under inclusion). Proof The A-algebra homomorphism A/P → A∗ /N (given by a + P → a + N for all a ∈ A) is evidently surjective, and it is also injective since N ∩ A = P. It follows that we can view A/P ⊆ B/N . (a) As D ⊂ B is inert and (B, N ) is quasi-local, D has (unique) maximal ideal N . Since D ∈ [A, B], we have A∗ = A + N ⊆ D. Also, as N must be the crucial maximal ideal of the inert (minimal ring) extension D ⊂ B, it follows that N = (D : B) and D/N ⊂ B/N B = B/N is a minimal field extension. In particular, B/N is algebraic, and hence integral, over D/N . Of course, D ⊂ B is also integral, since it is inert. (For an alternate proof that D ⊂ B is integral, one could apply [14, Corollary 1.5 (5)] to the pullback description D = D/N × B/N B; or, in view of the concrete nature of this pullback, one could instead give a straightforward calculational argument, as in the proof of [16, Lemma 4.6].) Similarly, as we have identified A∗ /N with A/P, an application of [14, Corollary 1.5 (5)] (or the reasoning in [16, Lemma 4.6]) to the pullback description A∗ = A + N = A/P × D/N D shows that D is integral over A∗ if and only if D/N is integral over A/P. In view of what has already been shown, the “Furthermore” assertion now follows from standard facts about integrality. (b) We will show first that the ring A∗ (= A + N ) is quasi-local. Observe the pullback description A∗ = A/P × B/N B. By applying [14, Theorem 1.4] to this pullback, we obtain a description of Spec(A∗ ) up to homeomorphism (in the Zariski topology). The order-theoretic impact of that information is the following description of Spec(A∗ ) as a poset (under inclusion): Spec(A∗ ) can be obtained by placing/gluing Spec(A/P) atop Spec(B), with 0 ∈ Spec(A/P) identified with (that is, glued to) N ∈ Spec(B). Since A is quasi-local, so is A/P. As (B, N ) is also quasi-local, it now follows from the order-theoretic interpretation of the gluing process that A∗ is quasi-local.

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Let Q denote the maximal ideal of A∗ . We have that N ⊆ Q. Suppose first that M ⊆ N . Then P (= N ∩ A) = M. Hence A∗ /N = A/P = A/M, which is a field. Thus N must be the unique maximal ideal of A∗ ; that is, N = Q (if M ⊆ N ). To complete the proof of (b), it suffices to show that if M  N , then Q ⊃ N ; that is, that Q = N ; that is, that the poset Spec( A/P) is not a singleton set; that is, that the domain A/P is not a field; that is, that P = M. This last condition does hold, for if P = M, then M = N ∩ A ⊆ N , contrary to hypothesis. (c) Combine the definition of S with the first assertion in (a). (d) Recall that we have identified A∗ /N with A/P. Since N is a common ideal of A∗ and B, it follows from a standard homomorphism theorem that the assignment E → E/N gives a bijection [A∗ , B] → [A/P, B/N ] (cf. [10, Lemma II.3]). This is in fact an order-isomorphism when [A∗ , B] and [A/P, B/N ] are each regarded as posets under inclusion. By (c) (or the first assertion in (a)), S ⊆ [A∗ , B]. When one restricts the above order-isomorphism [A∗ , B] → [A/P, B/N ] to S, the second assertion in (a) shows that the image of that restriction is a subset of T . Therefore, it will suffice to prove that if F is a minimal subfield of B/N such that (A/P ∼ =) A∗ /N ⊆ F (that is, if F ∈ T ), then F = D/N for some D ∈ [A∗ , B] such that D ⊂ B is inert (that is, such that D ∈ S). By the above-noted bijection, F = E/N for some E ∈ [A∗ , B]. It remains only to prove that E ∈ S, that is, that E ⊂ B is inert. By reasoning with pullbacks as above (using [14, Corollary 1.5 (5)] or [16, Lemma 4.6]), E ⊂ B inherits integrality from E/N = F ⊂ B/N . As (B, N ) is quasi-local, it follows from this integrality that (E, N ) is also quasi-local (cf. [1, Corollary 5.8 and Theorem 5.10]). It now suffices to prove that E ⊂ B is a minimal ring extension (for its crucial maximal ideal would then have to be N and it would follow that E ⊂ B is inert). That, in turn, follows since the above-noted order-isomorphism [A∗ , B] → [A∗ /N , B/N ] = [A/P, B/N ] restricts to a bijection [E, B] → [E/N = F, B/N ] and F ⊂ B/N is a minimal ring/field extension. The proof is complete.  Proposition 2.2 Let (R1 , M1 ) ⊆ (R2 , M2 ) be quasi-local domains which are subrings of a field K and are such that R2 contains the quotient field L of R1 . Let (B, N ) be any valuation domain which is not a field but is of the form B = K + N (where N = 0). Put Ai := Ri + N for i = 1, 2. Then: (a) The ring L + N is quasi-local, with unique maximal ideal N . The field extension (L + N )/N ⊆ B/N can be canonically identified with L ⊆ K . (b) Let i ∈ {1, 2}. Then B is integral over Ai ⇔ K is integral over Ri ⇔ Ri is a field and K is algebraic over Ri . (c) Suppose, in addition, that R1 is not a field (that is, R1 ⊂ L; equivalently, M1 = 0). Then M1  N , B is not integral over R1 , and A1 /N ∼ = R1 /(N ∩ R1 ) ∼ = R1 canonically. Proof Much of the proof of (a) and (b) will involve citing a couple of well known properties of the classical (D + M)-construction from [15]. The interested reader may replace those citations with appropriate pullback-theoretic citations from [14].

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(a) As (L , 0) is quasi-local, L + N is quasi-local with maximal ideal 0 + N = N , by [15, Exercise 12 (1), (2), p. 202]. Since K ∩ N = 0, the second assertion in (a) is clear. (b) Since K + N is a direct sum, the first equivalence follows from [15, Exercise 11 (2), p. 202]. The second equivalence in (b) is standard: cf. [1, Proposition 5.7]. (c) We have N ∩ R1 ⊆ N ∩ L = 0. If M1 ⊆ N , then M1 ⊆ N ∩ R1 = 0, a contradiction to the hypothesis that M1 = 0. Thus M1  N . If B is integral over R1 , then the unique maximal ideal of B lies over the unique maximal ideal of R1 (cf. [1, Corollary 5.8]), that is, N ∩ R1 = M1 , whence M1 ⊆ N , a contradiction. Thus, B is not integral over R1 . The final assertions follow from a standard isomorphism  theorem and the fact that N ∩ R1 = 0. The proof is complete. Remark 2.3 The above two foundational results could be used to give conceptually based proofs of some of the examples in [9]. In our opinion, such reworking of examples from [9] makes for a more unified presentation and more transparent proofs, while possibly also pointing the way to additional conjectures. We would encourage the reader to begin carrying out the details of such reworkings by using Theorem 2.1 and Proposition 2.2 to revisit [9, Example 2.1]. This completes the remark. Before further broadening the context, we pause to give two reasons to explain why our focus here has been primarily on ring extensions that have quasi-local base rings. By applying a classical result on Artinian rings (cf. [20, Theorem 3, p. 205]), one knows that any (nonzero) n finite (not necessarily local) base ring A is canonically Ai of (nonzero finite) local rings Ai ; fortunately, [6, a finite direct product i=1 Lemma 2.2 (b)] easily reduces questions about the nature (ramified, decomposed or inert) of a minimal ring extension of A or the (cardinal) number of the set of A-algebra isomorphism classes represented by a specific kind of minimal ring extension of A to the corresponding questions where A is a (nonzero, finite and) local ring (namely, one of the Ai ). Furthermore, attention to the case of a quasi-local ring is also warranted because of the result [11, Proposition 4.6] that if A ⊂ B is a (not necessarily integral) minimal ring extension with crucial maximal ideal M, then A M ⊂ B M (:= B A\M ) is a minimal ring extension and it is the same kind of minimal ring extension (that is, integrally closed, ramified, decomposed or inert) as A ⊂ B. Although [9] and the above results in this section have gone far beyond the context of [6, Theorem 2.5 (b)], matters can become even more complex than has been indicated so far. In particular, the final result of this section will give a generalization of [9, Example 2.8] in which the base ring A is not quasi-local. Example 2.4 Let 1 ≤ n ≤ ∞. Let A be a domain, with quotient field K , such that A is not quasi-local, dim(A) = n, and there exists a minimal field extension K ⊂ L. (For instance, take A to be the polynomial ring in n algebraically independent commuting indeterminates over a field, let X be one of those indeterminates, and √ take L := K ( X ).) Choose (B, N ) to be any valuation domain which is not a field but is of the form B = L + N . Put D := K + N . Then each M ∈ Max(A) satisfies M  N (and so there does not exist an A-algebra homomorphism A/M → B/N ), D ∈ [A, B], and D ⊂ B is an inert (minimal ring) extension.

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Proof Fix M ∈ Max(A). Then M = 0 (since dim(A) = n > 0). But, reasoning as in the proof of Proposition 2.2 (c), we have N ∩ A ⊆ N ∩ K ⊆ N ∩ L = 0, with the equality holding since L + N is a direct sum. Consequently, M  N . As in the discussion prior to [9, Example 2.5], it follows that there does not exist an Aalgebra homomorphism A/M → B/N . Of course, A ⊂ D ⊂ B. Also, by reasoning as in the proof of Proposition 2.2 (b), one can see that B is integral over D (the point being that L is integral over K ). It remains only to show that the extension D ⊂ B is inert. In a sense, one can say that this, in turn, follows in the spirit of Proposition 2.2 (a). Indeed, this does follow from the implication (5) ⇒ (1) in [9, Proposition 2.9 (b)], since the extension D/N ⊂ B/N can be identified with the minimal field extension  K ⊂ L.

3 Some Recent History and Some Open Questions Near the end of this section, I will give a list of suggestions for proposed further work on the subject of minimal ring extensions A ⊂ B. As the context for those suggested problems features base rings A that are finite local rings that are not fields, it seems appropriate for motivational purposes to mention that much is known about the possible existence of the various kinds of integral minimal ring extensions (that is, ramified, decomposed or inert). For information along these lines, see the Introductions of [8, 9]. In the following paragraphs, we will be interested in a (nonzero) finite (not necessarily local) base ring A. Recall that any such A can be viewed canonically as n A where each Ai is a nonzero (finite) local ring. Recall also that [6, Lemma i i=1 2.2 (b)] can be used to reduce questions about the (ramified, decomposed or inert) nature of a minimal ring extension of A or the (cardinal) number of the set of Aalgebra isomorphism classes represented by a specific kind of minimal ring extension of A to the corresponding questions where A is replaced by one of the Ai . Although the preceding statement is intuitively satisfying, the published proof of [6, Lemma 2.2 (b)] was far from straightforward, as it involved citations of [10, Lemma III.3], the proof of [3, Lemma 14] and, most significantly, [5, Theorem 2.2 and Corollary 2.3]. The overall enterprise that is contemplated below might be entitled “Competitions for prevalence among the minimal ring extensions of a finite commutative ring.” In view of the above comments concerning [6, Lemma 2.2 (b)], one loses no generality in henceforth restricting attention to base rings A that are finite local rings but not fields. By combining [2, Corollary 2.5] and [19, Lemma 2.1], any such A has a ramified extension, and it also has a decomposed extension [12, p. 805, lines 1–2], but it may not have an inert extension [3, Proposition 8]. Also, by the generator-andrelations characterizations of ramified extensions and of decomposed extensions in

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[12, Proposition 2.12] (see also [4, Lemma 2.1]), we have: for any (A, M) as above, the set of A-algebra isomorphism classes represented by (equivalently, consisting of) ramified (resp., decomposed) extensions of A is finite. Moreover, by [8, Theorem 3.4], the set of A-algebra isomorphism classes represented by ramified extensions of A has cardinality at least 2, with this inequality being sharp because of the behavior of SPIRs such as Z/4Z [3, Proposition 12 (e)]. Furthermore, the number of A-algebra isomorphism classes that can be represented by an inert extension of A must be finite [7, Theorem 2.3 (a)]. In addition, it was shown in [8, Example 4.4] (resp., [8, Example 3.8]; resp., [8, Example 2.10]) that, as A varies over the set of finite local rings that are not fields, there is no finite absolute upper bound on the number of A-algebra isomorphism classes represented by an inert (resp., a ramified; resp., a decomposed) extension of A. Initial evidence for the possible existence of what may be termed a “prevalence pattern” was given in [3] and, later, more definitively in [6] (leading to the title of the latter paper), as follows. By combining [3, Propositions 8, 10 and 12] and [6, Theorem 3.4 (f), (g)], one gets that if A is a finite SPIR but not a field, then at least 2/3 of the A-algebra isomorphism classes that are represented by minimal ring extensions of A consist of ramified extensions of A, and this assertion is best possible. However, any hope that this “pattern” would persist well beyond the realm of SPIRs was shattered by [8, Theorem 4.2], which gave an example of a local ring A of cardinality 8 which is not a field and has the property that the A-algebra isomorphism classes represented by a ramified extension of A constitute at most 4/9 of the (finitely many) A-algebra isomorphism classes represented by a minimal ring extension of A. With this admittedly limited amount of data in hand, we are now ready to move to a description of the proposed program of study. My suggestions begin with six questions, listed in no particular order, whose answers would advance (and, in a sense, complete) our work on the above-mentioned competitions for prevalence. Following that list, we describe a more ambitious project (whose successful conclusion would certainly complete the above-mentioned work). We hope that at least some readers will find the following comments to be interesting and that any work resulting therefrom would be both fruitful and useful. Here are six questions, to be considered when working with the (A-algebra isomorphism classes represented by the) minimal ring extensions of a finite local ring (A, M) which is not a field: 1. Does there exist a ring A of the type in question for which the decomposed extensions are in the majority? 2. Does there exist a ring A of the type in question for which the inert extensions are in the majority? 3. Does there exist a ring A of the type in question for which there exist more decomposed extensions than ramified extensions? 4. Does there exist a ring A of the type in question for which there exist more decomposed extensions than inert extensions? 5. Does there exist a ring A of the type in question for which there exist more inert extensions than ramified extensions?

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6. Does there exist a ring A of the type in question for which there exist more inert extensions than decomposed extensions? If the above six questions have been answered, then a probably much longer project would be to determine the set of ordered triples (n r , n d , n i ) of nonnegative integers such that there exists a final local commutative ring (A, M) which is not a field such that n r (resp., n d ; resp., n i ) is the number of A-algebra isomorphism classes represented by (equivalently, consisting of) ramified (resp., decomposed; resp., inert) extensions of A.

References 1. M.F. Atiyah, I.G. Macdonald, An Introduction to Commutative Algebra (Addison-Wesley, Reading, 1969) 2. D.E. Dobbs, Every commutative ring has a minimal ring extension. Comm. Algebra 34, 3875– 3881 (2006) 3. D.E. Dobbs, On the commutative rings with at most two proper subrings. Int. J. Math. Math. Sci. 2016, Article ID 6912360, 13 (2016). https://doi.org/10.1155/2016/6912360 4. D.E. Dobbs, Certain towers of ramified minimal ring extensions of commutative rings. Comm. Algebra 46(8), 3461–3495 (2018). https://doi.org/10.1080/00927872.2017.1412446 5. D.E. Dobbs, Is A × B isomorphic to B × A? Far East J. Math. Sci. 108(2), 217–228 (2018). https://doi.org/10.17654/MS108020217 6. D.E. Dobbs, A minimal ring extension of a large finite local prime ring is probably ramified. J. Algebra Appl. 19(1), 2050015 (27 pages) (2020). https://doi.org/10.1142/ S0219498820500152 7. D.E. Dobbs, Characterizing finite fields via minimal ring extensions. Comm. Algebra 47(12), 4945–4957 (2019). https://doi.org/10.1080/00927872.2019.1603303 8. D.E. Dobbs, On the nature and number of isomorphism classes of the minimal ring extensions of a finite commutative ring. Comm. Algebra 48(9), 3811–3833 (2020). https://doi.org/10. 1080/00927872.2020.1748193 9. D.E. Dobbs, Where some inert minimal ring extensions of a commutative ring come from. Kyungpook. Math. J. 60(1), 53–69 (2020). https://doi.org/10.5666/KMJ.2020.60.1.53 10. D.E. Dobbs, B. Mullins, G. Picavet, M. Picavet-L’Hermitte, On the FIP property for extensions of commutative rings. Comm. Algebra 33, 3091–3119 (2005) 11. D.E. Dobbs, G. Picavet, M. Picavet-L’Hermitte, J. Shapiro, On intersections and composites of minimal ring extensions. JP J. Algebra Number Theory Appl. 26, 103–158 (2012) 12. D.E. Dobbs, J. Shapiro, A classification of the minimal ring extensions of certain commutative rings. J. Algebra 308, 800–821 (2007) 13. D. Ferrand, J.-P. Olivier, Homomorphismes minimaux d’anneaux. J. Algebra 16, 461–471 (1970) 14. M. Fontana, Topologically defined classes of commutative rings. Ann. Mat. Pura Appl. 123(4), 331–355 (1980) 15. R. Gilmer, Multiplicative Ideal Theory (Dekker, New York, 1972) 16. B. Greenberg, Coherence in cartesian squares. J. Algebra 50, 12–25 (1978) 17. I. Kaplansky, Commutative Rings, rev. (University Chicago Press, Chicago, 1974) 18. B.R. McDonald, Finite Rings with Identity (Dekker, New York, 1974) 19. G. Picavet, M. Picavet-L’Hermitte, Modules with finitely many submodules. Internat. Electron. J. Algebra 19, 119–131 (2016) 20. O. Zariski, P. Samuel, Commutative Algebra, vol. I (Van Nostrand, Princeton, 1958)

A Survey on EM Conditions Emad Abuosba and Manal Ghanem

Abstract This is a survey for all the work done so far on EM-rings, their extensions, and some of their generalizations. Keywords Polynomial ring · Annihilating content polynomial · EM-ring 2010 Mathematics Subject Classification 13A15 · 13B25 · 13E05 · 13F20 · 13F25

1 Introduction All rings are assumed to be commutative with unity 1. If R is a commutative ring, then T (R) denotes the total quotient ring of R, Z (R) denotes the set of all zerodivisor elements of R, and Z (R)∗ = Z (R) − {0}. We survey in this article all the work we have done so far on EM-rings, their relations with other rings, some of their extensions and some of their generalizations. We survey the work we have achieved together with D. D. Anderson, Osama Alkam and Heba Abdelkarim. McCoy [10, Theorem 2] proved that if R is a commutative ring and f (x) ∈ Z (R[x])∗ , then a f (x) = 0 for some a ∈ Z (R)∗ . If f (x) ∈ R[x] − Z (R[x]), then f (x) is called a regular polynomial of R[x]. The idea of an annihilating content is a way to factor a zero-divisor polynomial into a product of a zero-divisor element of R and a regular polynomial in R[x]. In Sect. 2, we present the idea of annihilating content, give examples of polynomials with annihilating content and polynomials without annihilating content. We also discuss the uniqueness of the annihilating content. E. Abuosba (B) · M. Ghanem Department of Mathematics, The University of Jordan, Amman, Jordan e-mail: [email protected] M. Ghanem e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. Badawi and J. Coykendall (eds.), Rings, Monoids and Module Theory, Springer Proceedings in Mathematics & Statistics 382, https://doi.org/10.1007/978-981-16-8422-7_7

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In Sect. 3, we define the EM-rings, in which all polynomials have annihilating contents. We give several equivalent definitions for EM-rings and relate them to some famous rings as PP-rings, Bézout rings, rings with a.c. condition and Armendariz rings. In Sect. 4, we give more investigation of an EM-ring if it is Noetherian. In Sect. 5, we study when an extension of an EM-ring is an EM-ring and conversely. We study the polynomial ring R[x], localization S −1 R, the group ring RG, the idealization R(+)R and the direct limit lim Rα . −→ In Sect. 6, we study some generalizations of EM-rings such as locally EM-rings, EM-Hermite rings, K-Hermite rings and their relations. We give several open questions that we are still working on them.

2 Annihilating Content The concept of an annihilating content of a polynomial was first defined in [2]. Definition 1 Let R be a ring and let f (x) ∈ R[x]. If we can write f (x) = a f 1 (x) with a ∈ R and f 1 (x) is a regular polynomial in R[x], then a is called an annihilating content for f (x). Recall that a zero-divisor graph of a commutative ring R is the simple graph (R) whose vertices are the nonzero zero-divisors of R, with r, s adjacent if r = s and r s = 0, see [7]. If S is a nonempty set of positive integers and G S the graph whose vertices are the elements of S such that two distinct vertices a, b are adjacent if a|b or b|a. A graph G is called a divisor graph if there is a set of positive integers S such that G  G S , see [2]. The authors in [2] proved that if R is a commutative finite principal ideal ring, then every polynomial in R[x] has an annihilating content. So if f (x), g(x) ∈ R[x] with f (x) = a f f 1 (x) and g(x) = ag g1 (x), then f g = 0 if and only if a f ag = 0. The authors were interested in characterizing when the zero divisor graph of a commutative finite principal ideal ring is a divisor graph, and using the idea of annihilating content they proved the following Theorem. Theorem 2 ([2]) Let R be a finite commutative principal ideal ring with unity. Then the zero-divisor graph (R) is a divisor graph if and only if (R[x]) is. Further investigations on annihilating contents were carried out in [3, 6]. The following is an example of a polynomial ring, R[x], that contains an element f (x) such that f (x) does not have an annihilating content. Example 3 ([3]) Let R = Z4 [x]/(x 2 ) = {a0 + a1 X : ai ∈ Z4 and X 2 = 0}. Then R is an Artinian local ring with maximal ideal 2R + X R. If f (y) = 2 + X y ∈ R[y], then f (y) ∈ Z ∗ (R[y]), since 2X f (y) = 0. But f (y) has no annihilating content, n  bi y i ∈ r eg(R[y]), then 2 = ab0 and since if f (y) = a f 1 (y), a ∈ R and f 1 (y) = i=0

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X = ab1 , and so 2R + X R ⊆ a R ⊂ R which implies that 2R + X R = a R, a contradiction. The following is an example of a polynomial ring, R[x], that contains an element f (x) such that f (x) has at least two distinct annihilating contents (i.e. annihilating content of an element in R[x] needs not be unique). Example 4 ([3]) Let R = Z × Z and f (x) = (6, 0) + (12, 0)x. Then f (x) = (2, 0)((3, 1) + (6, 1)x) = (3, 0)((2, 1) + (4, 1)x). Note that Ann((2, 0)) = {0} × Z = Ann((3, 0)), but ((2, 0))R = 2Z × {0} = 3Z × {0} = ((3, 0))R. It is clear if a and b are annihilating contents for a polynomial f (x), then Ann R (a) = Ann R ( f (x)) = Ann R (b), and so a R ≈ R/Ann R (a) = R/Ann R (b) ≈ b R as R-modules. It was shown in [6] that if a R = b R and a is an annihilating content for a polynomial f (x), then so is b, but if a and b are annihilating contents for f (x), then it is not necessarily that a R = b R. If r ∈ R is regular and r | a with a an annihilating content for f (x), then ar is also an annihilating content for f (x). Moreover a ∈ R is an annihilating content for f (x) if and only if the content ideal c( f ) = a J where J is a finitely generated ideal of R with Ann(J ) = 0. A question was raised while we were working on Bézout rings (every finitely generated ideal is principal), concerning the degrees of the polynomials. It is clear that if f (x) = ag(x) with g(x) is regular, then deg( f ) ≤ deg(g). Is it necessary that deg( f ) = deg(g)? In fact the answer was also no, and an example for such a case was given in [4, p. 95]. In case the ring is an a.c. ring (Ann(α, β) = Ann(γ )), then we can choose a regular polynomial g such that deg( f ) ≤ deg(g) ≤ deg( f ) + 1, n m   since if ai x i = a bi x i , then ai = abi for 0 ≤ i ≤ n and 0 = abi for n + 1 ≤ i=0

i=0

i ≤ m . Since R is an a.c. ring, we have Ann(bn+1 , . . . , bm ) = Ann(d). Thus n  ai x i = ad = 0 and {0} = Ann(b1 , . . . , bm ) = Ann(b1 , . . . , bn , d). Therefore a(

n 

bi x + d x i

n+1

i=0

) with

n 

i=0

bi x + d x i

n+1

is a regular polynomial.

i=0

3 EM-Rings In this section, we investigate rings in which all polynomials have annihilating contents. It was proved in [2] that every polynomial over a finite principal ideal ring has an annihilating content. In [3] we proved that any principal ideal ring and more generally, any Bézout ring have also this property. This motivates us to give the following definition using our initials (Emad and Manal). Definition 5 ([3]) A ring R is called an EM-ring if every polynomial over R has an annihilating content.

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In [6] more investigation was done on EM-rings. The following Theorem gives alternative definitions. Theorem 6 ([6]) The following statements are equivalent: (1) R is an EM-ring. (2) If b + cx ∈ R[x], then there exists a ∈ R and a regular polynomial g(x) such that b + cx = ag(x). (3) For any finitely generated ideal I there exist a ∈ R and a finitely generated ideal J such that I = a J with Ann(J ) = {0}. In [3] we related EM-rings to some famous rings as shown in the following diagram: PP-ring

Bézout 

EM-ring





a.c. ring

Armendariz

Recall that a ring R is called a PP-ring if any principal ideal in R is projective, n m   and it is called Armendariz ring if whenever ( ai x i )( bi x i ) = 0, then ai b j = 0 i=0

i=0

for all i, j.

4 The Noetherian Case In this section, we investigate more properties of an EM-ring R given that R is a Noetherian ring. The following Theorem was first proved in [3] for reduced rings. Then a general version was established in [6]. Theorem 7 ([6]) Let R be a Noetherian ring. Then R is an EM-ring if and only if each associated prime of R is principal. Theorem 8 ([6]) Let R be a finite ring. Then R is an EM-ring if and only if R is a principal ideal ring. In the following Theorem, we give several equivalent conditions for a Noetherian ring. Some implications are in general true, but we couldn’t prove or disprove the equivalence in the general case. Theorem 9 ([6]) For a Noetherian ring R the following are equivalent: (1) R is an EM-ring. (2) R[X ] is an EM-ring. (3) R[[X ]] is an EM-ring. (4) R[X, X −1 ] is an EM-ring.

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Theorem 10 ([3, 6]) Let R be a Noetherian ring. Then the following are equivalent: (1) R is a strongly EM-ring (For every power series f there exist a ∈ R and a regular power series g such that f = ag). (2) R is an EM-ring. (3) R is a generalized morphic ring (The annihilator of each element in R is principal). (4) Every ideal in Z (R) is contained in a principal ideal in Z (R). If R was not Noetherian, then the above conditions need not be equivalent. For an example of an EM-ring that is not strongly EM, see Example 4.6 in [3] and for an example of an EM-ring that is not generalized morphic see Example 5.4 in [9]. Question: We found many examples of EM-rings that are not generalized morphic rings, but is there a ring R that is a generalized morphic but not an EM-ring?

5 Some Extensions of EM-Rings In this section, we investigate when an extension of an EM-ring is an EM-ring and vice versa.  It is clear that Ri is an EM-ring if and only if Ri is an EM-ring for each i. Unfortunately, the other cases are not so smooth. Theorem 11 ([9, Theorem 4.1]) If R is an EM-ring, then so is R[x], if R is reduced, then the converse is also true. We couldn’t yet prove or disprove that if R[x] is a nonreduced EM-ring, then R is an EM-ring, except for the case when R is Noetherian. Corollary 12 If R is an EM-ring, then so is R[x1 , x2 , . . . , xn ]. Theorem 13 ([3, Theorem 3.6]) Assume R is an EM-ring, and S is a multiplicatively closed subset of R. Then S −1 R is an EM-ring. In particular, if R is an EM-ring, then the total quotient ring T (R) is an EM-ring. Recall that a ring R is called von Neumann regular ring if for each a ∈ R there exists b ∈ R such that a = aba. For an article on von Neumann regular and related elements in commutative rings, see [8]. Example 14 ([3, Example 3.11]) Let R = Z6 [x, y]/(x y). Then R is a reduced Noetherian ring, and so T (R) is von Neumann regular ring, and hence it is a Bézout EM-ring. But R is not an EM-ring, since Ann(2y) = 3R + x R, which is not principal, and this prevents R to be an EM-ring. Hence T (R) is an EM-ring while R is not. Definition 15 A ring R is called weakly EM-ring if T (R) is an EM-ring.

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Theorem 16 ([3, Theorem 3.10]) Assume R has property A (ifI = (a1 , . . . , an ) ⊆ Z (R), then Ann(I ) = 0). If R is weakly EM-ring, then T (R) is a Bézout ring. That is T (R) is an EM-ring if and only if it is a Bézout ring. Question: What extra conditions must be added to a weakly EM-ring to become an EM-ring? Theorem 17 Let {Rα }α∈ be a direct system of commutative rings such that each Rα is an EM-ring. For α, β ∈  with α < β, let ϕβα : Rα −→ Rβ be the corresponding ring homomorphism (which also extends to ϕβα : Rα [X ] −→ Rβ [X ]). For α, β ∈  with α < β, suppose that if f ∈ Rα [X ] is regular in Rα [X ], then ϕβα ( f ) ∈ Rβ [X ] is regular in Rβ [X ] (equivalently; if a1 , . . . , an ∈ Rα with Ann Rα {a1 , . . . , an } = 0, then Ann Rβ {ϕβα (a1 ), . . . , ϕβα (an )} = 0). Then R = lim Rα is an EM-ring. −→

Theorem 18 ([6]) If R is an EM-ring and G is a torsion free Abelian group, then the group ring RG is an EM-ring. If G is not torsion free, then the above theorem needs not be true, since Z4 is an EM-ring, while the group ring Z4 C4 is not. Recall that if R is a ring, and M is an R-module, then the idealization ring (another name trivial extension of R) R(+)M is the set of all ordered pairs (r, m) ∈ R × M, equipped with addition defined by (r, m) + (s, n) = (r + s, m + n) and multiplication defined by (r, m)(s, n) = (r s, r n + sm). Theorem 19 ([9, Theorem 3.4] and [1, Theorem 2.5]) The following statements are equivalent for a ring R : (1) R[x]/(x n+1 ) is a generalized morphic ring for each n ∈ N (The annihilator of each element in R is principal). (2) R[x]/(x n+1 ) is an EM-ring for each n ∈ N. (3) R(+)R is a generalized morphic ring. (4) R(+)R is an EM-ring. (5) R is a PP-ring.

6 Some Generalizations In this section, we preview some extensions and generalizations of EM-rings. Definition 20 A ring R is called locally EM-ring if for each prime ideal P, the localization R P is an EM-ring. It follows by Theorem 13 that if R is an EM-ring, then it is locally EM-ring. The converse is not in general true. For an example of a locally EM-ring that is not an EM-ring, see [5]. Question: What extra conditions must be added to a locally EM-ring to become an EM-ring?

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We said earlier that if f (x) = ag(x) with g(x) regular, then deg( f ) ≤ deg(g) and that if R is an a.c. ring (in particular if R is an EM-ring), then we can choose g such that deg( f ) ≤ deg(g) ≤ deg( f ) + 1. But when we have equal degrees. This motivated us to define the EM-Hermite rings. Definition 21 ([4]) A ring R is called an EM-Hermite ring if for each f (x) = n  ai x ∈ R[x] there exist a ∈ R and a polynomial of the same degree g(x) = i=0 n 

bi x i ∈ R[x] such that f (x) = ag(x) and the ideal c(g) is a regular ideal, where

i=0

c(g) is the ideal in R generated by the coefficients of g. In fact this definition was motivated by the definition of the well known K-Hermite ring defined by I. Kaplansky, which we can redefined as: A ring R is called a Kn  Hermite ring if for each f (x) = ai x ∈ R[x] there exist a ∈ R and a polynomial of the same degree g(x) =

n 

i=0

bi x i ∈ R[x] such that f (x) = ag(x) and c(g) = R.

i=0

It is clear that if R is an EM-Hermite ring, then it has property A, while an EM-ring is incomparable with rings with property A. Theorem 22 ([4, Theorem 2.9]) A ring R is a K-Hermite ring if and only if R is a Bézout EM-Hermite ring. Motivated by this Theorem the authors in [4] showed that if X = [0, ∞) × [−1, 1], then C(β X − X ) is an EM-ring that is not EM-Hermite ring. We manage to characterize some cases at which EM-rings and EM-Hermite rings are equivalent. The following theorems illustrate this. Theorem 23 ([4, Theorem 3.5]) Let R be a Noetherian ring. Then R is an EM-ring if and only if R is an EM-Hermite ring. Theorem 24 ([1, Theorem 2.5]) The following statements are equivalent for a ring R: (1) R[x]/(x n+1 ) is an EM-ring for each n ∈ N. (2) R[x]/(x n+1 ) is an EM-Hermite ring for each n ∈ N. (3) R(+)R is an EM-ring. (4) R(+)R is an EM-Hermite ring. Question: What extra conditions must be added to an EM-ring to become an EM-Hermite ring? Question: What is the relation between the locally EM-rings and the weakly EM-rings?

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The following diagram illustrates the relation between these rings. K-Hermite







EM-Hermite

Bézout EM-ring with property A ↓ EM-ring

Locally EM-ring

 ............. ? ...................

Weakly EM-ring

References 1. H. Abdelkarim, E. Abuosba, M. Ghanem, Idealization of EM-Hermite rings. Comm. Korean Math. Soc. 35(1), 13–20 (2020) 2. E. Abu Osba, O. Alkam, When zero-divisor graphs are divisor graphs. Turkish J. Math. 41(4), 797–807 (2017) 3. E. Abuosba, M. Ghanem, Annihilating content in polynomial and power series rings. J. Korean Math. Soc. 56(5), 1403–1418 (2019) 4. E. Abuosba, M. Ghanem, EM-Hermite rings. Int. Electron. J. Algebra 27, 88–101 (2020) 5. E. Abuosba, M. Ghanem, Prüfer conditions vs EM conditions (Submitted) 6. D.D. Anderson, E. Abuosba, M. Ghanem, Annihilating content polynomials and EM-rings. J. Alg. Appl. (To appear in) 7. D.F. Anderson, P.S. Livingston, The zero-divisor graph of a commutative ring. J. Algebra 217, 434–447 (1999) 8. D.F. Anderson, A. Badawi, Von Neumann regular and related elements in commutative rings. Algebra Colloquium 19(Spec 1), 1017–1040 (2012) 9. M. Ghanem, E. Abuosba, Some extensions of generalized morphic rings and EM-rings. Analele Stiintifice ale Universitatii Ovidius Constanta (Seria Matematica) 26(1), 111–123 (2018) 10. N. McCoy, Remarks on divisors of zero. Amer. Math. Monthly 49, 286–295 (1942)

Some Remarks on the D + M Construction David F. Anderson

Abstract Let T = K + M be an integral domain, where the field K is a subring of T and M is a nonzero maximal ideal of T , and let D be a subring of K . Then R = D + M is a subring of T . This “D + M” construction has proved very useful for constructing examples since ring-theoretic properties of R are often determined by those of T and D. In this note, we investigate to what extent K and M are determined by T . Keywords D + M construction 2010 Mathematics Subject Classification 13A15 · 13B99 · 13F99 · 13G05 Let T be an integral domain of the form K + M, where the field K is a subring of T and M is a nonzero maximal ideal of T , and let D be a subring of K . Then R = D + M is a subring of T with the same quotient field as T . This “D + M” construction has proved very useful for constructing examples since ring-theoretic properties of R are often determined by those of T and D. The “classical” case, when T is a valuation domain, was first studied systematically in [7, Appendix II] (also see [4, 8]). In [5], the “generalized” D + M construction as above was introduced and investigated. Of course, one can also study more general pullbacks where the residue field K = T /M is not a subring of T (and in some cases the results are quite different). The simplest examples of integral domains of the form K + M are K [[X ]] = K + X K [[X ]] and K [X ] = K + X K [X ], the power series ring and polynomial ring over a field K . Also, for any field K and any totally orderd abelian group G, there is a valuation domain of the form K + M with value group G; and any complete local ring which contains a field has the form K + M [12, Theorem 31.1, p. 106]. Also, Dedicated to my twin brother Daniel D. Anderson. D. F. Anderson (B) Department of Mathematics, The University of Tennessee, Knoxville, TN 37996-1300, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. Badawi and J. Coykendall (eds.), Rings, Monoids and Module Theory, Springer Proceedings in Mathematics & Statistics 382, https://doi.org/10.1007/978-981-16-8422-7_8

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integral domains of the form A + X B[X ] ⊆ B[X ] and A + X B[[X ]] ⊆ B[[X ]], where A ⊆ B is an extension of integral domains, have been useful in constructing examples (see [1–3, 11, 14]). In fact, these above constructions are some of the most basic tools in a commutative ring-theorist’s toolbox. For more details, examples, and references, see [9] and the two survey articles [6, 11]. In this note, we are interested in investigating to what extent the field K and maximal ideal M are determined by T . In other words, can we have T = K + M = F + N for distinct fields K and F or distinct maximal ideals M and N ? We will see that neither the fields K and F nor the maximal ideals M and N need be uniquely determined; in fact, we need not even have htM = htN . Although the ideas and examples we present here are for the most part well known, we thought it would be useful to give them some more exposure. When we write T = K + M, we will always mean that T is an integral domain, K is a field which is a subring of T , and M is a nonzero maximal ideal of T . Since K ∩ M = {0}, each x ∈ T may be written uniquely as x = a + m with a ∈ K and m ∈ M. Throughout, U (R), R ∗ , and q f (R) will denote the group of units, the set of nonzero elements, and the quotient field of an integral domain R; and X, Y, Z , X i , Z i will always denote indeterminates. An overring of an integral domain R is a subring of q f (R) containing R. As usual, Z, Q, and R will denote the integers, rationals, and real numbers, respectively. For any other undefined notation or terminology, see [8] or [10]. We begin by showing that it is possible to have T = K + M = F + M for distinct (necessarily incomparable) fields K and F of T . In fact, we may have T = K + M = F + N for distinct fields K and F and distinct maximal ideals M and N . Example 1 (a) Let T = Q(X )[[Y ]] = Q(X ) + Y Q(X )[[Y ]], a DVR with maximal ideal M = Y Q(X )[[Y ]]. Then it is easily verified that T = Q(X + Y ) + Y Q(X )[[Y ]]. In fact, T = K + Y Q(X )[[Y ]] for infinitely many incomparable subfields K of Q(X, Y ). This example may be viewed as illustrating the nonuniqueness of the coefficient field of a complete local ring (cf. [12, pp. 110–111]). (b) Suppose that R = K + M = F + M for distinct fields K and F contained in R. Let T = R[X ] = R + X R[X ] = R + (X − 1)R[X ]. Then T = K + (M + X R[X ]) = F + (M + (X − 1)R[X ]) for distinct fields K and F contained in T and distinct maximal ideals M + X R[X ] and M + (X − 1)R[X ] of T . More generally, T = R + (X − a)R[X ] for any a ∈ R, and the maximal ideals M + (X − a)R[X ] and M + (X − b)R[X ] are distinct for distinct a, b ∈ K (or distinct a, b ∈ F). (c) Let R = Q(X )[[Y ]] and T = R[Z ]. By the above comments, we have T = K α + Mα for infinitely many distinct fields K α and maximal ideals Mα . Next we give some positive results. Remark 2 (a) Suppose that T = K + M = F + M for fields F and K . Then the natural map K → K + M = F + M → F induced by inclusion and projection is an isomorphism. In particular, if T is quasilocal, then the maximal ideal M is uniquely determined and the field K is determined up to isomorphism.

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(b) Let T = K + M, and let D1 and D2 be subrings of K . Then it is easily shown that D1 + M = D2 + M if and only if D1 = D2 . More generally, let D1 and D2 be subrings of T with D1 ∩ M = D2 ∩ M = {0}. If D1 + M = D2 + M and D1 and D2 are contained in a subring D of q f (T ) with D ∩ M = {0}, then D1 = D2 . In particular, D1 = D2 if D1 and D2 are comparable. (c) Suppose that T = K + M = F + N . If K and F are subfields of a field L ⊆ q f (T ) with L M ⊆ M and L N ⊆ N , then F = K . In particular, if F and K are comparable, then F = K . Thus, if U (T ) = K ∗ , then F ∗ ⊆ U (T ); so F ⊆ K , and hence F = K . Also, if either F or K is algebraic over F ∩ K , then F = K (let L be the field generated by F and K ). Thus K is uniquely determined if it is an algebraic extension of Q or Z/ pZ. (d) Let A, B, C, D be subrings of a field L with A ⊆ B and C ⊆ D. Then A + X B[X ] = C + X D[X ] ⇔ A + X B[[X ]] = C + X D[[X ]] ⇔ A = C and B = D. (However, see Example 1.) The following elementary example again shows that we may have T = K + Mα for infinitely many distinct maximal ideals Mα . Example 3 Let K be any field and T = K [X 1 . . . , X n ]. Then T = F + M for a field contained in T and a maximal ideal M of T if and only if F = K and M = (X 1 − a1 , . . . , X n − an ) for some a1 , . . . , an ∈ K . We must have F = K since U (T ) = K ∗ . If T = K + M, then each X i = ai + m i for some ai ∈ K and m i ∈ M, and thus M = (X 1 − a1 , . . . , X n − an ). Hence T = K [X 1 , . . . , X n ] = K + M for each maximal ideal M of T if and only if K is algebraically closed, and T = K + M for infinitely many distinct maximal ideals M of T if and only if K is infinite. As in Example 3 above, we have T = R[X ] = R + X R[X ] = R + (X − 1)R[X ] for distinct maximal ideals X R[X ] and (X − 1)R[X ] of T . However, note that the two subrings R1 = Q + X R[X ] and R2 = Q + (X − 1)R[X ] are not comparable. So in the D + M construction, the choice of the maximal ideal M is crucial. This observation is a special case of the next theorem. Theorem 4 Suppose that T = K + M = K + N for distinct maximal ideals M and N of T , and let D1 and D2 be subrings of the field K . Then R1 = D1 + M and R2 = D2 + N are comparable if and only if D1 = K or D2 = K (i.e., R1 = T or R2 = T ). Moreover, R1 = R2 if and only if D1 = D2 = K (i.e., R1 = R2 = T ). Proof Suppose that R1 ⊆ R2 . Then T = M + N ⊆ R2 ; so D2 + N = R2 = T =  K + N . Thus D2 = K . If R1 = R2 , then also D1 = K . The following result is sometimes useful for determining when an integral domain has the form K + M. Theorem 5 Let R be an integral domain which contains a field K as a subring. Then the following statements are equivalent. 1. R = K + M for some maximal ideal M of R.

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2. R has an overring of the form T = K + N for some maximal ideal N of T . 3. R has a quasilocal overring of the form T = K + N for N the maximal ideal of T . 4. R has a quasilocal extension ring of the form T = K + N for N the maximal ideal of T . 5. R has an extension ring of the form T = K + N for some maximal ideal N of T. Moreover, if any of (2)−(5) hold, then we may choose M = N ∩ R in (1). Proof (1) ⇒ (2), (3) ⇒ (4), and (4) ⇒ (5): Each implication is obvious. (2) ⇒ (3): Suppose that T = K + N is an overring of R. Then TN = K + N N is the desired quasilocal overring of R. (5) ⇒ (1): Let M = N ∩ R. We show that R = K + M, and hence M is a maximal ideal of R. The “⊇” inclusion is clear. For “⊆”, let x ∈ R ⊆ T . Then x = a + m for some a ∈ K and m ∈ N . Thus x − a = m ∈ N ∩ R = M, and hence x = a + m ∈ K + M. The “moreover” statement is clear by the proof of (5) ⇒ (1).  The construction in Theorem 6 will be useful for giving examples (cf. [10, Theorem 105, p. 77]; the new twist here is that each Ri = K + Mi ). Theorem 6 Let K be a field and R1 = K + N1 , . . . , Rn = K + Nn be quasilocal integral domains contained in a field L, with maximal ideals N1 , . . . , Nn , respectively. Let R = R1 ∩ · · · ∩ Rn and Mi = Ni ∩ R for each 1 ≤ i ≤ n, and assume that the Mi ’s are incomparable. Then R is semiquasilocal with maximal ideals M1 , . . . , Mn , and R = K + M1 = · · · = K + Mn . Proof It is easy to verify that R \ U (R) = M1 ∪ · · · ∪ Mn , and hence M1 , . . . , Mn are precisely the maximal ideals of R. By Theorem 5, we have R = K + Mi for each 1 ≤ i ≤ n.  Clearly (Krull) dim(K + M) = htM when K + M is quasilocal. We next give examples where T = K + M = K + N such that the maximal ideals M and N have different heights (cf. [2, Remark 2.7(b)], [9, Examples 28 and 29, pp. 62–65]). Example 7 (a) Let K be any field, n ≥ 2 an integer, and A = K [X 1 , . . . , X n ]. Set Rn = A N = K + N N , where N = (X 1 − 1, X 2 , . . . , X n ). Choose α1 , . . . , αn ∈ Z K [[Z ]] to be algebraically independent with α1 = Z , and α2 , . . . , αn each transcendental over K (Z ). Define ψ : A −→ K [[Z ]] by ψ(X i ) = αi for 1 ≤ i ≤ n and ψ(a) = a for all a ∈ K ; clearly ψ is an injective homomorphism. Extend ψ to an injective homomorphism ϕ : K (X 1 , . . . , X n ) −→ K [[Z , Z −1 ]], and set V = ϕ −1 (K [[Z ]]). Then V is a DVR of the form K + M with (X 1 , . . . , X n ) ⊆ M. Let R = V ∩ Rn , M1 = M ∩ R, and Mn = N N ∩ R. By Theorem 6, R = K + M1 = K + Mn . Note that R M1 = V and R Mn = Rn (cf. [13, Theorem 3]); so htM1 = 1 and htMn = n. Also, R is locally a Noetherian UFD, and hence a Noetherian UFD since R is semiquasilocal.

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We could also let A = K [{X n | 1 ≤ n < ∞}]. In this case, a similar construction to that above would yield R = K + M1 = K + M∞ , a nonNoetherian UFD with htM1 = 1 and htM∞ = ∞. (b) Let K be any field and n ≥ 2 an integer. Construct valuation domains (with a common quotient field) Ri = K + Ni with htNi = i for each 1 ≤ i ≤ n. Set R = R1 ∩ · · · ∩ Rn and Mi = Ni ∩ R for each 1 ≤ i ≤ n. If the Mi ’s are incomparable ´ domain, M1 , . . . , Mn are precisely and each Ri is an overring of R, then R is a Bezout the maximal ideals of R, and R Mi = Ri for each 1 ≤ i ≤ n [10, Theorem 107, p. 78]. Thus R = K + Mi with htMi = i for each 1 ≤ i ≤ n by Theorem 6. Such valuation domains Ri may be constructed as follows. Let A = K [X 1 , . . . , X n ] and Bi = K [[Z 1 , . . . , Z i ]] for each 1 ≤ i ≤ n. Define injective homomorphisms ψi : A −→ Bi such that ψi (X j ) = Z j + 1 for 1 ≤ j ≤ i, ψi (X j ) ∈ Bi \ U (Bi ) for i + 1 ≤ j ≤ n, and ψi (a) = a for all a ∈ K . Extend each ψi to an injective homomorphism ϕi : q f (A) −→ q f (Bi ). Let Vi be the valuation overring of Bi determined by the usual (lexicographic) order valuation on q f (Bi ) with value group Zi (i.e., Z 1α1 · · · Z iαi → (α1 , . . . , αi )). Set Ri = ϕi−1 (Vi ). Then each Ri is an i-dimensional valuation domain of the form K + Ni . Note that the Mi ’s are incomparable and A ⊆ Ri ⊆ q f (A) for each 1 ≤ i ≤ n since (X 1 − 1, . . . , X i − 1, X i+1 , . . . , X n ) ⊆ Mi .

References 1. D.D. Anderson, D.F. Anderson, M. Zafrullah, Rings between D[X ] and K [X ]. Houston J. Math. 17, 109–129 (1991) 2. D.F. Anderson, G.W. Chang, J. Park, Weakly Krull and related domains of the form D + M, A + X B[X ], and A + X 2 B[X ]. Rocky Mountain J. Math 36, 1–22 (2006) 3. D.F. Anderson, D.N. El Abidine, Factorization in integral domains, III. J. Pure Appl. Algebra 135, 107–127 (1999) 4. E. Bastida, R. Gilmer, Overrings and divisorial ideals of rings of the form D + M. Michigan Math. J. 20, 79–95 (1973) 5. J. Brewer, E.A. Rutter, D + M constructions with general overrings. Michigan Math. J. 23, 33–42 (1976) 6. S. Gabelli, E. Houston, Ideal theory in pullbacks, in Non-Noetherin Ring Theory, Mathematics and Its Applications, vol. 520 (Kluwer Academic Publishers, Dordrecht, 2000), pp. 199–227 7. R. Gilmer, Multiplicative Ideal Theory, Queen’s Papers in Pure and Applied Mathematics, No. 12 (Queen’s University Press, Kingston, 1968) 8. R. Gilmer, Multiplicative Ideal Theory (Marcel Dekker, New York, 1972) 9. H.C. Hutchins, Examples of Commutative Rings (Polygonal Publishing House, Passaic, 1981) 10. I. Kaplansky, Commutative Rings, revised. (University of Chicago Press, Chicago, 1974) 11. T.G. Lucas, Examples built with D + M, A + X B[X ], and other pullback constructions, in Non-Noetherin Ring Theory, Mathematics and Its Applications, vol. 520 (Kluwer Academic Publishers, Dordrecht, 2000), pp. 341–368 12. M. Nagata, Local Rings (Interscience Publishers, New York, 1962) 13. B. Prekowitz, Intersections of quasi-local domains. Trans. Amer. Math. Soc. 181, 329–339 (1973) 14. M. Zafrullah, Facets of rings between D[X ] and K [X ], in Lecture Notes in Pure and Applied Mathematics, vol. 231 (Marcel Dekker, New York, 2002)

On a Problem About Lowest Terms Domains Posed by D. D. Anderson Roy O. Quintero Contreras

Abstract In a recent paper, D. D. Anderson et al. introduced a list of different types of integral domains related to the process of putting fractions in lowest terms. They proved that the class of lowest terms domains contains the class of reduced to lowest terms domains. They also asked whether that containment is strict. In this paper, we prove that the domain candidate proposed by them certainly answers the question positively. Our techniques involve a particular type of factorization into irreducible elements and basic properties of the domain under question from the algebraic standpoint. Keywords Integral domains · Rings extensions · Lowest terms

1 Introduction In [2], the concepts of an LT domain and an RLT domain were studied. In order to make this article a little more self-contained, we will begin by recalling the definitions of these classes of domains. Definition 1.1 ([2, Definitions 1, 2]) Let D be an integral domain and let a, b, c, d, e ∈ D  . We say that a/b can be put in lowest terms if a/b = c/d where gcd D [c, d] = 1 and that a/b can be reduced to lowest terms if a/b = c/d where c = a/e and d = b/e for some common divisor e of a and b and gcd D [c, d] = 1. The integral domain D is a lowest terms (LT) domain if each nonzero fraction a/b (a, b ∈ D  ) can be put in lowest terms and is a reduced to lowest terms (RLT) domain if each nonzero fraction a/b (a, b ∈ D  ) can be reduced to lowest terms. It is evident that any RLT domain is an LT domain. The article ends with a couple of questions: • Question 1: Must an LT domain be an RLT domain? R. O. Quintero Contreras (B) Department of Mathematics, University of Southern California, Los Angeles, CA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. Badawi and J. Coykendall (eds.), Rings, Monoids and Module Theory, Springer Proceedings in Mathematics & Statistics 382, https://doi.org/10.1007/978-981-16-8422-7_9

149

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• Question 2: Must a ULT domain be a GCD domain? The authors presented the integral domain  D := k [X, Y, Z , T ]

   X Y X Y , , , , T T Z n Z n n≥1

where k is a field and X , Y , Z , and T are indeterminates over k and showed that D is not an RLT domain. The single goal of this paper is to demonstrate that D also works to answer Question 1 negatively [Theorem 3.2]. Well, let us start. First, observe that D is a proper ring extension of the well-known ring of polynomials in X , Y , Z , and T with coefficients in k, which we denote by R. So, R := k [X, Y, Z , T ]. Second, since X Y X Y the set of generators of D, i.e., , , n , n for every positive integer n, are T T Z Z some fractions involving the indeterminates, the domain D is properly contained in the quotient field of R, namely F := k(X, Y, Z , T ). Thus, we have the following containment sequence R  D  F. A general homogeneous element or monomial of D, say h = h(X, Y, Z , T ), has the form (up to order of factors and up to associates)  X

m1

Y

m2

Z

m3

T

m4

X T

m 5  m 6     Y X m7 Y m8 , T Z n1 Z n2

where m i ∈ Z≥0 for i = 1, . . . , 8 and n j ∈ Z>0 for j = 1, 2. Observe that h can be rewritten as X m 1 +m 5 +m 7 Y m 2 +m 6 +m 8 Z m 3 −(n 1 m 7 +n 2 m 8 ) T m 4 −(m 5 +m 6 ) , where the exponents m 1 + m 5 + m 7 and m 2 + m 6 + m 8 are non-negative integers and m 3 − (n 1 m 7 + n 2 m 8 ) and m 4 − (m 5 + m 6 ) can be negative integers. Moreover, when m 4 − (m 5 + m 6 ) is negative the sum of the exponents of X and Y cannot be less than or equal to m 4 . The remaining sections of this paper are: 2. Basic Properties of D, 3. Main Results, 4. Acknowledgements, and finally References. The title of each section is self-explanatory of its content.

2 Basic Properties of D The main goals of this section are to describe a general representation of each element of D and prove a result about factorizations in D of irreducible polynomials in R.

On a Problem About Lowest Terms Domains Posed by D. D. Anderson

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First, we characterize when an element h of the form X α Y β Z γ T δ is a homogeneous element of D by imposing some conditions on the integers α, β, γ , and δ. Clearly, the first condition is α, β ∈ Z≥0 . Let us divide the argument into four cases. 1 1st case: α = 0, β = 0. In this case is evident that γ , δ ∈ Z≥0 (because −γ and Z 1 ∈ / D when γ , δ are negative integers). T −δ 2nd case: α = 0, β ≥ 1. We proceed by considering the following four sub-cases: 2.1 γ ≥ 0, δ ≥ 0. No more restrictions are required because h ∈ k[Y, Z , T ] ⊂ D. 2.2 γ < 0, δ ≥ 0. No more restrictions are required because Y ∈ D. Z −γ

h = Y β−1 T δ

2.3 γ ≥ 0, δ < 0. In this sub-case we need the condition β ≥ −δ to express h=Y

β+δ

 −δ Y Z ∈ D. T γ

2.4 γ < 0, δ < 0. In this sub-case we need the condition β ≥ 1 − δ to express h=Y

β+δ−1

 −δ Y Y ∈ D. T Z −γ

3rd case: α ≥ 1, β = 0. In this case we have the four sub-cases: 3.1 γ ≥ 0, δ ≥ 0. No more restrictions are required because h ∈ k[X, Z , T ] ⊂ D. 3.2 γ < 0, δ ≥ 0. No more restrictions are required because X ∈ D. Z −γ

h = X α−1 T δ

3.3 γ ≥ 0, δ < 0. In this sub-case we need the condition α ≥ −δ to express h=X

α+δ

Z

γ



X T

−δ

∈ D.

3.4 γ < 0, δ < 0. In this sub-case we need the condition α ≥ 1 − δ to express h=X

α+δ−1



X T

−δ

X ∈ D. Z −γ

4th case: α ≥ 1, β ≥ 1. In this case we also have four sub-cases: 4.1 γ ≥ 0, δ ≥ 0. No more restrictions are required because h ∈ k[X, Y, Z , T ] ⊂ D.

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4.2 γ < 0, δ ≥ 0. No more restrictions are required because h = X α−1 Y β T δ

X ∈ D. Z −γ

Observe that h can also be written as h = X α Y β−1 T δ

Y ∈ D. Z −γ

4.3 γ ≥ 0, δ < 0. In this sub-case we need the condition α + β ≥ −δ to express ⎧  −δ X ⎪ ⎪ ∈D ⎨ X α+δ Y β Z γ T  α  −(α+δ) h= ⎪ X Y ⎪ ⎩Y α+β+δ Z γ ∈D T T

if − δ ≤ α

.

if α < −δ ≤ α + β

4.4 γ < 0, δ < 0. In this sub-case we need the condition α + β ≥ 1 − δ to express ⎧  −δ X X ⎪ α+δ−1 β ⎪ Y ∈D ⎨X −γ  Tα  Z−(α+δ) h= ⎪ X Y Y ⎪ ⎩Y α+β+δ−1 ∈D T T Z −γ

if − δ < α

.

if α ≤ −δ ≤ α + β − 1

Notice that the 13 sub-cases shown above can be summarized in the following proposition. Proposition 2.1 The monomial X α Y β Z γ T δ with α, β, γ , and δ ∈ Z is an element of D, if and only if, any of the following conditions is true: 1. 2. 3. 4. 5.

α = β = 0 and γ , δ ∈ Z≥0 . α, β ∈ Z≥0 not simultaneously zero and γ ≥ 0, δ ≥ 0. α, β ∈ Z≥0 not simultaneously zero and γ < 0, δ ≥ 0. α, β ∈ Z≥0 not simultaneously zero, γ ≥ 0, δ < 0, and α + β ≥ −δ. α, β ∈ Z≥0 not simultaneously zero, γ < 0, δ < 0, and α + β ≥ 1 − δ.

Before continuing, we set up certain notation. For every integer n ≥ 1 let An := {(α, β) ∈ Z≥0 × Z≥0 : α + β < n}. In Fig. 1, you can see A1 , A2 , and A3 . Notice that A1 = {(0, 0)}, A2 = A1 ∪ {(0, 1), (1, 0)}, A3 = A2 ∪ {(0, 2), (1, 1), (2, 0)}, .. . An = An−1 ∪ {(0, n − 1), (1, n − 2), . . . , (n − 2, 1), (n − 1, 0)}.

On a Problem About Lowest Terms Domains Posed by D. D. Anderson

153

Fig. 1 An for n = 1, 2, 3

Now, let us define the sets U1 , U2 , U3 , and U4 as follows: U1 :={0} × {0} × Z≥0 × Z≥0 , U2 :=(Z≥0 × Z≥0 \ A1 ) × Z × Z≥0 , U3 := U4 :=

∞ δ=1 ∞

((Z≥0 × Z≥0 \ Aδ ) × Z≥0 × {−δ}), and ((Z≥0 × Z≥0 \ Aδ+1 ) × Z0 )2 , and α = m1 + m5 + m7, β = m2 + m6 + m8, γ = m 3 − (m 7 n 1 + m 8 n 2 ), and δ = m 4 − (m 5 + m 6 ).

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R. O. Quintero Contreras

Observe that two distinct 10-tuples in E 10 may generate like terms or even the same terms. Here is an example: if we take as coefficients q1 and q2 (q1 = q2 ) and the two 10-tuples (1, 1, 1, 1, 2, 1, 2, 4, 5, 4) and (0, 1, 1, 1, 1, 2, 4, 3, 2, 6), respectively, we get the like terms q1 X 5 Y 6 Z −25 T −2 and q2 X 5 Y 6 Z −25 T −2 . Of course, we get the same terms when q1 = q2 . Moreover, notice that q X α Y β Z γ T δ = q if and only if (m 1 , m 2 , m 3 , m 4 , m 5 , m 6 , m 7 , m 8 , n 1 , n 2 ) = (0, . . . , 0, n 1 , n 2 )

for every n 1 , n 2 positive integers. Returning to the representation of f = 0, it can be expressed-after collecting like terms-as n0

qi X αi Y βi Z γi T δi , f = i=1

where qi = 0 and (αi , βi , γi , δi ) ∈ U1 ∪ U2 ∪ U3 ∪ U4 for every i = 1, . . . , n 0 (See Lemma 2.2). Considering that the four sets U1 , U2 , U3 , and U4 are pairwise disjoint any element f in D  can be thought of as a sum of four elements fUi for i = 1, 2, 3, 4, each of which is zero or a finite sum of different monomials of the form q X α Y β Z γ T δ , where q ∈ k and (α, β, γ , δ) ∈ Ui for i = 1, 2, 3, 4 (not all fUi ’s can be zero). It clearly follows from its definition that fU1 is any polynomial p(Z , T ) ∈ k[Z , T ]. Thus, any f ∈ D can be written as f = p(Z , T ) + fU2 + fU3 + fU4 ,

where p(Z , T ) =

qi Z γi T δi

i∈{1,...,n 0 } (0,0,γi ,δi )∈U1

and fU j =

qi X αi Y βi Z γi T δi

i∈{1,...,n 0 } (αi ,βi ,γi ,δi )∈U j

for j = 2, 3, 4. We have seem that every exponent of X or Y is non-negative, but it might happen that some exponents of Z or T are negative. If some exponents of Z are negative, let

On a Problem About Lowest Terms Domains Posed by D. D. Anderson

155

us define the following set A := {i ∈ {0, 1, . . . , n 0 } : γi < 0}. If some exponents of T are negative, let us define the set B := {i ∈ {0, 1, . . . , n 0 } : δi < 0}. By following ideas given in [1], we are in position to prove the following lemma. Lemma 2.3 If f ∈ D, then there exist non-negative integers p and q such that Z p T q f ∈ R. Proof If f ∈ R, take p = q := 0. Otherwise, there are three possibilities: 1. A = ∅ and B = ∅. Take p := max{−γi : i ∈ A}, q := 0. Then, p = max{−γi : i = 0, 1, . . . , n 0 } > 0 and Z pT q f = Z p

n0

qi X αi Y βi Z γi T δi =

i=0

n0

qi X αi Y βi Z p+γi T δi ∈ R

i=0

because p + γi and δi are non-negative for every i. 2. A = ∅ and B = ∅. Take p := 0, q := max{−δi : i ∈ B}. Then, q = max{−δi : i = 0, 1, . . . , n 0 } > 0 and Z T f =T p

q

q

n0

αi

βi

γi

qi X Y Z T

δi

=

i=0

n0

qi X αi Y βi Z γi T q+δi ∈ R

i=0

because γi and q + δi are non-negative for every i. 3. A = ∅ and B = ∅. Take p := max{−γi : i ∈ A}, q := max{−δi : i ∈ B}. Then, p = max{−γi : i = 0, 1, . . . , n 0 } > 0, q = max{−δi : i = 0, 1, . . . , n 0 } > 0 and Z T f =Z T p

q

p

q

n0

i=0

αi

βi

γi

qi X Y Z T

δi

=

n0

qi X αi Y βi Z p+γi T q+δi ∈ R

i=0

because p + γi and q + δi are non-negative for every i.



Based on Lemma 2.3, we will proceed to prove that any irreducible polynomial f in 1 R but reducible in D may have only factors of the form Z k T l and k l f , where k Z T and l are non-negative integers. Before proving any result in this direction, let us recall some basic facts about posets, (i.e., partially ordered sets). The set P := N × N endowed with the product order  defined as (m, n)  (m , n ) iff m ≤ m and n ≤ n is a poset. Given a non-empty set S ⊆ P, an element (k, l) ∈ S is called a maximal element of S if for all (m, n) ∈ S such that (k, l)  (m, n) implies m = k and n = l. Given any irreducible polynomial f in R but reducible in D, by S f we denote the set

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R. O. Quintero Contreras

{(k, l) ∈ P \ {(0, 0)} : Z k T l | f in D}. Notice that S f might be an empty set, but we affirm that it is always (by Proposition 2.4 part 1) non-empty. Now, let us see some examples. First of all, it is not hard to check that the following polynomials X − Y 2 , Z 2 − X T , X Z + T , Z 2 T − X Y and Z T − X are irreducible in R = k[X,Y, Z , T ], but they are reducible as elements of D =  X Y X Y . In fact, , , R , T T Z n Z n n≥1  X − Y2 = T

Y X −Y T T





X Y −Y n n Z Z

= Zn

 for every n ≥ 1,

  X Z2 − XT = Z2 1 − T 2 , Z  XZ + T = T

Z

 X +1 , T

    X Y X Y 2 Z T − XY = Z T 1 − 2 , and = Z T 1− Z T T Z2 2

2



X ZT − X = Z T − Z



 =T

X Z− T

 .

As you can easily check, all of these factorizations are permissible in D. In the first case, both factors T and TX − Y YT are irreducible in D, but in the second factorization the factor ZXn − Y ZYn is still reducible in D because it can be expressed as  Z

X Z n+1

−Y

Y Z n+1



 or even as Z

X Z n+1

−

Y Y Z Zn

 .

Further, Z T does not divide X − Y 2 . In the second case, Z 2 is the highest power of Z that divides Z 2 − X T and Z and 1 − T ZX2 are irreducible in D and Z 2 − X T is not divisible by T . In the next case, T and Z TX + 1 are irreducible in D and X Z + T is not divisible by Z . In the fourth case, Z 2 T − X Y is divisible by Z , T and Z T . Moreover, Z , T , 1 − ZX2 YT , and 1 − TX ZY2 are irreducible in D. Finally, Z T − X is divisible by Z and T but not by Z T . In this case, Z and T − XZ are irreducible as elements of D and T and Z − TX too. Based on these examples, we present the following proposition. Proposition 2.4 Let f be any irreducible polynomial in R. Then, 1. If f is reducible in D, then S f = ∅.

On a Problem About Lowest Terms Domains Posed by D. D. Anderson

157

1 2. If f is reducible in D and (k, l) is a maximal element of S f , then k l f is Z T irreducible in D. Proof 1. Assume that f is an irreducible polynomial in R and f can be written as f = f1 f2 , where f 1 and f 2 are non-constant elements in D. By Lemma 2.3 there exist nonnegative integers p1 , q1 , p2 , and q2 such that g1 := Z p1 T q1 f 1 and g2 := Z p2 T q2 f 2 belong to R. Observe that not all the integers p1 , q1 , p2 , and q2 can be zero. Otherwise, f would be reducible in R. Then, Z p1 + p2 T q1 +q2 f = g1 g2 . But R is a UFD (Unique factorization domain) and the polynomials Z , T and f are irreducible in R, so f must divide one of the two polynomials g1 and g2 . Without loss of generality we can assume that f |g1 . So g1 = Z p3 T q3 f and g2 = Z p1 + p2 − p3 T q1 +q2 −q3 (up to associates), where p3 and q3 are integers satisfying 0 ≤ p3 ≤ p1 + p2 and 0 ≤ q3 ≤ q1 + q2 . Thus, f2 =

1 1 p1 + p2 − p3 q1 +q2 −q3 T = Z p1 − p3 T q1 −q3 . q2 g2 = p2 q2 Z Z T Z T p2

Set k := p1 − p3 and l := q1 − q3 . Since f 2 ∈ D is not constant, k and l can not 1 be both zero. Therefore, k l f = f 1 ∈ D which shows that Z k T l divides f in Z T D and S f = ∅. 1 2. Assume that k l f is reducible in D, so there exist non-constant elements Z T g, h ∈ D such that 1 f = gh. (1) k Z Tl Then, f = (Z k T l g)h. By part 1, there exist non-negative integers k , l (not both 1 1 zero) such that Z k T l g = k l f or h = k l f (up to associates). We will Z T Z T prove that each case implies a contradiction. On the one hand, if the first equation 1 is true, then g = k+k l+l f ∈ D. So, (k, l)  (k + k , l + l ), but (k, l) is a Z T maximal element of S f . Thus, k = l = 0, which is a contradiction. On the other hand, the second equation is equivalent to say that Z k T l divides f . This implies 1 (k , l ) ∈ S f . After replacing h by k l f in equation (1) and simplifying we get Z T g = Z k −k T l −l and this implies that (k, l)  (k , l ) because g ∈ D. But again, for being (k, l) a maximal element of S f , we conclude that k = k and l = l and g = 1, which is another contradiction. 

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Based on Proposition 2.4, the next goal is to study expressions in D of the form 1 f , where f is an irreducible polynomial in R. ZkT l n 0 Assume that f = q0 + i=1 qi X αi Y βi Z γi T δi , where q0 ∈ k, qi ∈ k , (αi , βi , 4 γi , δi ) ∈ Z≥0 for i = 1, . . . , n 0 and (αi1 , βi1 , γi1 , δi1 ) = (αi2 , βi2 , γi2 , δi2 ) if i 1 = i 2 . The first fact is the following: Since Z k T l | f (in D) and either k or l is nonzero, 1 1 q0 = 0. Otherwise, k l = q0 Z −k T −l would be in D, which is a contradiction. Z T q0 Before continuing, let us fix some notation. By E and F we denote the sets E := {i ∈ {1, . . . , n 0 } : αi = βi = 0}

and

F := {1, . . . , n 0 } \ E.

So f can be written as f = p1 (Z , T ) + p2 (X, Y, Z , T ), where p1 =

i∈E

qi Z γi T δi

and

p2 =

qi X αi Y βi Z γi T δi .

i∈F

At first sight, it seems that either E or F may be empty (but not both). A second fact is that p2 = 0. Otherwise, since Z k T l divides f = p1 , γi ≥ k and δi ≥ l for every i ∈ E and f wouldn’t be irreducible. Observe that this second result implies F = ∅. Therefore, we can divide our study in two cases: p1 = 0 and p1 = 0. We will see that in any case f can be factored as a (finite) product of powers of irreducible elements in D. In order to apply Proposition 2.4 part 2, we should describe S f in detail and try to see whether any S f has a maximal element. In Theorem 3.1, we formally state these facts.

3 Main Results We start this section with a result upon which we will base to prove that D is in fact an LT domain. First of all, let us set up the following notation: k0 := min{γi : i ∈ E}, l0 := min{δi : i ∈ E}, and m 0 := min{αi + βi + δi : i ∈ F}.

Notice that at least one of the integers k0 and l0 is positive and m 0 is always a positive integer. Also, we set up the additional notation k1 := min{γi : i ∈ J } where J := {i ∈ F : αi + βi + δi = m 0 }. Notice that J is a non-empty set. Also, we denote by M f the set of maximal elements of S f .

On a Problem About Lowest Terms Domains Posed by D. D. Anderson

159

Fig. 2 S f when p1 = 0, m 0 > l0

Now we proceed to prove the theorem mentioned at the end of the previous section under the condition k0 > 0, l0 > 0. The cases (k0 > 0, l0 = 0) and (k0 = 0, l0 > 0) will be considered later on. n 0 qi X αi Y βi Z γi T δi be any irreducible polynomial in R Theorem 3.1 Let f = i=1 but reducible in D. Assume that k0 and l0 are positive integers. Then 1.

2.

⎧ ⎪ S ⎪ ⎪ 1 ⎪ ⎨S 2 Sf = ⎪S ⎪ ⎪ 3 ⎪ ⎩ S4

:= {0, . . . , k0 } × {0, . . . , l0 } \ {(0, 0)} := {0, . . . , k0 } × {0, . . . , m 0 } \ {(0, 0)}

if p1 = 0, m 0 > l0 if p1 = 0, m 0 ≤ l0 , k0 ≤ k1

:= S2 \ {(i, m 0 ) : k1 < i ≤ k0 } := (N × {0, . . . , m 0 − 1} \ {(0, 0)}) ∪ ({0, . . . , k1 } × {m 0 })

if p1 = 0, m 0 ≤ l0 , k0 > k1 if p1 = 0

⎧ {(k0 , l0 )} ⎪ ⎪ ⎪ ⎨{(k , m )} 0 0 Mf = ⎪{(k1 , m 0 ), (k0 , m 0 − 1)} ⎪ ⎪ ⎩ {(k1 , m 0 )}

if if if if

p1 p1 p1 p1

.

= 0, m 0 > l0 = 0, m 0 ≤ l0 , k0 ≤ k1 . = 0, m 0 ≤ l0 , k0 > k1 =0

3. f can be factored as a finite product of powers of irreducible elements in D. Proof

1 (a) Assume that p1 = 0. Let (k, l) be any element of S f . Then, Z k T l divides p1 , which implies γi ≥ k and δi ≥ l for every i ∈ E. So, k ≤ k0 and l ≤ l0 . Thus, (k, l) ∈ S1 . This proves the containment S f ⊆ S1 .

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R. O. Quintero Contreras

Fig. 3 S f when p1 = 0, m 0 < l0 , k0 < k1

Fig. 4 S f when p1 = 0, m 0 = l0 , k0 > k1

From now on, we divide the situation into three sub-cases: (a.1) m 0 > l0 . It is clear that αi + βi + δi ≥ l0 + 1 for every i ∈ F. Based on this condition, we claim that S f = S1 . Observe that  it is enough  to 1 1 k0 −k l0 −l T f ∈D prove that (k0 , l0 ) ∈ S f , since k l f = Z Z T Z k0 T l0 for every (k, l) ∈ P \ {(0, 0)} such that (k, l)  (k0 , l0 ). In fact, f can be factored as

On a Problem About Lowest Terms Domains Posed by D. D. Anderson

f =Z T k0

l0



qi Z

γi −k0

T

δi −l0

+

i∈E

= Z k0 T l0





161

 αi

βi

qi X Y Z

γi −k0

T

i∈F

qi Z γi −k0 T δi −l0 +

i∈E

δi −l0

 qi h i (X, Y, Z , T )

i∈F

= Z k0 T l0 (q1 + f 1 ),

(2)

  where q1 = i∈E qi Z γi −k0 T δi −l0 ∈ k[Z , T ]  D and f 1 = i∈F qi h i is an element in D, since h i ∈ D for every i ∈ F by Proposition 2.1 (parts 2 and 3), when δi − l0 ≥ 0; otherwise, we divide the situation in two sub-cases: (1) γi − k0 ≥ 0, in this case, h i also belongs to D, by Proposition 2.1 part 4, due to fact that αi + βi = αi + βi + δi − δi ≥ m 0 − δi > l0 − δi = −(δi − l0 ). (2) γi − k0 < 0, in this case, h i ∈ D as well, by part 5 of the same proposition, due to fact that αi + βi = αi + βi + δi − δi ≥ m 0 − δi ≥ l0 + 1 − δi = 1 − (δi − l0 ). In Fig. 2, the set S f has been framed and its maximal element has been circled. (a.2) m 0 ≤ l0 , k0 ≤ k1 . We start by checking that (k0 , m 0 ) belongs to S f . In fact,  

k0 m 0 γi −k0 δi −m 0 αi βi γi −k0 δi −m 0 f =Z T qi Z T + qi X Y Z T i∈E

= Z k0 T m 0



i∈F

qi Z γi −k0 T δi −m 0 +

i∈E

= Z k0 T m 0 (q2 + f 2 ),

 qi hˆ i (X, Y, Z , T )

i∈F

(3)

  where q2 = i∈E qi Z γi −k0 T δi −m 0 ∈ k[Z , T ]  D and f 2 = i∈F qi hˆ i is an element in D, since hˆ i ∈ D for every i ∈ F by Proposition 2.1 (parts 2 and 3), when δi − m 0 ≥ 0; otherwise, we consider two subcases: (1) i ∈ J , in this case, due to the fact that k0 ≤ k1 , γi − k0 ≥ 0, but αi + βi = αi + βi + δi − δi = m 0 − δi = −(δi − m 0 ), by Proposition 2.1 part 4, hˆ i ∈ D. (2) i ∈ F \ J , in this case αi + βi = αi + βi + δi − δi ≥ m 0 + 1 − δi = 1 − (δi − m 0 ) no matter what sign has γi − k0 , by Proposition 2.1 parts 4 and 5, hˆ i ∈ D as well. So far this proves the containment S2 ⊆ S f . Now we will prove that S f = S2 by checking that S f ∩ (S1 \ S2 ) = ∅. Of course, we are done if m 0 = l0 . Otherwise, let (k, l) ∈ S1 \ S2 and i ∈ J , then δi < αi + βi + δi = m 0 < l. So, δi − l < 0, but γi − k ≥ 0 because k ≤ k0 ≤ k1 , which (by Proposition 2.1 part 4) would imply that m 0 = (αi + βi ) + δi ≥ −(δi − l) + δi = l if (k, l) ∈ S f , which is a contradiction. Therefore, (k, l) ∈ / S f and we are done. In Fig. 3, the set S f has been framed and its maximal element has been circled.

162

R. O. Quintero Contreras

(a.3) m 0 ≤ l0 , k0 > k1 . We start by checking that (k1 , m 0 ) and (k0 , m 0 − 1) belong to S f . In fact, observe that f =Z T k1

m0



qi Z

γi −k1

T

δi −m 0

+

i∈E

=Z T k1

m0



 αi

βi

qi X Y Z

γi −k1

T

i∈F

qi Z

γi −k1

T

δi −m 0

+

i∈E

δi −m 0

 qi h¯ i (X, Y, Z , T )

i∈F

= Z k1 T m 0 (q3 + f 3 ),

(4)

  where q3 = i∈E qi Z γi −k1 T δi −m 0 ∈ k[Z , T ]  D and f 3 = i∈F qi h¯ i is an element in D, since h¯ i ∈ D for every i ∈ F by Proposition 2.1 (parts 2 and 3), when δi − m 0 ≥ 0; otherwise, we consider two subcases: (1) i ∈ J , in this case, by definition of k1 , γi − k1 ≥ 0, but αi + βi = αi + βi + δi − δi =m 0 − δi = − (δi − m 0 ), by Proposition 2.1 part 4, h¯ i ∈ D. (2) i ∈ F \ J , in this case αi + βi = αi + βi + δi − δi ≥ m 0 + 1 − δi = 1 − (δi − m 0 ) no matter what sign has γi − k1 , by Proposition 2.1 parts 4 and 5, h¯ i ∈ D as well. On the other hand, f = Z k0 T m 0 −1



qi Z γi −k0 T δi −(m 0 −1) +

i∈E

= Z T k0

m 0 −1



= Z T

m 0 −1

 qi X αi Y βi Z γi −k0 T δi −(m 0 −1)

i∈F

qi Z

γi −k0

i∈E k0

(q4 + f 4 ),

T

δi −(m 0 −1)

+

 qi h˜ i (X, Y, Z , T )

i∈F

(5)

 q4 = i∈E qi Z γi −k0 T δi −(m 0 −1) ∈ k[Z , T ]  D and f 4 = ˜ ˜ i∈F qi h i is an element in D, since h i ∈ D for every i ∈ F by Proposition 2.1 (parts 2 and 3), when δi − (m 0 − 1) ≥ 0; otherwise, we consider two sub-cases: (1) γi − k0 ≥ 0, in this case, due to the fact that αi + βi = αi + βi + δi − δi ≥ m 0 − δi > m 0 − 1 − δi = −(δi − (m 0 − 1)), by Proposition 2.1 part 4, h˜ i ∈ D. (2) γi − k0 < 0, in this case αi + βi = αi + βi + δi − δi ≥ m 0 − δi = 1 − (δi − (m 0 − 1)), so by Proposition 2.1 part 5, h˜ i ∈ D as well. So far this proves the containment S3 ⊆ S f . Now we will prove that S f = S3 by checking that S f ∩ (S1 \ S3 ) = ∅. In fact, let (k, l) ∈ S1 \ S3 . We consider two sub-cases: (1) m 0 = l0 . So k1 < k ≤ k0 and l = m 0 . Since k > k1 , there exists i 0 ∈ J such that γi0 − k < 0. Moreover, δi0 − l < αi0 + βi0 + δi0 − l = m 0 − l = 0. Thus, if (k, l) would belong to S f , by Proposition 2.1 part 5, m 0 = αi0 + βi0 + δi0 ≥ 1 − (δi0 − l) + δi0 = 1 + l, which is a contradiction. Therefore, (k, l) ∈ / S f and we are done with this case. (2) m 0 < l0 . For this case, we have three possibilities: (2.1) k1 < k ≤ k0 and l = m 0 . We repeat the argument employed in the previous case. (2.2) 0 ≤ k ≤ k1 and m 0 < l ≤ l0 . Since k ≤ k1 , there where 

On a Problem About Lowest Terms Domains Posed by D. D. Anderson

163

exists i 1 ∈ J such that γi1 − k ≥ 0. Moreover, δi1 − l < δi1 − m 0 < αi1 + βi1 + δi1 − m 0 = 0. Thus, if (k, l) would belong to S f , by Proposition 2.1 part 4, m 0 = αi1 + βi1 + δi1 ≥ −(δi1 − l) + δi1 = l, which is a contradiction. Therefore, (k, l) ∈ / S f and we are done with this case too. 2.3) k1 < k ≤ k0 and m 0 < l ≤ l0 . Since k1 < k0 , there exists i 2 ∈ J such that γi2 − k < 0. Moreover, δi2 − l < δi2 − m 0 < αi2 + βi2 + δi2 − m 0 = 0. Thus, if (k, l) would belong to S f , by Proposition 2.1 part 5, m 0 = αi2 + βi2 + δi2 ≥ 1 − (δi2 − l) + δi2 = 1 + l, which is a contradiction. Therefore, (k, l) ∈ / S f and we are done with this case as well. In Fig. 4, the set S f has been framed and its two maximal elements have been circled. (b) Now, we assume that p1 = 0. The first additional condition that f satisfies is that there are subscripts i 1 and i 2 in F (they might coincide, but it is not relevant) such that γi1 = δi2 = 0. Otherwise, f would be reducible in R.  Since (k, l) ∈ S f , f can be expressed as f = Z k T l i∈F qi X αi Y βi Z γi −k T δi −l . Based on Proposition 2.1, additional conditions can emerge over the exponents of f if at least one γi − k or one δi − l is negative, which will always happen due to the fact γi1 = δi2 = 0. Now, we split F into two disjoint subsets J (a non-empty set) and K = F \ J . The second condition is that l cannot be greater than m 0 . Otherwise, l = m 0 + l for some positive integer l and f would be factored as f = ZkT l





qi X αi Y βi Z γi −k T δi −(m 0 +l ) +

i∈J

= ZkT l



i∈J

 qi

 qi X αi Y βi Z γi −k T δi −l

i∈K

X T

 αi  βi

Y γi −k −l αi βi γi −k δi −l Z T + qi X Y Z T , T i∈K 

which is not possible because the monomials

X T

αi  βi Y Z γi −k T −l ∈ / T

D for every i ∈ J . Now, let us see what other condition we can impose on k. From now on, we proceed by dividing the situation into two sub-cases: (b.1) l = m 0 . On the one hand, if i 1 ∈ J , then k1 = 0 and this implies k = 0.  αi1  βi 1 Y X Z −k ∈ / D, which is a conOtherwise, the monomial T T tradiction. On the other hand, if i 1 ∈ / J , then 0 ≤ k ≤ k1 . Otherwise, there would exist a subscript i ∈ J such that γi < k and the monomial  αi  βi Y X Z γi −k ∈ / D, which would be another contradiction. So T T far, we have proved that l = m 0 ⇒ k ≤ k1 (See Fig. 5). This is the third condition. (b.2) l < m 0 . In this case, we will prove that k could be any non-negative integer (See Fig. 5). This is the fourth and last condition. Its proof is

164

R. O. Quintero Contreras

straightforward. All that we need to prove, by Proposition 2.1, is that any monomial of the form gi = X αi Y βi Z γi −k T δi −l belongs to D for every i ∈ F. In fact, this is true, by Proposition 2.1 (parts 2 and 3), when δi − l ≥ 0. Otherwise, we consider two sub-cases: (1) γi − k ≥ 0, in this case αi + βi = αi + βi + δi − δi ≥ m 0 − δi > l − δi = −(δi − l) and by Proposition 2.1 part 4, gi ∈ D. (2) γi − k < 0, in this case αi + βi = αi + βi + δi − δi ≥ m 0 − δi ≥ l + 1 − δi = 1 − (δi − l) and by Proposition 2.1 part 5, gi ∈ D as well. So far, we have proved that S f ⊆ S4 . Notice that the reverse inclusion is also true because all the formulas shown work backwards, due to the way that the integers k1 and m 0 were defined (See Fig. 5). This completes the proof of part 1. To conclude, as we did with the case p1 = 0, we present the factorization of f when (k, l) = (k1 , m 0 ). In fact,

f = Z T k1

m0



αi

qi X Y Z

i∈J

= Z k1 T m 0



βi

 qi

i∈J

X T

γi −k1

αi 

Y T

T

βi

δi −m 0

+

 αi

βi

qi X Y Z

γi −k1

T

δi −m 0

i∈K

Z γi −k1 +

 qi h i (X, Y, Z , T )

i∈K

= Z k1 T m 0 ( p5 + f 5 ),

(6)

   X αi Y βi γi −k 1 evidently belongs to D and Z where p5 = i∈J qi T T  f 5 = i∈K qi h i is also another element in D, because h i = X αi Y βi Z γi −k1 T δi −m 0 belongs to D for every i ∈ K . Notice that in Eq. (6), at least one of the exponents of Z of the first term of the third factor is zero. 2. Clearly, each of the pairs (k0 , l0 ) (when p1 = 0, m 0 > l0 ), (k0 , m 0 ) (when p1 = 0, m 0 ≤ l0 , k0 ≤ k1 ), (k1 , m 0 ) and (k0 , m 0 − 1) (when p1 = 0, m 0 ≤ l0 , k0 > k1 ), and (k1 , m 0 ) (when p1 = 0) is a maximal element of S f (See Figs. 2, 3, 4, and 5). 3. By part 2 of Proposition 2.4, each of the factorizations given in Eqs. (2), (3), (4), (5), and (6) is a finite product of powers of irreducible elements in D.  



Remark The points 1 and 2 given in Theorem 3.1, when either k0 = 0 or l0 = 0, are the following: (a) If k0 > 0 and l0 = 0, then  1. S f =

 S1 = {(1, 0), . . . , (k0 , 0)} if p1 = 0 (k0 , 0) if p1 = 0 , 2. M f = . S4 (k1 , m 0 ) if p1 = 0 if p1 = 0

On a Problem About Lowest Terms Domains Posed by D. D. Anderson

165

Fig. 5 S f when p1 = 0

(b) If k0 = 0 and l0 > 0, then ⎧ ⎧ if p1 = 0, m 0 > l0 ⎪ ⎪ ⎨ S1 = {(0, 1), . . . , (0, l0 )} if p1 = 0, m 0 > l0 ⎨(0, l0 ) 1. S f = S2 = {(0, 1), . . . , (0, m 0 )} if p1 = 0, m 0 ≤ l0 , 2. M f = (0, m 0 ) if p1 = 0, m 0 ≤ l0 . ⎪ ⎪ ⎩ ⎩ S4 if p1 = 0 (k1 , m 0 ) if p1 = 0

From now on, we will use the symbol f [k,l] to denote the irreducible element in D 1 ZkT l

f,

when f is an irreducible polynomial in R but reducible in D, and [k, l] is one of the maximal elements of S f as indicated in Theorem 3.1 and the previous Remark. In particular, to avoid ambiguities, [k, l] = (k1 , m 0 ) when S f = S3 in Theorem 3.1. The next and final result establishes that the integral domain D is an LT domain, which is not an RLT domain. This will answer negatively the first of the two questions posed at the end of paper [2] and recalled in the Introduction. As mentioned in the Introduction, it was proved in [2, Example 3] that D is not an RLT domain. With this last result we are showing that the class of RLT domains is properly contained in the class of LT domains. Theorem 3.2 D is an LT domain. Proof Let g, h be two elements in D ∗ . By Lemma 2.3, there exist two couples of non-negative integers p1 , q1 and p2 , q2 such that Z p1 T q1 g and Z p2 T q2 h belong to R.

166

R. O. Quintero Contreras

Set p := max{ p1 , p2 }, q := max{q1 , q2 } and s := Z p T q . It is clear that sg, sh ∈ R. But R = k[X, Y, Z , T ] is a UFD. Then, there exist f 1 , . . . , fr a set of pairwise nonassociate irreducibles in R such that sg = f 1a1 · · · frar , sh = f 1b1 · · · frbr , with ai , bi ∈ N, and 

gcd R [sg, sh] = d R :=

f i min{ai ,bi } .

i∈{1,...,r }

Let I1 := {i ∈ {1, . . . , r } : f i is irreducible in D} and I2 := {1, . . . , r } \ I1 . From now on, we split the situation into two cases: 1. Neither Z nor T belongs to { f i : i ∈ I1 }. As an element of D, by Theorem 3.1, sg can be further factorized as follows: sg =



f i ai ·

i∈I1

=

 

i∈I1

=



f i ai

i∈I2

fi · ai

i∈I1

=

 

(Z ki T li f i[k ,l ] )ai i i

i∈I2

f i ai ·



(Z ki ai T li ai f i ai ) [ki ,li ]

i∈I2

Z ki ai ·

i∈I2

= Z m 1 T n1 ·



T li ai ·

i∈I2



 i∈I2

ki ai and n 1 :=

f i ai ·

i∈I1

f i ai ·

i∈I1

where m 1 :=



 i∈I2

 i∈I2

f i ai

[ki ,li ]

f i ai

[ki ,li ]

,



sh = Z m 2 T n 2

li ai . Similarly, we get   · f i bi · f i bi , i∈I2

i∈I1

i∈I2

[ki ,li ]

  where m 2 := i∈I ki bi and n 2 := i∈I li bi . 2 2 The next step is to prove that gcd D [sg, sh] = d := Z min{m 1 ,m 2 } · T min{n 1 ,n 2 } ·

 i∈I1

f i min{ai ,bi } ·

 i∈I2

f i min{ai ,bi } . [ki ,li ]

(7) In fact, clearly d|sg, sh in D. Now, assume that d ∈ D is any other common divisor of sg and sh (in D). By Lemma 2.3, there exist non-negative integers u and v such that Z u T v d ∈ R. Thus, Z u T v d |Z u T v sg, Z u T v sh in R. Then, Z u T v d |Z u T v d R because

On a Problem About Lowest Terms Domains Posed by D. D. Anderson

167

Z u T v gcd R [sg, sh] = gcd R [Z u T v sg, Z u T v sh] and

Z u T v d = Z u 1 T v1 f 1c1 · · · frcr

(up to associates) for some non-negative integers u 1 ≤ u, v1 ≤ v, and ci ≤ min{ai , bi } for every i ∈ {1, . . . , r }. Then, Z u T v d = Z u 1 T v1



f i ci ·

i∈I1 v1

m

f i ci

i∈I2

=Z T Z T · u1



n

 i∈I1

= Z m+u 1 T n+v1 ·





f i ci ·

f i ci

[ki ,li ]

i∈I2

f i ci ·

i∈I1



f i ci

[ki ,li ]

i∈I2

,

(8)



 ki ci ≤ min{m 1 , m 2 } and n := i∈I li ci ≤ min{n 1 , n 2 }. Due  2 c c to Eq. (8), we know that Z m+u 1 T n+v1 · i∈I f i i · i∈I f i[ki ,l ] is divisible by i i 1 2 Z u T v in D because d ∈ D. After canceling the factor Z u T v in D (D is an integral domain), we get that where m :=

i∈I2

d = Z m+u 1 −u T n+v1 −v ·



f i ci ·

i∈I1



f i ci

[ki ,li ]

i∈I2

.

Now observe that the exponents of Z and T satisfy the inequalities m + u 1 − u ≤ min{m 1 , m 2 } and n + v1 − v ≤ min{n 1 , n 2 }. We claim that both exponents are non-negative. Otherwise, Z or T should divide one of the irreducibles in the set { f i : i ∈ I2 }, but this fact contradicts the way in which ki and li were chosen, for all i ∈ I2 . Therefore, d |d and Eq. (7) is true. 2. Otherwise, we have three possible cases: (a) Z ∈ A := { f i : i ∈ I1 }, say Z = f i1 , and T ∈ / A. (b) Z ∈ / A and T ∈ A, say T = f i2 . (c) Z , T ∈ A, say Z = f i1 and T = f i2 (i 1 = i 2 ). If we define the integers m 1 , n 1 , m 2 , and n 2 as follows:   ai1 + i∈I ki ai if Z ∈ A 2 m 1 :=  if Z ∈ / A i∈I ki ai

2



bi + m 2 := 1

i∈I2

 i∈I2

ki bi

ki bi if Z ∈ A if Z ∈ / A

  ai2 + i∈I li ai if T ∈ A 2 , n 1 :=  , if T ∈ / A i∈I li ai

2

  bi2 + i∈I li bi if T ∈ A 2 , and n 2 :=  , l b if T ∈ / A i i i∈I

2

168

R. O. Quintero Contreras

it is straightforward to check that sg and sh can be factored in D as follows: ⎧ m n  ai  ai 1 1 if Z ∈ A, T ∈ / A ⎪ ⎨ Z T · i∈I1 \{i1 } f i · i∈I2 f i[ki ,li ] ai ai m 1 n1 if Z ∈ / A, T ∈ A , sg = Z T · i∈I \{i } f i · i∈I f i[k ,l ] i i 2 ⎪ ai  ai ⎩ m 1 n 1  1 2 Z T · i∈I \{i ,i } f i · i∈I f i[k ,l ] if Z ∈ A, T ∈ A i i 1 1 2 2 ⎧   b b m 2 n2 i i ⎪ if Z ∈ A, T ∈ / A ⎪ ⎨ Z T · i∈I1 \{i1 } f i · i∈I2 f i[ki ,li ] bi bi m 2 n2 sh = Z T · i∈I \{i } f i · i∈I f i[k ,l ] if Z ∈ / A, T ∈ A . i i 1 2 2 ⎪ ⎪ bi  bi ⎩ Z m 2 T n 2 ·  f · f if Z ∈ A, T ∈ A i∈I1 \{i 1 ,i 2 }

i∈I2

i

i [k ,l ] i i

The next step is to prove that







gcd D [sg, sh] = d := Z min{m 1 ,m 2 } · T min{n 1 ,n 2 } · d ·



f i min{ai ,bi } ,

i∈I2

where

⎧ min{ai ,bi } ⎪ ⎪ ⎨i∈I1 \{i1 } f i min{ai ,bi } d := i∈I1 \{i 2 } f i ⎪ ⎪ min{ai ,bi } ⎩ i∈I \{i ,i } f i 1

1

2

[ki ,li ]

if Z ∈ A, T ∈ / A if Z ∈ / A, T ∈ A . if Z ∈ A, T ∈ A

Of course, the proof follows in the same fashion as the previous case where d R is expressed as ⎧ min{ai ,bi } min{ai1 ,bi1 }  ⎪ · i∈{1,...,r }\{i } f i ⎪ ⎨Z 1  min{ai ,bi } d R := T min{ai2 ,bi2 } · i∈{1,...,r }\{i } f i 2 ⎪ ⎪ ⎩ Z min{ai1 ,bi1 } T min{ai2 ,bi2 } · 

i∈{1,...,r }\{i 1 ,i 2 }

if Z ∈ A, T ∈ / A fi

min{ai ,bi }

if Z ∈ / A, T ∈ A . if Z ∈ A, T ∈ A

So we have proved that d = gcd D [sg, sh] and finally  gcd D

 sg sh , = 1. d d

Therefore, D is an LT domain by [2, Theorem 1, part 1].



Acknowledgements First, my gratitude goes to Professor Daniel Anderson, who gave me access to the question formulated by him in one of his most recent papers on Divisibility in integral domains. After talking several times in his office about the nature of the domain under question and noting his clear belief that that domain was the answer to the aforementioned question, I made the decision of converting his problem into mine. Second, I also want to give my thanks to Professor Ryan Kinser for helping me to get a better presentation of this paper both on its mathematical content as well on its appearance. Finally, my thanks to Professor Teodoro Lara for his valuable suggestions.

On a Problem About Lowest Terms Domains Posed by D. D. Anderson

169

References 1. H.E. Adams, Factorization-prime ideals in integral domains. Pac. J. Math. 66(1) (1976) 2. D.D. Anderson, E. Hasse, Reducing fractions to lowest terms, in Rings, Polynomials, and Modules. ed. by M. Fontana, S. Frisch, S. Glaz, F. Tartarone, P. Zanardo (Springer, Cham, 2017) 3. T. Becker, V. Weispfenning, H. Kredel, Gröbner Bases: A Computational Approach to Commutative Algebra (Springer, New York, 1998). (Corrected second printing) 4. T.W. Hungerford, Algebra (Springer, New York, 1974)

Regularity and Related Properties in Tensor Products of Algebras Over a Field S. Kabbaj and F. Suwayyid

Abstract It has been established, quite recently, that a tensor product of k-algebras, if Noetherian, it inherits the concepts of Cohen–Macaulay ring, Gorenstein ring, and complete intersection ring. However, it is well known that a tensor product of two extension fields is not necessarily regular. In 1965, Grothendieck showed that it is, in fact, regular if one of the two fields is separable and finitely generated. Since then, many articles appeared in the literature featuring partial results on this topic. The problem on the transfer or defect of regularity in more general settings remains elusively open. This survey paper tracks and studies some of these works which deal with this problem; precisely, we shed brighter light on the main results and examples published, chronologically, in [6, 28, 7]. Keywords Tensor product of k-algebras · Regular ring · Embedding dimension · Complete intersection ring · Gorenstein ring · Cohen-Macaulay ring

1 Introduction Through this paper, all rings are commutative with unity and k will denote a field. Given a Noetherian local ring (R, m), the embedding dimension of R, denoted by embdim(R), is the dimension of m / m2 as an (R/ m)-vector space; equivalently, the cardinality of a minimal generating set (called, minimal basis) for m. The ring R is regular if its embedding dimension coincides with its Krull dimension. Recall, from [8], that “the notion of regularity was initially introduced by Krull and became prominent when Zariski showed that a local regular ring corresponds to a smooth point on an algebraic variety. Later, Serre proved that a ring is regular if and only if it has finite global dimension. This allowed to see that regularity is stable under S. Kabbaj (B) · F. Suwayyid Department of Mathematics, KFUPM, Dhahran 31261, Saudi Arabia e-mail: [email protected] F. Suwayyid e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. Badawi and J. Coykendall (eds.), Rings, Monoids and Module Theory, Springer Proceedings in Mathematics & Statistics 382, https://doi.org/10.1007/978-981-16-8422-7_10

171

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localization and then the definition got globalized.” Then, R is called a complete intersection ring if its m-completion is equal to the quotient of a local regular ring modulo an ideal generated by a regular sequence; and R is called Gorenstein if its injective dimension is finite. Finally, recall that R is Cohen–Macaulay if the height of m coincides of its grade. The definitions of all these notions carry over to localizations with respect to the prime ideals. For more details, we refer to the classic books [9, 10, 12, 14, 20, 21, 24, 32]. We have, for a ring R, the diagram of implications:  R regular ring  e.g., k[[X, Y ]] ⇓ R complete intersection ring  k[[X, Y ]]  e.g., (X Y ) ⇓ R Gorenstein ring   k[[X, Y, Z ]] e.g., (X Y, X Z , Y Z , X 2 − Y 2 , X 2 − Z 2 ) ⇓ R Cohen–Macaulay ring  k[X, Y ]  e.g., (X 2 , X Y, Y 2 ) ⇓ R Noetherian ring  k[X, Y ]  e.g., (X 2 , X Y ) It is, now, established that a Noetherian tensor product of k-algebras inherits the notions of Cohen–Macaulay ring, Gorenstein ring, and complete intersection ring [6, 17, 25, 26, 28]. However, it is well known that a tensor product of two extension fields is not necessarily regular. Note that tensor products of k-algebras subject to the aforementioned geometric concepts were used to delimit or widen the scope of validity of some homological conjectures [18, 19]. In 1965, Grothendieck proved that, given a field k, “the tensor product of two extension fields (of k) is regular provided one of them is finitely generated and separable over k” [16]. In 1969, Watanabe et al. investigated the transfer of Gorensteiness and turned out that “the tensor product of two regular k-algebras, one of them being finitely generated, is a complete intersection [31].” Since then, it has been proved that “a Noetherian tensor product of k-algebras A ⊗k B inherits from A and B the notions of locally complete intersection ring, Gorenstein ring, and Cohen–Macaulay ring [6, 17, 25, 26, 28].” In particular, “K ⊗k L is a locally complete intersection ring, for any two extension fields K and L of k such that K ⊗k L is Noetherian [28].” Recently, it has been proved that “a Noetherian tensor product of two k-algebras A and B is regular if and only if so are A and B in the special case where k is perfect (i.e., every (algebraic) extension of k is separable) [17, 28].” Moreover, it is established in [7] that, under geometrical regularity (i.e., A ⊗k F is regular for every

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finite extension F of k), “A ⊗k B is regular if and only if B is regular and A ⊗k B is Noetherian”, which allowed to establish transfer results for tensor product of two extension fields of k subject to separability conditions. The aforementioned works draw on basic results and use techniques and methods from prime spectra and dimension theory. For early and recent developments on these topics, we refer to [3–5, 7, 25–27, 29, 30] for special settings of tensor products of algebras, and to [1, 11, 15, 19–22] for the general context of commutative rings. The problem on the transfer of regularity in more general settings remains elusively open. This survey paper tracks and studies some of the main works which deal with this problem; precisely, we shed brighter light on the main results and examples published, chronologically, in [6] (co-authored with S. Bouchiba), [28] (authored by M. Tousi and S. Yassemi), and [7] (co-authored with S. Bouchiba).

2 Preliminaries In order to make this paper reasonably self-contained, this section recalls some notations and definitions as well as some basic relevant results, most of which deal particularly with various aspects of tensor products of algebras. These will be used through this paper implicitly without specific mention. Let R be a ring, S a multiplicatively closed subset of R, and M an R-module. It is customary to denote the localization of M with respect to S by S −1 M, and if S is a complement of a prime ideal P in R, the localization is denoted by M P instead. If no ambiguity arises, we denote either localizations by M S or M P . If T is another multiplicatively closed subset and T  is its image in R S and if S  is the image of S in RT , then (M S )T  ∼ = M ST = MT S = (MT ) S  [2, Exercise 3, p. 43]. We usually drop the apostrophe, even though the meaning might be different, but the algebraic properties remain unchanged. Further, if S consists only of units in R, then obviously ∼ RS ∼ = R and, as a consequence, if S ⊆ T , then RT ∼ = (R  xT ) S = (R S )T .  It is known that the prime ideals of R S have the form s | x ∈ p, s ∈ S , for some unique prime p in R [2, Proposition 3.11]. It is usually denoted by S −1 p and if S is a complement of a prime q, then it is denoted by p Rq . We also use the notation p S or pq as in [20, Sect. 1.4]. Throughout, (R, m) or (R, m, k) indicates that R is a local ring with maximal ideal m and k := R/ m is its residue field. Frequently, it is denoted by κ(m) and, for any prime ideal p of R, R p / p R p is denoted by κ R ( p). The module M is flat if, for every injective R-map f : E  −→ E, the induced map f ⊗ R 1 : E  ⊗ R M −→ E ⊗ R M is injective; and M is said to be faithfully flat if it is flat and whenever f ⊗ R 1 is injective (resp., surjective), f is injecitve (resp., surjective). In the sequel, Spec(R) (resp., Max(R)) will denote the set of all prime (resp., maximal) ideals of R.

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Proposition 2.1 ([21, Theorem 7.2]) “Let R be a ring and M an R-module. Then, the following statements are equivalent: (1) M is faithfully flat; (2) M is flat and, for every nonzero R-module N , M ⊗ R N = 0; (3) M is flat and, for each m ∈ Max(R), m M = M.” As a consequence of this proposition, free modules are faithfully flat, since if M is free, then M ⊗ R N is just a sum of copies of N . Let f : A −→ B be a ring homomorphism and let q ∈ Spec(B) with p := f −1 (q), which is usually (and, abusively, even if f is not one-to-one) denoted by A ∩ q. Then, f induces two natural ring homomorphisms g : A p −→ Bq , defined B A −→ , defined by h(a) = f (a). The extension of p to , and h : by g( as ) = ff (a) (s) p q B is denoted by p B := f ( p)B. Notice that we always have the following natural isomorphisms Bp ∼ = κ A ( p) ⊗ A B p Bp and hence

Bq ∼ = κ( p p ) ⊗ A p Bq p Bq

as, obviously, f (A \ p) ⊆ B \ q. The ring κ A ( p) ⊗ A B is called the fiber ring over p. Fiber rings over prime ideals play a primordial role in the study of the embedding dimension and Krull dimension of tensor products of algebras. Now, if A, B are R-algebras and S, S  are multiplicatively closed subsets of A and B, respectively, then T := {s ⊗ R s  | s ∈ S, s  ∈ S  } is a multiplicatively closed subset of A ⊗ R B with (A ⊗ R B)T ∼ = A S ⊗ R BS . Let A, B be k-algebras, and I , J ideals of A and B, respectively. Then, by faithful flatness, we have B A ⊗k B A ∼ (1) = ⊗k I ⊗k B + A ⊗k J I J and if I is proper in A then so is I ⊗k B in A ⊗k B. Also, throughout, we will identify A and B with their respective images in A ⊗k B. Finally, recall that f : A −→ B is said to be flat if the A-algebra B is flat over A (i.e., actually over f (A)); and if (A, m) and (B, n) are local rings, then f is said to be a local homomorphism if f (m) ⊆ n. Proposition 2.2 ([21, Theorem 15.1]) “Let f : A −→ B be a flat homomorphism of Noetherian rings and let q ∈ Spec(B) with p := A ∩ q. Then: dim(Bq ) = dim(A p ) + dim(κ A ( p) ⊗ A B p ).”

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An element x of R is a zero-divisor on M if there exists a nonzero element m of M such that xm = 0. Equivalently, the R-map f x : m −→ xm is not injective. We denote the collection of zero-divisors of M by Z(M). Recall that Z(M) is a union of primes and this union is unique if it is taken over the maximal primes among them [20, p. 34]. A nonzero-divisor of M is also called a regular element of M. An associated prime of M is a prime p of R such that there exists a nonzero element m of M with p = (0 : m) := {r ∈ R | r m = 0}. If R is Noetherian and M is finitely generated over R, the maximal primes of M are finite and thus Z(M) is a unique finite union of maximal primes of M, each of which is an associated prime of M [20, Theorem 80]. A sequence x1 , x2 , . . . , xn of elements of R is said to be regular on M or an M-sequence, provided (x1 , x2 , . . . , xn )M = M and xi+1 is regular on

M , for each i = 1, . . . , n − 1. (x1 , x2 , . . . , xi )M

Lemma 2.3 ([20, Theorem 116]) “Let R be a ring and M an R-module. Let x1 , . . . , xn ∈ R and 1 ≤ i ≤ n − 1. Then, the following statements are equivalent: (1) x1 , . . . , xn is an M-sequence; (2) x1 , . . . , xi is an M-sequence and xi+1 , . . . , xn is an

M -sequence.” (x1 , . . . , xi )M

Note that a regular sequence x1 , x2 , . . . , xn generates a strictly increasing sequence of ideals, (x1 )  (x1 , x2 )  · · ·  (x1 , x2 , . . . , xn ). Therefore, in Noetherian settings, regular sequences saturate. Northcott and Rees [23] proved that, given an Noetherian ring R, a finitely generated R-module M, and an ideal I of R with I M = M, then any two maximal M-sequences in I have the same length [20, Theorem 121].   M (cf. Then, an M-sequence in I is maximal if and only if I ⊆ Z (x1 , x2 , . . . , xn )M [20, Appendix 3-1, p. 101]). This gives rise to the concept of grade. Definition 2.4 “Let R be a Noetherian ring, M a finitely generated R-module, and I an ideal of R with I M = M. The length of a maximal M-sequence in I is called the grade of I on M, denoted G(I, M).” If I ⊆ Z(M), we write G(I, M) = 0; and if M = R we just write G(I ) instead. Some authors use the term ‘depth’ instead of ‘grade’ (cf. [21]). Proposition 2.5 ([20, Theorem 132]) “Let R be a Noetherian ring, M a finitely generated R-module, and I ⊆ J proper ideals of R with J M = M. Then: (1) G(I, M) ≤ G(J, M). (2) G(I ) ≤ ht(I ).” Definition 2.6 ([20, Theorems 136]) “A Noetherian ring R is Cohen–Macaulay if it satisfies any one of the following equivalent conditions:

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(1) G(m) = ht(m), for each m ∈ Max(R); (2) G(I ) = ht(I ), for each ideal I of R.” The Cohen–Macaulay property is stable under localization. In fact, it is a local property, as seen below. Theorem 2.7 ([20, Theorems 139 and 140]) “Let R be a Noetherian ring and let S be a multiplicatively closed subset S of R. Then: (1) If R is Cohen–Macaulay, then so is R S . (2) R is Cohen–Macaulay if and only if Rm is Cohen–Macaulay, for each m ∈ Max(R) if and only if R p is Cohen–Macaulay, for each p ∈ Spec(R)” Finally, it is worthwhile recalling that Cohen–Macaulay property is stable under polynomial and power series rings, as stated below. Proposition 2.8 ([20, Theorems 151 and 157]) “A Noetherian ring R is Cohen– Macaulay if and only if so is R[x] (resp., R[[x]]).” If (R, m) is local, then we write G(R) := G(m) and is called the grade of R [20]. For local flat homomorphisms, fiber rings over prime ideals are involved as follows: Proposition 2.9 ([21, Corollary, p. 181]) “Let f : (A, m) −→ (B, n) be a local flat homomorphism of Noetherian rings. Then: (1) G(B) = G(A) + G(κ(m) ⊗ A B). (2) A and κ(m) ⊗ A B are Cohen–Macaulay if and only if so is B.” Let (R, m, k) be a Noetherian local ring. Hence, m is finitely generated and so m / m2 is a finite-dimensional vector space over k. The embedding dimension of R is defined as embdim(R) := dimk (m / m2 ); which is equal to the cardinality of a minimal generating set (called, minimal basis) for m. We always have G(R) ≤ dim(R) ≤ embdim(R). Definition 2.10 “A Noetherian local ring R is regular if embdim(R) = dim(R). A Noetherian ring R is regular, if Rm is regular, for each m ∈ Max(R).” The transfer of regularity through flat local homomorphisms is of great significance for the study of regularity in tensor products of algebras. The following three relevant basic results will be used most often in the sequel. Theorem 2.11 ([21, Theorem 23.7]) “Let f : (A, m) −→ (B, n) be a local flat homomorphism of Noetherian rings. (1) If B is regular, then so is A. (2) If A and κ A (m) ⊗ A B are regular, then so is B. (3) Assume m B = n. Then, A is regular if and only if so is B.”

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The first result, on the transfer of regularity to the tensor product of two extension fields, is due to Grothendieck (1965) [16]. Lemma 2.12 ([16, Lemma 6.7.4.1]) “Let K and L be two extension fields of k. If either K or L is finitely generated and K is separable over k, then K ⊗k L is regular.” The injective dimension of an R-module M, denoted id R (M), is defined as the smallest number n such that there exists an injective resolution 0 → M → E0 → E1 → · · · → En and we write id R (M) = n. If no such resolution exists, we define id R (M) = ∞. Obviously, M is injective if and only if id R (M) = 0. For more details on this homological invariant, we refer to [24]. Definition 2.13 “A Noetherian local ring R is Gorenstein if id(R) < ∞. A Noetherian ring R is Gorenstein, if Rm is Gorenstein, for each m ∈ Max(R).” completion of a local ring (R, m) is defined as the inverse limit of   Then m-adic R/ m n≥0 ; that is,  = lim R . R ←− mn  with maximal ideal m  and =mR If (R, m) is a Noetherian local ring, then so is R  is flat. For instance, the power series ring the natural ring homomorphism R → R  k[[x]] is the xk[x](x) -adic completion of the ring k[x](x) ; that is, k[x] (x) = k[[x]], where k is a field. For more details on the m-adic topology, we refer to [2, Chap. 10]. Definition 2.14 “A Noetherian local ring (R, m) is a complete intersection if its  is equal to the quotient of a local regular ring modulo an ideal m-completion R generated by a regular sequence. A Noetherian ring R is a locally complete intersection, if Rm is a complete intersection, for each m ∈ Max(R).” We close this preliminary with some recalls on Gorenstein and complete intersection rings. Proposition 2.15 ([31, Corollary 1] and [21, Theorem 23.5]) “If a Noetherian local ring R is Gorenstein (resp., complete intersection), then so are R[x] and R[[x]].” For the Gorenstein and complete intersection notions, the fiber rings are involved as follows: Theorem 2.16 ([31, Theorem 1] and [12, Remark 2.3.5]) “Let f : (A, m) −→ (B, n) be a local flat homomorphism of Noetherian rings. Then: (1) A and κ(m) ⊗ A B are Gorenstein if and only if so is B.

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(2) A and κ(m) ⊗ A B are complete intersection if and only if so is B.” Proposition 2.17 ([31, Corollary 2] and [21, Theorem 23.6]) “Let A be a Gorenstein (resp., complete intersection) ring containing k and K a finitely generated extension field of k. Then, A ⊗k K is a Gorenstein (resp., complete intersection) ring.” Theorem 2.18 ([31, Theorem 2]) “Let B and C be A-algebras, with B being flat over A and C finitely generated over A. Then: (1) If A, B and C are Gorenstein rings, then B ⊗ A C is a Gorenstein ring. (2) If B is a complete intersection ring and A and C are regular, then B ⊗ A C is a complete intersection ring.”

3 Tensor Products of Cohen–Macaulay k-Algebras This section deals with the transfer of Cohen–Macaulayness to Noetherian tensor products of k-algebras. For this purpose, we first examine the grade and height of some special ideals which play a crucial role within the ideal structure of a tensor product of k-algebras. The first main theorem investigates the grade in Noetherian tensor products. Theorem 3.1 ([6, Theorem 1.1]) “Let A, B be two k-algebras such that A ⊗k B is Noetherian and I, J two proper ideals of A, B, respectively. Then: (1) G(I ⊗k B) = G(I ). (2) G(I ⊗k B + A ⊗k J ) = G(I ) + G(J ). (3) G(I ⊗k J ) = Min(G(I ), G(J )).” The proof of this result draws on the following lemmas. Lemma 3.2 Let A, B be two k-algebras and let x ∈ A. Then, x is an A-sequence if and only if x ⊗k 1 is an (A ⊗k B)-sequence. Proof Consider the linear map f : A → A, a → xa and note that B is faithfully flat (since a k-vector space). So, x is an A-sequence if and only if f is injective but not surjective if and only if f ⊗k 1 is injective but not surjective if and only if x ⊗k 1 is  an (A ⊗k B)-sequence. Recall that a regular sequence x1 , . . . , xn is said to be permutable if any permutation of the xi ’s is also regular. Lemma 3.3 Let A, B be two k-algebras, x an A-sequence, and y a B-sequence. Then, x ⊗k 1, 1 ⊗k y is a permutable (A ⊗k B)-sequence.

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Proof By Lemma 3.2, x ⊗k 1 is an (A ⊗k B)-sequence and 1 ⊗k y is an (

179

A ⊗k B)(x)

A A ⊗k B ⊗k B ∼ . Therefore, by Lemma 2.3, x ⊗k 1, 1 ⊗k y is = (x) (x) ⊗k B an (A ⊗k B)-sequence. Likewise, starting with y yields 1 ⊗k y, x ⊗k 1 is an (A ⊗k B)-sequence.  sequence. But,

Lemma 3.4 Let A, B be two k-algebras. If x1 , . . . , xn is an A-sequence, and y1 , . . . , ym is a B-sequence, then x1 ⊗k 1, . . . , xn ⊗k 1, 1 ⊗k y1 , . . . , 1 ⊗k ym is an (A ⊗k B)-sequence. Moreover, if x1 , . . . , xn and y1 , . . . , ym are permutable, so are their respective images. Proof First, we show that x1 ⊗k 1, . . . , xn ⊗k 1 is an (A ⊗k B)-sequence. Lemma 3.2 handles the case n = 1. We proceed by induction on n . Consider the ideal In−1 := A A ⊗k B ⊗k B ∼ , for any ideal I of A. (x1 , x2 , . . . , xn−1 ). Again, recall that = I I⊗B A A ⊗k B Since x¯n is an -sequence, x¯n ⊗k 1 is an - sequence. It follows, In−1 In−1 ⊗k B by induction, that x1 ⊗k 1, . . . , xn ⊗k 1 is an (A ⊗k B)-sequence. Similar arguA ments shows that 1 ⊗k y1 , . . . , 1 ⊗k ym is an ⊗k B-sequence. By Lemma 2.3, In x1 ⊗k 1, . . . , xn ⊗k 1, 1 ⊗k y1 , . . . , 1 ⊗k ym is an (A ⊗k B)-sequence. Permutability follows easily.  Lemma 3.5 ([6, Lemma 1.2]) Let A, B be two k-algebras and let x1 , . . . , xn and y1 , . . . , yn be two permutable sequences over A and B, respectively. Then, x1 ⊗ y1 , . . . , xn ⊗ yn is a permutable (A ⊗k B)-sequence. Proof Combine Lemma 3.4 with the general fact that if x1 , x2 , . . . , xn is a permutable  regular sequence, then so is (x1 x2 ), x3 , . . . , xn . Proof of Theorem 3.1 (1) If x1 , x2 , . . . , xn is a maximal A-sequence in I , then by Lemma 3.4, x1 ⊗k 1, . . . , xn ⊗k 1 is an A ⊗k B sequence in I ⊗k B. Since it is maximal, there exists a nonzero a¯ in (x1 ,x2A,...,xn ) such that I a¯ = 0 by [20, Theorem 82]. Hence, (I ⊗k B)(a¯ ⊗k 1) = I a¯ ⊗k B = 0 in (x1 ,x2A,...,xn ) ⊗k B. Since (x1 ,x2A,...,xn ) is a ¯ ⊗k B = 0. Therefore, G(I ⊗k B) = faithfully flat k-algebra, a¯ ⊗k 1 = 0; i.e., (a) G(I ). Similarly, G(A ⊗k J ) = G(J ). (2) Let x1 , x2 , . . . , xn be a maximal A-sequence in I and y1 , y2 , . . . , ym be a maximal B-sequence in J . Then x1 ⊗k 1, . . . , xn ⊗k 1 is a maximal (A ⊗k B)-sequence in I ⊗k B, and 1 ⊗k y1 , . . . , 1 ⊗k ym is a maximal (x1 ,x2A,...,xn ) ⊗k B-sequence in A ⊗k J by similar steps in (1). Therefore, by Lemma 2.3, x1 ⊗k 1, . . . , xn ⊗k (x1 ,x2 ,...,xn ) 1, 1 ⊗k y1 , . . . , 1 ⊗k ym is a maximal (A ⊗k B)-sequence in I ⊗k B + A ⊗k J . (3) Without loss of generality assume that G(I ) = n ≤ G(J ) = m. By [20, Exercise 23, p. 104], there exists a maximal permutable A-sequence x1 , x2 , . . . , xn in I and a maximal permutable B-sequence y1 , y2 , . . . , ym in J . By Lemma 3.5, x1 ⊗k y1 , . . . , xn ⊗k yn is a permutable (A ⊗k B)-sequence in I ⊗k J . Hence, n ≤ G(I ⊗k

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J ). Since I ⊗k J ⊆ I ⊗k A, G(I ⊗k J ) ≤ G(I ⊗k A) = n by Proposition 2.5.  Therefore, G(I ⊗k J ) = n. Now, we state below the second main theorem of this section. Theorem 3.6 ([6, Theorem 2.1]) “Let A, B be two k-algebras such that A ⊗k B is Noetherian. Then, the following statements are equivalent: (1) A ⊗k B is a Cohen–Macaulay ring; (2) G(I ⊗k B + A ⊗k J ) = ht(I ⊗k B + A ⊗k J ), for all proper ideals I, J of A, B, resp.; (3) G( p ⊗k B + A ⊗k q) = ht( p ⊗k B + A ⊗k q), for all prime ideals p, q of A, B, resp.; (4) A and B are Cohen–Macaulay rings.” Remark 3.7 In general, even if A and B are Noetherian k-algebras, A ⊗k B does not have to be Noetherian [29]. But, if A ⊗k B is Noetherian, then so are A and B through faithfully flatness [21, Change of coefficient rings, p. 46 and Exercise 7.9, p. 53]. Moreover, if A ⊗k B is Noetherian, then the transcendence degree of A or B over k must be finite. Indeed, let t. degk (A) denotes the transcendence degree of A over k. Then, by [30, p. 392], 

 A | p ∈ Spec(A) . t. degk (A) := sup t. degk p Since A and B are Noetherian, they have a finite number of minimal prime ideals. Hence, there exist primes p in A and q in B such that t. degk (A) = t. degk ( Ap ) and t. degk (B) = t. degk ( Bq ). But, κ A ( p) ⊗k κ B (q) is Noetherian, as a localization of the B ∼ A A ⊗k B ⊗k . Therefore, by [29, Corollary 10], Noetherian ring = p q p ⊗k B + A ⊗k q t. degk (κ A ( p)) < ∞ or t. degk (κ B (q)) < ∞. To prove Theorem 3.6, we first establish the following results. Lemma 3.8 ([6, Lemma 2.2]) Let T and V be two extension fields of k such that T ⊗k V is Noetherian. Then, T ⊗k V is a Cohen–Macaulay ring. Proof Since T ⊗k V is Noetherian, then without loss of generality, we may assume that the transcendence degree of T is finite; say, t. degk (T ) = n < ∞, with x1 , x2 , . . . , xn being algebraically independent over k. Observe that T ⊗k V ∼ = T ⊗k(x1 ,x2 ,...,xn ) S −1 V [x1 , x2 , . . . , xn ] where S := k[x1 , x2 , . . . , xn ] \ {0} [9, Proposition 2.6]. Now, T is algerbaic over k(x1 , x2 , . . . , xn ) and S −1 V [x1 , x2 , . . . , xn ] is a Cohen–Macaulay ring by Proposition 2.8 and Theorem 2.7. Therefore, we reduce the proof to the case of a tensor product of an algebraic extension and a Cohen–Macaulay ring.

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Next, let T be an algebraic extension field of k such that T ⊗k A is Noetherian. Notice first that T ⊗k A is an integral extension of A; that is, faithfully flat over A since T is algebraic over k. Let Q be a prime ideal of T ⊗k A with p := A ∩ Q. Then, since T ⊗k A is integral over A and T ⊗k p ⊆ Q, by Proposition 2.5, Theorem 3.1 and [20, Theorem 48], we get ht(Q) ≤ ht( p) = G( p) = G(T ⊗k p) ≤ G(Q) ≤ ht(Q). Therefore, G(Q) = ht(Q) and so A ⊗k B is Cohen–Macaulay by Theorem 2.7.  Proposition 3.9 ([6, Proposition 2.3]) Let A and B be k-algebras such that A ⊗k B is Noetherian and let Q be a prime of A ⊗k B with p := A ∩ Q and q := B ∩ Q. Then:   Q . (1) ht(Q) = ht( p) + ht(q) + ht p⊗k B+A⊗ kq   Q (2) G(Q Q ) = G( p p ) + G(qq ) + ht p⊗k B+A⊗k q . Proof (1) Notice that ht(Q) = ht(Q Q ) = dim(A ⊗k B) Q and consider the flat local homomorphism A p −→ (A ⊗k B) Q . By Proposition 2.2, we have that  dim(A ⊗ B) Q = dim(A p ) + dim

A ⊗k B p

Q p⊗k B

.

(2)

Now, consider the flat local homomorphism Bq −→ ( Ap ⊗k B) Q . Again by Propop⊗k B sition 2.2,   A B A ⊗k B ⊗k dim = dim(Bq ) + dim . Q Q p p q p⊗k B

p⊗k B+A⊗k q

Therefore  dim

A ⊗k B p



Q p⊗k B

 = dim(Bq ) + ht

Q p ⊗k B + A ⊗k q

.

(3)

Combining (2) and (3) yields to (1)  ht(Q) = ht( p) + ht(q) + ht

Q p ⊗k B + A ⊗k q

.

(2) By similar steps and by using Proposition 2.9 and replacing ht(−) with G(−), we get that

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A ⊗k B p  B A ⊗k = G(Bq ) + G p q



G(A ⊗ B) Q = G(A p ) + G  G

A ⊗k B p

Q p⊗k B

Q p⊗k B

Q p⊗k B+A⊗k q

Let S := (A \ p) ⊗ (B \ q) and H :=

,

(4)

.

(5)

S −1 Q . Then, we have the p A p ⊗ Bq + A p ⊗ q Bq

natural isomorphism 

A B ⊗k p q

Q p⊗k B+A⊗k q

∼ = (κ A ( p) ⊗k κ B (q)) H .

Further, Lemma 3.8 implies that κ A ( p) ⊗k κ B (q) is a Cohen–Macaulay ring and therefore G(κ A ( p) ⊗k κ B (q)) H = dim(κ A ( p) ⊗k κ B (q)) H  Q . = ht p ⊗k B + A ⊗k q Hence, by combining (4), (5) and (6) we get (2).

(6) 

Proof of Theorem 3.6 The implications (1)⇒(2)⇒(3) are obvious. Assume that (3) holds. Let p and q be prime ideals of A and B, respectively. Notice that flat homomorphisms satisfy GD (Going Down) [21, Theorem 15.1] and under GD, obviously, minimal primes contract to minimal primes. So, minimal primes of A ⊗k B ∼ A ⊗ B contract to the null ideal in the domains A and B . That = p ⊗k B + A ⊗k q p q p q is, minimal primes of A ⊗ B over p ⊗k B + A ⊗k q contract to p and q in A and B, respectively. Therefore, by Proposition 3.9, ht( p ⊗k B + A ⊗k q) = ht( p) + ht(q). By Theorem 3.1, we obtain ht( p) + ht(q) = ht( p ⊗k B + A ⊗k q) = G( p ⊗k B + A ⊗k q) = G( p) + G(q). It follows that ht( p) − G( p) = G(q) − ht(q). The left side is a non-negative quantity while the other side is a non-positive one by Proposition 2.5. Therefore, G( p) = ht( p) and ht(q) = G(q). Hence, A and B are Cohen–Macaulay rings by Theorem 2.7. Assume now that (4) holds. Let Q be a prime ideal of A ⊗k B with p := A ∩ Q and q := B ∩ Q. Since A and B are Cohen–Macaulay rings, G( p) = G( p p ) and G(q) = G(qq ). Therefore, by Proposition 3.9,

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 Q dim((A ⊗k B) Q ) = ht(Q Q ) = ht( p) + ht(q) + ht p ⊗k B + A ⊗k q  Q = G((A ⊗k B) Q ). = G( p) + G(q) + ht p ⊗k B + A ⊗k q Hence, (A ⊗k B) Q is a Cohen–Macaulay ring, for every prime ideal Q. Therefore,  by Theorem 2.7, A ⊗k B is a Cohen–Macaulay ring.

4 Regularity, Gorensteiness, and Complete Intersection in Tensor Products This section deals with the transfer of regularity, Gorensteiness, complete intersection, and Serre’s conditions to Noetherian tensor products of k-algebras. Given a Noetherian ring A and a positive integer n, the so-called Serre’s conditions are defined as follows: (Rn ) A p is regular, for every p ∈ Spec(A) with ht( p) ≤ n. (Sn ) G(A p ) ≥ Min(dim(A p ), n), for each p ∈ Spec(A). Notice that A satisfies (Rn ) for each n is equivalent to A being a regular ring. Similarly, A satisfies (Sn ) for each n is equivalent to A being a Cohen–Macaulay ring [21, p. 183]. Serre’s conditions transfer through flat homomorphisms of Noetherian rings, as shown below. Proposition 4.1 ([12, Propositions 2.1.16 and 2.2.21]) Let f : A −→ B be a flat homomorphism of Noetherian rings. Let Q ∈ Spec(B) and set q := Q ∩ A. Then: (1) If B Q satisfies (Sn ) (resp., (Rn )), then so does Aq . (2) If A and κ A ( p) ⊗ A B, for all p ∈ Spec(A), satisfy (Sn ) (resp., (Rn )), then so does B. I t is known that a Noetherian tensor product of two extension fields is Gorenstein [26]. This section features the extension of this result to a more general context of tensor products of k-algebras, in addition to the transfer of complete intersection and Serre’s conditions. The main theorem reads as follows: Theorem 4.2 ([28, Theorem 6]) “Let A, B be k-algebras such that A ⊗k B is Noetherian. Then: (1) A ⊗k B is a locally complete intersection (resp., Gorenstein) ring if and only if so are A and B. (2) A ⊗k B satisfies (Sn ) if and only if so do A and B. (3) If A ⊗k B is regular, then so are A and B. (4) If A ⊗k B satisfies (Rn ), then so do A and B. (5) The converses of (3) and (4) hold if k is perfect.”

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Recall that k is said to be perfect if every (algebraic) extension of k is separable;  equivalently, either char(k) = 0 or char(k) = p with k = a p | a ∈ k [9, Sect. 4, Definition 2] and [9, Sect. 7, Proposition 2 and Corollary]. The proof of the theorem requires the following preliminary results. Lemma 4.3 ([28, Corollary 2]) Let f : A −→ B be a flat homomorphism of Noetherian rings. Then: (1) If A and the fibers κ A ( p) ⊗ A B, for all p ∈ Spec(A), are regular (resp., locally complete intersection, Gorenstein) rings, then so is B. (2) If B is a locally complete intersection (resp., Gorenstein) ring, then so are the fibers κ A ( p) ⊗ A B, for all p ∈ Spec(A). Proof (1) Let Q be a prime ideal of B and set q := Q ∩ A. Consider the induced flat local homomorphism Aq −→ B Q and note that BQ ∼ = q BQ



Bq q Bq

Qq q Bq

∼ = (κ A (q) ⊗ A B) Qq . q Bq

Since A and κ A (q) ⊗ A B are regular (resp., locally complete intersections, Gorenstein), Aq and (κ A (q) ⊗ A B) Qq are regular (resp., complete intersections, Gorenq Bq

stein). By Theorems 2.11 and 2.16, B Q is regular (resp., a complete intersection, Gorenstein). Since Q is an arbitrary prime ideal of B, B is regular (resp., locally complete intersection, Gorenstein). (2) Let p be a prime ideal of A. If p is not a contraction of any prime of B then p B = B (since f satisfies GD) and hence there is nothing to show. Next, suppose that p = A ∩ q, for some prime ideal q of B. We show that T := κ A ( p) ⊗ A Bp q B∼ is a locally complete intersection (resp., Gorenstein) ring. Let p Bp p be = p Bp a prime ideal of T and notice that q p ∩ A p = p p ; that is, q  ∩ A = p. Consider the induced homomorphism A p −→ Bq  . Since B is locally complete intersection Bq  ∼ B p ) q  is a complete (resp., Gorenstein), then so is Bq  . By Theorem 2.16, =( p Bq  p B p p Bpp intersection (resp., Gorenstein) ring. Since this holds for arbitrary prime ideals of T , so T is a locally complete intersection (resp., Gorenstein) ring, proving (2).  Lemma 4.4 Let f : A −→ B be a faithfully flat homomorphism of Noetherian rings. If B is locally complete intersection (resp., Gorenstein), then so is A. Proof Assume that B is locally complete intersection (resp., Gorenstein). Since f is faithfully flat, Spec(B) → Spec(A) is subjective [2, Exercise 16, p. 45]. Let p be a prime ideal of A and let q a prime ideal of B such that p = A ∩ q. Then, Bq is a complete intersection (resp., Gorenstein) ring. Consider the induced flat local homomorphism A p −→ Bq . By Theorem 2.16, A p is a complete intersection (resp., Gorenstein) ring, completing the proof.  The following lemma handles the converse of Theorem 2.11 and Proposition 4.1.

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Lemma 4.5 ([28, Corollary 4]) Let f : A −→ B be a faithfully flat homomorphism of Noetherian rings. Then: (1) If B is regular (resp., a complete intersection, Gorenstein), then so is A. (2) If B satisfies (Sn ) (resp., (Rn )), then so does A. Proof The proof steps are similar to those in Lemma 4.3.



Lemma 4.6 Let f : A −→ B be a flat homomorphism of Noetherian rings. Let q ∈ Spec(B) and set p := A ∩ q. Assume that p B = q. Then Bq is a Gorenstein (resp., complete intersection) ring if and only if so is A p . Proof Consider the flat local homomorphism A p −→ Bq and observe that

Bq ∼ = p Bq

Bq = κ B (q), as p B = q. Therefore, Theorem 2.16 leads to the conclusion. qq



Lemma 4.7 ([26, Theorem 2.2]) Let K and L be two extension fields of k such that K ⊗k L is Noetherian. Then, K ⊗k L is Gorenstein. Proof Let q be a prime ideal of K ⊗k L. Since K ⊗k L is Noetherian, q is finitely K ⊗k L for i = 1, 2, . . . , n for some n such generated; that is, there exist m i ∈ i α ji ⊗k β ji for some α ji ∈ K and β ji ∈ L. that q = (m 1 , . . . , m n ). Each m i = nj=1 n Let S := i=1 {α1i , . . . , αnii } and let V := k(S) the extension field of k generated by S. Notice that V ⊆ K and, since L is flat over k, V ⊗k L ⊆ K ⊗k L. Let p := (V ⊗k L) ∩ q and notice that q = p(K ⊗k L) as q’s generators are in V . Since K is faithfully flat over V , K ⊗k L ∼ = K ⊗V (V ⊗k L) is faithfully flat over V ⊗k L by [21, p. 46]. By Lemma 4.6, (K ⊗k L)q is Gorenstein if and only if so is (V ⊗k L) p . Further, by Corollary 2.17, since V is finitely generated over k and L is Gorenstein, V ⊗k L is Gorenstein, and so is (V ⊗k L) p . Hence (K ⊗k L)q is Gorenstein. Since  q is arbitrary, we showed that K ⊗k L is Gorenstein. Let us recall again Grothendieck’s result that the tensor product of two extension fields of k is regular provided one of them is finitely generated and separable over k [16]. Lemma 4.8 ([28, Proposition 5]) Let K and L be two extension fields of k such that K ⊗k L is Noetherian. Then: (1) K ⊗k L is a locally complete intersection ring. (2) If k is perfect, then K ⊗k L is regular. Proof (1) Mimic Lemma 4.7 after substituting ‘(locally) complete intersection’ for ‘Gorenstein’. (2) Here too mimic Lemma 4.7 after substituting ‘regular’ for ‘Gorenstein’ and use Grothendieck’s aforementioned result. 

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Proof of Theorem 4.2 (1) Let Q be a prime of A ⊗k B with p := A ∩ Q and q := B ∩ Q. By similar steps of Proposition 3.9 and using Theorem 2.16, one can see that (A ⊗k B) Q is a complete intersection (resp., Gorenstein) ring if and only if A p , Bq and (κ A ( p) ⊗k κ B (q)) H are complete intersection (resp., Gorenstein) rings, S −1 Q . By Lemma 4.8 where S := (A \ p) ⊗ (B \ q) and H := p A p ⊗ Bq + A p ⊗ q Bq (resp., Lemma 4.7), κ A ( p) ⊗k κ B (q) is a locally complete intersection (resp., Gorenstein) ring and so is (κ A ( p) ⊗k κ B (q)) H . It follows that (A ⊗k B) Q is a complete intersection (resp., Gorenstein) ring if and only if so are A p and Bq . So, if A and B are locally complete intersection (resp., Gorenstein) rings, so is A ⊗k B. Conversely, since A ⊗k B is faithfully flat over A (resp., B), then the transfer of locally complete intersection and Gorensteiness to A and B is guaranteed by Lemma 4.4. (2) If A ⊗k B satisfies (Sn ), then A and B satisfy (Sn ) by Lemma 4.5. Conversely, following similar steps of (1), we have κ A ( p) ⊗k κ B (q) is Cohen–Macaulay by Theorem 3.6 and therefore it satisfies (Sn ). Further, since A and B satisfy (Sn ), so do A p and Bq . Finally, by Proposition 4.1, A ⊗k B satisfies (Sn ). (3) and (4) Similar arguments of (1) lead to the conclusion via Lemma 4.5. (5) Suppose k is perfect; i.e., every extension field of k is separable. Let Q be a prime ideal of A ⊗k B with p := A ∩ Q and q := B ∩ Q. Since A and B are regular, A p and Bq are regular. By Lemma 4.8, since k is perfect and κ A ( p) ⊗k κ B (q) is Noetherian, κ A ( p) ⊗k κ B (q) is regular. Therefore, (κ A ( p) ⊗k κ B (q)) H is regular. By similar steps of Proposition 3.9 and Lemma 4.3, it is easily seen that since A p , Bq and (κ A ( p) ⊗k κ B (q)) H are regular, (A ⊗k B) Q is regular, completing the proof of the theorem  A counter-example to the converse of Theorem 4.2(3) is given below. At this point, recall that a regular local ring is an integral domain [20, Theorem 164]. Example 4.9 ([28, Remark 7]) Let k be an imperfect field with char(k) = 3. Hence, k[x] = k(β), there exists a ∈ k such that x 3 − a has no root in k. Then, K := 3 (x − a) K [x] for β := x, ¯ is a splitting field of x 3 − a = (x − β)3 . Therefore, K ⊗k K ∼ = 3 (x − a) is not a regular ring , since the localization with respect to (x − β) is not an integral domain (e.g., , (x − β)3 = 0).

5 Regularity of Tensor Products of Extension Fields and Applications This section deals with the transfer of regularity to tensor products of extension fields. Precisely, it establishes necessary and sufficient conditions for the tensor product of two extension fields of k to inherit regularity under various assumptions of separa-

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bility. Then, among the applications, the results are extended to residually separable k-algebras. Recall that, in contrast to the notions of Cohen–Macaulay, Gorenstein, and complete intersection rings, the tensor product of two extension fields of k need not be regular, in general (see Example 4.9). Several cases have been covered regarding the transfer of regularity to tensor products. Grothendieck proved that the tensor product of two extension fields K and L is regular provided that either K or L is finitely generated over k and if K is separable over k (see Lemma 2.12). In [28], the authors showed that for two k-algebras A and B, if A ⊗k B is Noetherian, then A ⊗k B is regular if and only if A and B are regular (Theorem 4.2) in the special case where k is perfect (i.e., every (algebraic) extension of k is separable). Later, [7] tackled more general settings; including, the cases where one of the extension fields K or L is purely inseparable, normal, or algebraic over k, or a product of the separable closure and the purely inseparable closure of k. The case where, for instance, K is purely inseparable or normal over k, or product of the separable closure and the purely inseparable closure of k is handled in Theorem 5.1. The case where K is algebraic is established in Corollary 5.7. Recall that a Noetherian k-algebra A is said to be geometrically regular over k if A ⊗k F is regular for every finite extension field F of k; in particular, A is regular. The authors of [7] defined a new class of k-algebras called residually separable (the definition will be recalled later, see Definition 5.8). It turns out that regular residually separable k-algebras are geometrically regular. The converse, in general, is not true as illustrated by Example 5.11. With this class, new cases have been investigated in Theorem 5.10 which generalizes Theorem 4.2. Suppose k has characteristic p. Let K and L be extension fields of k such that K is (algebraic) purely inseparable over k; i.e., for every x ∈ K there exists an integer n n such that x p ∈ k (cf. [9, p. V.24]). Then, by [9, Proposition 5, p. V.26], K can be embedded in an algebraic closure of L via a unique homomorphism u that acts as an identity map on k. Further, u(K ) is purely inseparable. Therefore, in such a case, we can assume that K and L are contained in a common field. We denote the separable closure and the purely inseparable closure of k in K by K s and K i , respectively. Recall further, if K and L are subfields of a common field , then K and L are linearly disjoint if every free subset of K (resp., L) over k is also free over L (resp., K ); equivalently, if K ⊗k L is a domain (cf. [9, p. V.13]). We now announce the first main theorem of this section. Theorem 5.1 ([7, Theorem 2.4]) “Let K and L be two extension fields of k such that K ⊗k L is Noetherian. Assume that K = K i K s and let K i = k(S) for some generating subset S of K i . Then, the following assertions are equivalent: (1) (2) (3) (4) (5)

K ⊗k L is regular; K i ⊗k L is a domain; K i ⊗k L is a field; [k(S  ) : k] = [L(S  ) : L], for each finite subset S  of S; K i ∩ L(S  ) = k(S  ), for each finite subset S  of S.”

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For the proof of this theorem, we need the following lemmas. Lemma 5.2 ([7, Lemma 2.1]) Let A be a geometrically regular k-algebra and K an extension field of k such that A ⊗k K is Noetherian. Then, A ⊗k K is regular. Proof Let Δ denote the set of all finitely generated extension fields of k contained in K and let R := A ⊗k K = lim R(E) → E∈Δ

where R(E) := A ⊗k E, for each E ∈ Δ. Next, let P ∈ Spec(R) with PE := P ∩ R(E), for each E ∈ Δ. Step 1: We show that if F ∈ Δ such that PE R = PF R for each E ∈ Δ containing F, then P = PF R. Indeed, let F ∈ Δ such that PE R = PF R for each E ∈ Δ containing F. Let x ∈ P. Then, there exists E  ∈ Δ such that x ∈ R(E  ), and thus x ∈ PE  R. Whence, x ∈ PF(E  ) = PF R, where F(E  ) denotes the composite field of F and E  in K . It follows that P = PF R, proving the claim. Step 2: We claim that there exists E ∈ Δ with P = PE R. Indeed, assume, for contradiction, that PE R  P for any E ∈ Δ (notice that under this hypothesis K is necessarily infinitely generated over k; i.e., K ∈ / Δ). Choose E 1 ∈ Δ. By Step 1, there exists E 2 ∈ Δ containing E 1 such that PE1 R  PE2 R. Iterating this process yields the following infinite chain of ideals in R PE1 R  · · ·  PEn R  · · ·  P, where the E j s ∈ Δ. This is absurd, as R is Noetherian. Step 3: We show that P R P is generated by an R P -regular sequence. By Step 2, P = PE R for some E ∈ Δ. Note that R P := (A ⊗k K ) P ∼ = (R(E) PE ⊗ E K ) P and P R P ∼ = (PE R(E) PE ⊗ E K )R P with PE R(E) PE being the maximal ideal of R(E) PE . As E is finitely generated over k, R(E) is regular (recall that A is geometrically regular). Hence R(E) PE is a regular local ring. By [20, Theorem 169], PE R(E) PE is generated by an R(E) PE -regular sequence x1 , . . . , xr . Further, it sequence of is easily seen that x1 ⊗  k 1, . . . , xr ⊗ E 1 is an R(E) PE x⊗1 ⊗EE 1K -regular PE R(E) PE ⊗ E K . As PE R(E) PE ⊗ E K R P ∼ = P R P , 1 , . . . , xn ⊗1 E 1 is an R P regular sequence of P R P . Now, since PE R(E) PE = (x1 , . . . , xn )R(E) PE , we get  P R P = x1 ⊗1 E 1 , . . . , xn ⊗1 E 1 R P , establishing the last step.  It follows, by [20, Theorem 160], that R P is regular, completing the proof. Lemma 5.3 ([7, Lemma 2.1]) Let A and B be k-algebras such that A is geometrically regular. Then, A ⊗k B is regular if and only if B is regular and A ⊗k B is Noetherian. Proof Necessity holds by Theorem 4.2(3). For sufficiency, let P ∈ Spec(A ⊗k B) with q := B ∩ P. We show that (A ⊗k B) P is regular. Notice that we always have the induced flat local homomorphism Bq → (A ⊗k B) P . Since B is regular, then so is Bq . Hence, it is enough to show that (A ⊗k κ B (q)) Pq is regular which would imply A⊗k qq

that (A ⊗k B) P is regular by Theorem 2.11. But, this is a localization of a Noetherian tensor product of A with an extension field of k. So, it is regular by Lemma 5.2. 

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In particular, if K is a separable extension field of k; i.e., K is geometrically regular by Lemma 2.12 (cf. [21, p. 198]), the above lemma asserts that K ⊗k A is regular if and only if A is regular and K ⊗k A is Noetherian. Proof of Theorem 5.1 Without loss of generality we may assume that char(k) = p = 0 and that k is imperfect by Lemma 4.8. We reduce the proof to the case where K is purely inseparable by the following steps. Since K s is a separable extension field of k, K s ⊗k K i is reduced [32, Chap. III, Sect. 15, Theorem 39]. Moreover, since K i is algebraic, dim(K s ⊗k K i ) = 0 by Remark 3.7 and therefore it has one unique minimal prime ideal by [29, Proposition 2(c)]. Hence, K s ⊗k K i is local. But since it is reduced, i.e., Nil(K s ⊗k K i ) = 0, it is a field. Next, consider the homomorphism φ : K s ⊗k K i −→ K defined by a ⊗k b → ab using the bilinear map (a, b) → ab. Since K = K s K i , φ is surjective, and since it is not a zero map and K s ⊗k K i is a field, it is injective. Therefore, K ∼ = K s ⊗k K i . Now, K ⊗k L is regular if and only if K s ⊗k (K i ⊗k L) is regular. Since K s is separable, this holds if and only if K i ⊗k L is regular by Lemma 5.3. Hence, without loss of generality, we may assume K is purely inseparable over k. Therefore, by similar argument as above, K ⊗k L is an Artin local ring. It follows then (1) ⇒(2)⇒(3)⇒(1) by [20, Theorem 164] (which states that a regular local ring is a domain). Furthermore, (2)⇔(4) by [9, Proposition 5(a), p. V.13] and that K ⊗k L ∼ = K ⊗k(S  ) (k(S  ) ⊗k L) ∼ = K ⊗k(S  ) L(S  ) since the fact K ⊗k L is a domain is equivalent to K and L are linearly disjoint over k. Also, by a similar argument, (2)⇒(5). (5)⇒(3) Let Δ be the collection of all finite subsets of S and notice that K = lim S  ∈Δ k(S  ). Since K ⊗k L ∼ = lim S  ∈Δ (k(S  ) ⊗k L) by [2, Exercise 20, p. 33], it is − → − →  enough to show that k(S ) ⊗k L is a field for every finite subset S  of S. Let x ∈ S and, without loss of generality, assume that x ∈ / k. Since K is purely inseparable over k and char(k) = p = 0, [k(x) : k] = p m for some m ≥ 1 and x has a minimal m m polynomial y p − a, where a = x p . Observe that x ∈ S ⊆ K , and by minimality of pr / K ∩ L = k, for each integer r < m. Notice also that the polynomial and (5), x ∈ m r r m−r X p − a = (X p − x p ) p , for each integer r < m, and so it is irreducible over L. ] ∼ Therefore, k(x) ⊗k L ∼ = (XL[X p m −a) = L(x) is a field. We proceed by induction on the   length of S . Let S = {x1 , x2 , . . . , xn } and notice that k(x1 , x2 , . . . , xn ) ⊗k L ∼ = k(x1 , x2 , . . . , xn ) ⊗k(x1 ,x2 ,...,xn−1 ) (k(x1 , x2 , . . . , xn−1 ) ⊗k L).

By induction, k(x1 , x2 , . . . , xn−1 ) ⊗k L ∼ = L(x1 , x2 , . . . , xn−1 ). Further, by (5) we have k(x1 , x2 , . . . , xn−1 ) ⊆ k(x1 , x2 , . . . , xn ) ∩ L(x1 , x2 , . . . , xn−1 ) ⊆ K ∩ L(x1 , x2 , . . . , xn−1 ) = k(x1 , x2 , . . . , xn−1 ). We obtain

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k(x1 , x2 , . . . , xn−1 )(xn ) ∩ L(x1 , x2 , . . . , xn−1 ) = k(x1 , x2 , . . . , xn−1 ). Therefore by similar steps replacing k by k(x1 , x2 , . . . , xn−1 ), we get that k(x1 , x2 , . . . , xn ) ⊗k L ∼ = k(x1 , x2 , . . . , xn−1 )(xn ) ⊗k(x1 ,x2 ,...,xn−1 ) L(x1 , x2 , . . . , xn−1 ) ∼ = L(x1 , x2 , . . . , xn )

completing the proof of the theorem.



Next, we provide a simple illustrative example of the previous theorem. Example 5.4 (See also [7, Example 2.5]) Let p be a prime integer, x and y two Z indeterminates over Z p := . Let k := Z p (x p ), K := k(x) and L := k(y). Then, pZ K and L are both finitely generated extension over k and hence K ⊗k L is Noetherian. Notice also that K is purely inseparable over k, and that K ∩ L = k and K ∩ L(x) = k(x). Therefore, by Theorem 5.1(5), K ⊗k L is regular. It is also possible to construct examples of tensor products, of two extension fields of k, that are locally complete intersection but not regular by a simple violation of Theorem 5.1(5); i.e., when k  K ⊆ L such that K is purely inseparable over k. Example 5.5 (See [7, Example 2.7]) Let x be an indeterminate over Z p , k := Z p (x p ), and K := L := Z p (x). Then, K ⊗k L is Noetherian, and therefore it is locally complete intersection by Lemma 4.8. Since K ∩ L = k, K ⊗k L is not regular by Theorem 5.1(5). In [28], the authors provided an example of a tensor product of two fields in the form K ⊗k K for which it is locally complete intersection but not regular (Example 4.9). The next corollary provides necessary and sufficient conditions when such a form of tensor products is regular. Corollary 5.6 ([7, Corollary 2.6]) “Let K be an extension field of k. The following conditions are equivalent: (1) (2) (3) (4) (5)

K K K K K

⊗k K is regular; ⊗k K is Noetherian and K is separable over k; is finitely generated and separable over k; ⊗k L is regular, for each extension field L of k; is finitely generated over k and a projective K ⊗k K -module.”

Proof (2)⇒(3) follows from Remark 3.7, (3)⇒(4) is handled by Lemma 2.12, (4)⇒(1) is trivial (with L = K ), and (3)⇔(5) is a particular case of [13, Theorem 7.10, p. 179]. (1)⇒(2) Assume that K ⊗k K is regular. Then, it is Noetherian and therefore by Remark 3.7, it is a finitely generated extension field of k. We show that K is separable over k. First, let E be an extension field of k in K and notice that

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K ⊗k K ∼ = K ⊗ E (E ⊗k K ) ∼ = K ⊗ E (K ⊗k E) ∼ = (K ⊗ E K ) ⊗k E. It follows, by Theorem 4.2(3), that K ⊗ E K is regular. Let B be a finite transcendence basis of K over k and let E be the algebraic separable closure of k(B). Hence, by [9, Proposition 13, p. V.44], K is purely inseparable over E. By Theorem 5.1(5), with L = K and K i = K , K = K i ∩ L(S) = E(S) = E. So, K is separable over k.  The investigation of the case when K or L is algebraic over k is handled below below, and it generalizes slightly [29, Proposition 8], since if K is separable over k, then K ⊗k L is reduced [32, Chap. III, Sect. 15, Theorem 39]. Corollary 5.7 ([7, Corollary 2.8]) “Let K and L be two extension fields of k such that K ⊗k L is Noetherian. Assume that K is algebraic over k. Then, the following assertions are equivalent: (1) K ⊗k L is (von Neumann) regular; (2) K ⊗k L is reduced; (3) K ⊗k L is a finite product of fields. If, in addition, K is separable and L is Galois over k such that K and L are contained in an algebraic closure of k, then the above statements are equivalent to: (4) n := [(K ∩ L) : k] < ∞. Moreover, K ⊗k L is isomorphic to the product of n copies of the field K (L).” Proof Recall that a regular ring is reduced by a combination of [20, Theorem 164] and [20, Theorem 168]. On the other hand, a reduced zero-dimensional ring is von Neumann regular by [20, Exercise 22, p. 64]. Further, by [29, Lemma 0], if a zerodimensional von Neumann ring is Noetherian, it is a finite product of fields. By [12, Corollary 2.2.20], a finite product of regular rings is regular. Therefore, on a zerodimensional Noetherian ring, the above notions are equivalent. Now, if K and L are extension fields of k such that K is algebraic over K and K ⊗k L is Noetherian, then dim(K ⊗k L) = 0 by [27, Theorem 3.1], and therefore (1)⇔(2)⇔(3). Further, if the last assertion holds, then by [29, Proposition 8], we get (3)⇔(4).  Definition 5.8 “A k-algebra R is said to be residually separable, if κ R ( p) is separable over k, for each p ∈ Spec(R).” Several examples of residually separable k-algebras can be easily constructed. For instance, let K be a separable extension field of k and let A := K [x](x) . Then, A is local (in fact, valuation) ring with only two prime ideals; namely, 0 and x A. Their residue fields are K (x) and K , respectively, and so both are separable over k. In order to emphasize the importance of this class of k-algebras, next we show that, in this class, regularity coincides with geometrical regularity.

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Lemma 5.9 Let A be a residually separable k-algebra. Then A is regular if and only if A is geometrically regular. Proof As shown in the introduction of this section, if A is geometrically regular, then it is regular. Now, assume that A is regular and let K be a finitely generated extension field of k. Hence, A ⊗k K is Noetherian. Let P be a prime ideal of A ⊗k K and set p := A ∩ P. Since κ A ( p) is separable and K is finitely generated, κ A ( p) ⊗k K is regular Lemma 2.12. Since A p is regular and (κ A ( p) ⊗k K ) H is regular, where P H = p p ⊗pk K , (A ⊗k K ) P is regular by Theorem 2.11. Since P is arbitrary, A ⊗k K is regular. Further, since K is arbitrary, A is geometrically regular.  Next, we announce the second theorem of this section. Theorem 5.10 ([7, Theorem 2.11]) “Let A and B be k-algebras such that A ⊗k B is Noetherian. Consider the following assertions: (1) (2) (3) (4) (5)

A, B and κ A ( p) ⊗k κ B (q) are regular, for each ( p, q) ∈ Spec(A) × Spec(B); B and A ⊗k κ B (q) are regular, for each q ∈ Spec(B); A and κ A ( p) ⊗k B are regular, for each p ∈ Spec(A); A ⊗k B is regular; A and B are regular.”

Then (1)⇒(2) (resp., (1)⇒(3)) ⇒ (4) ⇒ (5). If A (or B) is residually separable, then all assertions are equivalent Proof The first statement is handled by a combination of Lemmas 4.3, 4.5 and Theorem 4.2. Next, assume that A and B are regular and A is residually separable. Let ( p, q) ∈ Spec(A) × Spec(B). Then, κ A ( p) ⊗k κ B (q) is Noetherian (through quotient and localization) and hence it is regular by Lemma 5.3, which also make κ A ( p) ⊗k B and A ⊗k κ B (q) regular by the first statement. Therefore, the assertions in the theorem fall down to A ⊗k B is regular if and only if A and B are regular.  The following example shows that the above implications are not reversible in general. It also shows that Lemma 5.3 does not hold, in general, for purely inseparable extensions. It also provides an example of a geometrically regular k-algebra that is not residually separable. Example 5.11 ([7, Example 2.12]) Let k be an imperfect field of characteristic p and K a purely inseparable extension field of k. Let u ∈ K such that e [k(u) : k] = p e for some e ≥ 2. Then, a := u p ∈ k. Let r ∈ {1, . . . , e − 1} and e−r A := k[x](x pe−r −a) . Notice that x p − a is irreducible by similar argument in the e−r last part of Theorem 5.1 and therefore (x p − a) is a maximal ideal in k[x]. Hence, e−r A is local regular with maximal ideal m := (x p − a)A since k[x] is regular. Fur∼ ther, k(u) ⊗k A = k(u) ⊗k k[x] ⊗k[x] (k[x](x pe−r −a) ) ∼ = S −1 k(u)[x] is regular by [20, e−r Theorem 171 and Exercise 9, p. 121], where S := k[x] \ (x p − a). Moreover, A ∼ k[x] r ∼ = pe−r = k(u p ), which is finitely generated and purely inseparable over m (x − a)

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k. This shows that A is not residually separable since the only nonzero prime ideal A r r is not regular by Theoof A is m. Since k(u) ∩ k(u p ) = k(u p ) = k, k(u) ⊗k m rem 5.1(5). Also, for any finitely generated extension field L of k, A ⊗k L ∼ = S −1 L[x] is regular. Therefore A is geometrically regular that is not residually separable.

References 1. D.F. Anderson, A. Bouvier, D.E. Dobbs, M. Fontana, S. Kabbaj, On Jaffard domains. Expo. Math. 6, 145–175 (1988) 2. M.F. Atiyah, I.G. Macdonald, Introduction to Commutative Algebra (Westview Press, Boulder, 1969) 3. S. Bouchiba, D.E. Dobbs, S. Kabbaj, On the prime ideal structure of tensor products of algebras. J. Pure Appl. Algebra 176, 89–112 (2002) 4. S. Bouchiba, F. Girolami, S. Kabbaj, The dimension of tensor products of AF-rings, in Lecture Notes in Pure Applied Mathematics, vol. 185, (Dekker, New York, 1997), pp. 141–154 5. S. Bouchiba, F. Girolami, S. Kabbaj, The dimension of tensor products of k-algebras arising from pullbacks. J. Pure Appl. Algebra 137, 125–138 (1999) 6. S. Bouchiba, S. Kabbaj, Tensor products of Cohen-Macaulay rings. Solution to a problem of Grothendieck. J. Algebra 252, 65–73 (2002) 7. S. Bouchiba, S. Kabbaj, Regularity of tensor products of k-algebras. Math. Scand. 115, 5–19 (2014) 8. S. Bouchiba, S. Kabbaj, Embedding dimension and codimension of tensor products of algebras over a field, in Rings, Polynomials, and Modules. (Springer, New York, 2017), pp. 53–77 9. N. Bourbaki, Algèbre, Chapitres 4–7 (Masson, Paris, 1981) 10. N. Bourbaki, Algèbre Commutative, Chapitres 8–9 (Masson, Paris, 1981) 11. A. Bouvier, D.E. Dobbs, M. Fontana, Universally catenarian integral domains. Adv. Math. 72, 211–238 (1988) 12. W. Bruns, J. Herzog, Cohen-Macaulay Rings (Cambridge University Press, Cambridge, 1993) 13. H. Cartan, S. Eilenberg, Homological Algebra (Princeton University Press, Princeton, 1956) 14. D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry. Graduate Texts in Mathematics, vol. 150 (Springer, New York, 1995) 15. M. Fontana, S. Kabbaj, Essential domains and two conjectures in dimension theory. Proc. Amer. Math. Soc. 132(9), 2529–2535 (2004) 16. A. Grothendieck, Eléments de géométrie algébrique. Institut des Hautes Etudes Sci. Publ. Math. No. 24, Bures-sur-yvette (1965) 17. H. Haghighi, M. Tousi, S. Yassemi, Tensor product of algebras over a field, in Commutative Algebra: Noetherian and Non-Noetherian Perspectives (Springer, New York, 2011), pp. 181– 202 18. C. Huneke, D.A. Jorgensen, Symmetry in the vanishing of Ext over Gorenstein rings. Math. Scand. 93, 161–184 (2003) 19. P. Jaffard, Théorie de la dimension dans les anneaux de polynômes. Mém. Sc. Math. vol. 146 (Gauthier-Villars, Paris, 1960) 20. I. Kaplansky, Commutative rings (University of Chicago Press, Chicago, 1974) 21. H. Matsumura, Commutative Ring Theory (Cambridge University Press, Cambridge, 1986) 22. M. Nagata, Local Rings (Robert E. Krieger Publishing Co., Huntington, 1975) 23. D.G. Northcott, D. Rees, Extensions and simplifications of the theory of regular local rings. J. London Math. Soc. 32, 367–374 (1957) 24. J.J. Rotman, An Introduction to Homological Algebra (Springer, New York, 2009)

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25. R.Y. Sharp, The effect on associated prime ideals produced by an extension of the base field. Math. Scand. 38, 43–52 (1976) 26. R.Y. Sharp, Simplifications in the theory of tensor products of field extensions. J. Lond. Math. Soc. 15, 48–50 (1977) 27. R.Y. Sharp, The dimension of the tensor product of two field extensions. Bull. Lond. Math. Soc. 9, 42–48 (1977) 28. M. Tousi, S. Yassemi, Tensor products of some special rings. J. Algebra 268, 672–676 (2003) 29. P. Vamos, On the minimal prime ideals of a tensor product of two fields. Math. Proc. Camb. Philos. Soc. 84, 25–35 (1978) 30. A.R. Wadsworth, The Krull dimension of tensor products of commutative algebras over a field. J. Lond. Math. Soc. 19, 391–401 (1979) 31. K. Watanabe, T. Ishikawa, S. Tachibana, K. Otsuka, On tensor products of Gorenstein rings. J. Math. Kyoto Univ. 9, 413–423 (1969) 32. O. Zariski, P. Samuel, Commutative Algebra, vol. I (Van Nostrand, Princeton, 1960)

Tame-Wild Dichotomy for Commutative Noetherian Rings—A Survey Lee Klingler, Roger Wiegand, and Sylvia Wiegand

Abstract We survey what is known about tame-wild dichotomy for commutative Noetherian rings. That is, we summarize terminology and results relating to the questions: “For which Noetherian rings does the category of finitely generated (or finite-length) modules have tame representation type?” and “For which rings do these categories have wild representation type?” We also address the question: “Can a Noetherian ring have both tame and wild representation type?” We provide an outline for showing that many tame rings are not wild. Keywords Representation type · Tame · Wild · Noetherian rings 2010 Mathematics Subject Classification 13H15

1 Introduction Let R be a ring and C a full subcategory of mod-R that is closed under direct summands, finite direct sums, and isomorphism. Here mod-R is the category of finitely generated right R-modules. In other words, the following hold, for finitely generated R-modules M and N :

RW was partially supported by Simons Collaboration Grant 426885. L. Klingler (B) Florida Atlantic University, Boca Raton, FL 33431-6498, USA e-mail: [email protected] R. Wiegand · S. Wiegand University of Nebraska, Lincoln, NE 68588-0130, USA e-mail: [email protected] S. Wiegand e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. Badawi and J. Coykendall (eds.), Rings, Monoids and Module Theory, Springer Proceedings in Mathematics & Statistics 382, https://doi.org/10.1007/978-981-16-8422-7_11

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(a) If M, N ∈ C, then HomC (M, N ) = Hom R (M, N ). (b) If M ∈ C and M ∼ = N , then N ∈ C. (c) M ⊕ N ∈ C ⇐⇒ M ∈ C and N ∈ C. Informally speaking, C has tame representation type provided there is a description or classification of all of the modules in C. Also, C has wild representation type provided such a description is deemed to be impossible. Somewhat more formally, wild representation type means that there exists a field k with the property that a classification of the modules in C would lead to a classification of all modules of finite k-dimension over the non-commutative polynomial ring kx, y, a classification deemed to be impossible. A more precise definition of wildness in our setting is given in Sect. 4. A tame-wild dichotomy for C is a statement either that C is tame but not wild, or that C is wild but not tame. Alas, in the context we are concerned with, where C ⊆ mod-R for a fixed commutative Noetherian ring R, there seems to be no satisfactory definition of tameness, other than to classify all of the indecomposable finitely generated modules and their direct-sum relations in C, and then to declare that this means that C is tame. The ring R is called tame (wild) if C = mod-R is tame (wild). From now on, the term Noetherian ring always means commutative Noetherian ring. We should mention that some authors refer to “trichotomy”, adding a third possibility—finite representation type (meaning that there are, up to isomorphism, only finitely many indecomposable modules in C), and including “infinite representation type” in the definition of tameness. For our purposes, however, we’ll just think of finite representation type as a special case of tame representation type. In this paper our goal (not quite fulfilled) is to prove, for every Noetherian ring R, tame-wild dichotomy for the class mod-R of finitely generated R-modules. This brings into focus the Dedekind-like rings studied by Klingler and Levy in a series of long papers [12, 13, 13, 15, 17]; see Definition 2.1. In these papers, they classify all the finitely generated modules over almost every Dedekind-like ring, the only possible exceptions being associated with imperfect fields of characteristic two. Thus most Dedekind-like rings, and consequently their homomorphic images, have tame representation type. There’s another kind of ring, rather small and called, appropriately, a Klein ring, that has tame representation type; see Definition 2.5. By Theorem 4.3, every Noetherian ring that is neither a Klein ring nor a homomorphic image of a Dedekind-like ring has wild representation type. The missing piece of dichotomy here is a theorem saying that no Noetherian ring can have both tame and wild representation type. The new results in this note make some progress on this problem. In Sect. 2, we discuss Dedekind-like rings and tameness. Section 3 contains Theorem 3.3, a dichotomy theorem due to Klingler and Levy: every Noetherian ring is either (1) a Klein ring or a homomorphic image of a Dedekind-like ring or (2) has one of two special types of well-defined rings as a homomorphic image. In Sect. 4, using our definition of wildness and relying on Klingler and Levy’s work, we prove Theorem 4.3, that the type (2) Noetherian rings are wild. In Sect. 5, we discuss the Brauer group of a field and give Theorem 5.1: For certain local Dedekind-like rings that are tame, if the Brauer group of the residue field is non-trivial, then R is not wild.

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In Sect. 6, we provide an outline with some justification for showing that many tame Dedekind-like rings are not wild. As parts of this outline, we give some problems for future work and we prove Proposition 6.9 (a fun computation): Every field k has an extension field  with non-trivial Brauer group.

2 Tameness Dedekind-like rings are the main source of tameness, for finitely generated modules over commutative Noetherian rings. We will not attempt a formal definition of tameness in the general context of commutative Noetherian rings. The theorems in [14] describe, for almost every Dedekind-like ring, all the finitely generated modules and the direct-sum relations among them. On the basis of these theorems, we declare mod-R to be tame for most Dedekind-like rings. These theorems are extremely long and difficult, and even the descriptions of the modules are highly intricate. There is one class of Dedekind-like rings, termed “exceptional” and defined in Definition 2.3, to which the theorems in [14] do not apply. Definition 2.1 A commutative ring R is Dedekind-like provided it satisfies the following six properties: (a) R is Noetherian. (b) R is reduced; that is, R has no non-zero nilpotent elements. (c) The integral closure R of R in its total quotient ring T (T is obtained by inverting all non-zerodivisors of R) is the direct product of finitely many Dedekind domains. Often we refer to R as the normalization of R. (d) For each maximal ideal m of R, either Rm = R m or (R/R)m is a simple Rmodule. (e) For each maximal ideal m of R, mRm is the Jacobson radical of R m . (f) Every maximal ideal of R has height one. The normalization R is not required to be a finitely generated R-module. (See [10, Theorem 4.4] for a proof that the normalization of a Dedekind-like ring need not be finitely generated.) However, for each maximal ideal m of a Dedekind-like ring R, the normalization of Rm is generated by either one or two elements (by one element if Rm is a discrete valuation ring, and by two elements otherwise). It follows that R is finitely generated as an R-module if and only if R has only finitely many singular maximal ideals (maximal ideals m for which Rm is not a discrete valuation ring).

2.1 The Local Case By a local ring we always mean a Noetherian ring with exactly one maximal ideal. Remarks 2.2 gives a characterization and some notes concerning Dedekind-like rings among local rings:

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Remarks 2.2 (I) A local ring (R, m, k) with normalization R is Dedekind-like if and only if either R is a discrete valuation ring, or else all of the following five conditions hold: (i) R is reduced. (ii) dim(R) = 1. (iii) R is either a discrete valuation ring or a principal ideal ring with exactly two maximal ideals. (iv) R/R is a simple R-module. (v) m is the Jacobson radical of R. (II) A Noetherian ring R is Dedekind-like if and only if Rm is Dedekind-like for every maximal ideal m of R. Moreover, a local ring (R, m, k) is Dedekind-like if  is Dedekind-like; see [15, Proposition 5.1] and and only if its m-adic completion R [14, Lemma 11.8]. (III) It follows from (I) that exactly one of the following things happens for (R, m, k) a local Dedekind-like ring with normalization R = R: (1) R is the direct product of two discrete valuation domains. In this case one says that R is strictly split. (2) R is a domain with exactly two maximal ideals, say n1 and n2 , and both R n1 and R n2 are discrete valuation domains. In this case one says that R is nonstrictly split. (3) R is a discrete valuation domain whose residue field K has degree 2 over k. In this case one says that R is unsplit. In cases (1) or (2), R is called a split local Dedekind-like ring. If, in case (3), K is a purely inseparable extension of k, then the local unsplit Dedekind-like ring R is said to be exceptional Dedekind-like.

2.2 Global Tameness In this discussion, R is not necessarily local. First we define “exceptional” Dedekindlike rings more generally. Definition 2.3 A Dedekind-like ring R is said to be exceptional provided it has at least one maximal ideal m for which Rm is exceptional. That is, the normalization Rm is a discrete valuation domain whose residue field is a purely inseparable extension of degree 2 over the residue field of Rm . At this point it is not known, for an exceptional Dedekind-like ring R, whether modR is tame, wild, neither, or both! However, for a Dedekind-like ring R that is not exceptional, the finitely generated R-modules are classified, and their direct-sum relations are described explicitly, in the long and difficult paper [14]. On the basis of these descriptions, we declare:

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Theorem 2.4 [14] Let R be a Dedekind-like ring, and assume R is not exceptional. Then mod-R, the category of finitely generated R-modules, has tame representation type. Moreover, if I is an ideal of R and C is a class of finitely generated (R/I )modules that is closed under direct summands, finite direct sums, and isomorphism, then C has tame representation type. Thus all non-exceptional Dedekind-like rings and their homomorphic images are tame. There is another family of tame rings, namely, Klein rings: Definition 2.5 A local ring (R, m, k) is a Klein ring provided the minimal number of generators of m is two, m2 is a non-zero principal ideal, m3 = 0, and every element of m has square 0. The group algebra of the Klein 4-group over a field of characteristic 2 is a Klein ring, whence the name. If (R, m, k) is a Klein ring, then k has characteristic 2, and R has characteristic 2 or 4 [12, Lemma 2.9]. Some, but not all, Klein rings are homomorphic images of Dedekind-like rings [12, Theorem 5.2]: Remark 2.6 These are equivalent, for a Klein ring (R, m, k): (i) R is a homomorphic image of a Dedekind-like ring. (ii) R is a homomorphic image of a local (necessarily unsplit) Dedekind-like ring. (iii) k is imperfect. Proof The equivalence of (ii) and (iii) is [12, Theorem 5.2], where it is tacitly assumed that the Dedekind-like ring in question is local. To see that (i) =⇒ (ii), suppose  is Dedekind-like and ϕ :   R is surjective. Let n be the unique maximal ideal of  containing ker(ϕ). Then ϕ induces a surjective homomorphism from the  local Dedekind-like ring n onto R. For a Klein ring (R, m, k), [13, Theorem 11.3] tells us that R/m2 is a homomorphic image of a strictly split local Dedekind-like ring and that every finitely generated R-module is the direct sum of a free R-module and an R/m2 -module. Using this and Theorem 2.4, we get the following: Corollary 2.7 If (R, m, k) is a Klein ring, then mod-R, the class of finitely generated R-modules, has tame representation type.

3 A Dichotomy Theorem In this brief section we describe the dichotomy theorem for Noetherian rings, due to Klingler and Levy. The basic idea is that a Noetherian ring that is neither a Klein ring nor a homomorphic image of a Dedekind-like ring must itself possess, as a homomorphic image, at least one of two very special types of “minimally wild” rings. These two special types of rings are defined next, and their finite-length wildness is proved in the next section.

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Definition 3.1 A local ring (R, m, k) is an Artinian triad provided the minimal number of generators of m is three, and m2 = 0. Definition 3.2 A local ring (R, m, k) is a Drozd ring provided the minimal number of generators of m is two; the minimal number of generators of m2 is two; m3 = 0; and there is an element x ∈ m \ m2 such that x 2 = 0. The prototypical example of an Artinian triad is k[x, y, z]/(x, y, z)2 , while k[x, y]/(x 2 , x y 2 , y 3 ) is the easiest example of a Drozd ring. Every Drozd ring has composition length 5 [12, (2.4.1)]. The ring Z[x]/(x 2 , p 2 x, p 3 ), where p is any prime number, is a Drozd ring that does not contain a field [12, Example 6.1]. The following “ring-theoretic” dichotomy theorem is established in [14, Theorem 14.3]. Theorem 3.3 Let R be a commutative Noetherian ring. Then exactly one of the following two possibilities occurs. (i) R is a Klein ring or a homomorphic image of a Dedekind-like ring. (ii) R has an Artinian triad or a Drozd ring as a homomorphic image. Note that this theorem is a genuine dichotomy: Exactly one of the cases (i) or (ii) occurs, but not both. Theorem 2.4 and Corollary 2.7 show that the rings in (i) have tame representation type, provided the relevant Dedekind-like rings are not exceptional. In the next section we give the promised formal definition of wildness from [12], and prove Theorem 4.3, that the rings in (ii) have wild representation type. To have a “tame-wild” dichotomy theorem for all Noetherian rings that are not exceptional Dedekind-like rings, one must then show that the relevant rings in (i) are not wild, and the rings in (ii) are not tame.

4 Wildness There are many definitions of wild representation type in the literature, for a commutative ring R and a subcategory C of mod-R. Always the subcategory C is assumed closed under direct summands, finite direct sums, and isomorphism. It is not clear whether or not the definitions are equivalent; proving equivalence of two typical definitions appears to be a formidable task. Nevertheless all of these definitions say, more or less, that a classification of the modules in C would give rise to a classification of all kx, y-modules of finite k-dimension, where k is a residue field of R, and x and y are non-commuting indeterminates. This would yield, for every finite-dimensional k-algebra A, a classification of all A-modules of finite k-dimension; see Remark 4.2. Most of these definitions involve some full, additive subcategory W of C and a notion of representation equivalence from W to kx, y-mod. Unfortunately, even the definition of “representation equivalence” has several variations, and we shall choose one that behaves well for our purposes and requires stronger assumptions

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than most versions in the literature. Of course, we must be sure that the definition we use for our proof of non-wildness also works for wildness of Drozd rings and Artinian triads. Another notion of wildness, the existence of a “representation embedding” from kx, y-mod to C, was used by Crabbe and Leuschke [5] to verify wildness of hypersurfaces of dimension at least 2 and multiplicity at least 4. A representation embedding is an exact functor preserving non-isomorphism and indecomposability. This notion was recently used also by Bongartz [1]. While the tame-wild dichotomy theorem for finite-dimensional algebras is proven for algebraically closed fields [6], the tame versus wild distinction makes sense for infinite fields in general [5]. For finite fields, however, it does not. Moreover, the functorial embedding by which one normally establishes wild representation type does not work for commutative rings which are not algebras over a field. Our main focus will be on the category of modules of finite length over a Noetherian ring. Definition 4.1 Let R be a Noetherian ring, k a residue field of R, and C a subcategory of mod-R closed under finite direct sums, direct summands, and isomorphism. We say that C has wild representation type with respect to k provided there exist a full, additive subcategory W of C, closed under finite direct sums, direct summands, and isomorphism, and a representation equivalence  from W to the category of finite-dimensional right kx, y-modules. This means that  is an additive functor that (i) is dense (i.e., maps onto all isomorphism classes), (ii) reflects isomorphism (i.e., M ∼ = N if and only if (M) ∼ = (N )), and (iii) is full (i.e., is surjective on homomorphism groups). It follows easily that  preserves and reflects indecomposability; closure of W under finite direct sums is needed for this. Closure of W under direct summands is used in the proof of Theorem 5.1. If F has wild representation type, where F is the category of R-modules of finite length, one says that R is finite-length wild. If C and C  are subcategories of mod-R closed under finite direct sums, direct summands, and isomorphism, such that C is contained in C  , and if C has wild representation type with respect to a residue field of R, then so has C  . In particular, if R is finite-length wild, then mod-R has wild representation type. If I is an ideal of R and R/I is finitelength wild, then so is R, since every finite-length R/I -module is also a finite-length R-module. Remark 4.2 A well-known trick of Brenner [3, Theorem 3] implies, for every finitely generated k-algebra A, that the category of finite-dimensional kx, ymodules has a full subcategory that is equivalent to the category of finite-dimensional A-modules. Thus, any classification of all isomorphism classes of all finitedimensional kx, y-modules would yield, for every finitely generated k-algebra A, a classification of all isomorphism classes of all finite-dimensional A-modules, a seemingly impossible task.

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From [9, Lemma 3] and [12, Theorem 4.9], with some enhancement, to allow for our stronger notion of wildness, we obtain the following wildness theorem, which establishes part of the tame-wild dichotomy for commutative Noetherian rings. Theorem 4.3 If (R, m, k) is an Artinian triad or a Drozd ring, then R is finite-length wild with respect to its residue field k. More generally, every Noetherian ring that has R as homomorphic image is finite-length wild with respect to k. Proof We examine the proofs of wildness of Artinian triads [9] and Drozd rings [12] separately, to ensure that the subcategories W defined there are indeed closed under direct summands and finite direct sums. For an Artinian triad (R, m, k), Guralnick, Levy and Warfield [9] write m = (a, b, c), and define a finitely generated R-module M to be a-translatable provided multiplication by a induces an isomorphism between the k-vector spaces M/mM and mM. (The assumption m2 = 0 makes mM a vector space.) The set W consists precisely of the a-translatable modules. A direct argument shows that, for R-modules M1 and M2 , the direct sum M1 ⊕ M2 is in W if and only if both M1 and M2 are in W. For a Drozd ring (R, m, k), Klingler and Levy [12, Definition 4.6] describe a class of R-modules called Ringel modules, adapted from [18]. By combining [12, Lemma 4.5 and Theorem 4.9], we obtain an additive category W  of Ringel modules that is a full subcategory of the category of finite-length R-modules, and a representation equivalence  from W  to the category of finite-dimensional modules over kx, y. The category W  is easily seen to be closed under finite direct sums. We need to check that it is closed under direct summands. For a module M in W  , let E(M) denote the endomorphism ring of M as an R-module; this is the same as the endomorphism ring of M as an object in the full subcategory W  . Let E((M)) be the endomorphism ring of (M) as a kx, y-module. By assumption,  defines a surjective ring homomorphism  M from E(M) onto E((M)). It follows from [12, Lemma 4.8] that ker( M ) is nilpotent. Since orthogonal idempotents lift modulo a nilpotent ideal, primitive orthogonal idempotents in E(M) correspond to primitive orthogonal idempotents in E(M)/ ker( M ) ∼ = E((M)). Therefore, if M = M1 ⊕ . . . ⊕ Mn , for some indecomposable R-modules M1 , . . . , Mn , the corresponding primitive orthogonal idempotents in E(M) map to primitive orthogonal idempotents in E((M)). Thus (M) = X 1 ⊕ . . . ⊕ X n , for some indecomposable kx, y-modules X 1 , . . . , X n . Since  is dense, there are modules Y1 , . . . , Yn in W  such that (Y j ) ∼ = X j , for each index j, so that (Y1 ⊕ . . . ⊕ Yn ) ∼ = X 1 ⊕ . . . ⊕ X n . Hence M ∼ = Y1 ⊕ . . . ⊕ Yn , because  reflects isomorphism. The endomorphism ring of each X j contains no nontrivial idempotents, so the same is true of each Y j . Therefore each Y j is an indecomposable R-module. Finally, by the Krull-Remak-Schmidt Theorem, we can assume that Y j ∼ = M j , for each index j. Hence every direct summand of M is isomorphic to a direct sum of some subset of the modules Y1 , . . . , Yn and therefore is a  module in W  .

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5 Non-wildness of Dedekind-Like Rings A complete proof of tame-wild dichotomy for finitely generated modules over Noetherian rings would require the following (in addition to what we have already covered in this paper): (i) Decide whether exceptional Dedekind-like rings have tame representation type, wild representation type, both, or neither. (ii) Show that tame Dedekind-like rings and Klein rings do not have wild representation type. Unfortunately, we have nothing to add regarding (i), other than to say that it appears to be a difficult problem. Here, and in the next section, we shall make some progress on (ii), for tame Dedekind-like rings, and also indicate some possible approaches to further progress. To our knowledge, the non-wildness of Klein rings is still open, though we do not expect that a proof would be difficult.

5.1 Brauer Groups The Brauer group Br(k) of a field k is the set of equivalence classes of finitedimensional central simple k-algebras, with the operation given by ⊗k . A k-algebra A is central simple provided it is a simple ring with center k. By the Wedderburn Structure Theorem, every finite-dimensional central simple k-algebra is of the form Mat n (D), for some finite-dimensional division algebra D, where Mat n denotes the ring of n × n matrices. The equivalence relation is: Mat p (D) ∼ Matq (D), for every such division ring D and each pair p, q of positive integers. Thus Br(k) parametrizes the set of finite-dimensional division algebras over k. In particular, Br(k) is non-trivial if and only if k has a non-commutative finite-dimensional division algebra. In the proof of the following theorem, the main idea comes from Hyeja Byun’s paper [4], based on her Wisconsin Ph.D. dissertation, supervised by Lawrence Levy. The results we need are [4, Theorem 2.18 and Theorem 3.15]. Theorem 2.18 applies when the module M in the proof below has finite length, and Theorem 3.15 applies when M has infinite length. In the statements of these theorems, the local rings R P are split Dekekind-like as in Remarks 2.2(III), though the word “split” is not mentioned in [4]. In fact, at that time all Dedekind-like rings were assumed to be split. The unsplit ones came along later. Theorem 5.1 Let (R, m, k) be a split local Dedekind-like ring, and assume that Br(k) = 0. Then mod-R is not wild. Proof Suppose the assertion is false. Let W be a full subcategory of mod-R, closed under finite direct sums, direct summands, and isomorphism; let  : W → modkx, y be a representation equivalence. Since Br(k) = 0, there exists a finitedimensional division algebra D over k that is non-commutative. By Brenner’s

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trick from Remark 4.2, the category mod-kx, y has a full subcategory D that is equivalent to mod-D. Let D  be the object of D corresponding to the D-module D, and choose, by density, an R-module M with (M) isomorphic to D  . Then Endkx,y D  ∼ = EndD D  ∼ = End D (D), a division ring, and hence D  is indecomposable in mod-kx, y. Then M is indecomposable in W and, since W is closed under direct summands, M is indecomposable in mod-R. Now E := End R M maps onto Endkx,y D  ∼ = D, a non-commutative division ring, say, E/I ∼ = D, where I is a right ideal of E. Let N be the nilradical of E. By [4, Theorem 2.18 and Theorem 3.15], E/N is either a simple algebraic field extension K of k, or the endomorphism ring H of an ideal of R. By Remark 5.2 below, H is commutative. Since E/I ∼ = D, one has I ⊇ N . This provides a surjective ring homomorphism from the commutative ring E/N onto E/I ∼ = D, a contradiction since D is non-commutative.  The next result is surely well known, but we give a detailed proof just for fun. Remark 5.2 Let I be an ideal in a reduced Noetherian ring R. Then End R (I ) is commutative. Proof Let J be the annihilator of I . If x ∈ R, n ∈ N, and x n ∈ J , then (x I )n = x n I n = 0, whence x I = 0. This shows that R/J is reduced. Moreover, I ∩ J = 0, and hence the composition I → R  R/J is injective; that is, I may be regarded as an ideal of R/J . Since End R (I ) ∼ = End R/J (I ), we may assume at the outset that I is faithful. We claim that I contains a non-zerodivisor of R. If not, then by [11, Sec. 2-2, Exercise 12], I is contained in the union P1 ∪ · · · ∪ Ps of the minimal prime ideals of R and hence in one of them, say P1 . But P1 is annihilated by the non-zero ideal (P2 ∩ · · · ∩ Pn ), contradicting faithfulness of I . Let x ∈ I be a non-zerodivisor of R. Given any R-endomorphism ϕ of I , put ϕ(x) = y. Then, for each z ∈ I , we have xϕ(z) = ϕ(zx) = zϕ(x) = zy = x

y x

z,

  and hence ϕ(z) = xy z. This shows that every endomorphism of I is given by multiplication by a suitable element of the total quotient ring of R and hence that End R (I ) is commutative.  The field of real numbers has a non-commutative division algebra, namely, the quaternions. Hence a split local Dedekind-like ring with the real numbers as residue field is not wild. The literature is full of computations of Brauer groups, and there are many classes of fields with non-trivial Brauer groups. In the next section we give more examples.

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6 What’s Left? Much of this section of the paper is speculative. The following conjecture helps to focus this discussion. Conjecture 6.1 Let (R, m, k) be a non-exceptional local Dedekind-like ring. Then R is not finite-length wild. Outline 6.3 gives a possible approach to attack Conjecture 6.1, at least for R a split local Dedekind-like ring. As we discuss this approach, motivated by Theorem 5.1, we indicate various problems that need to be solved for a complete proof. We use the following fact: Fact 6.2 Suppose that (R, m, k) is a local ring and  is a field extension of k. There exists a local ring (S, n, ) and a flat local homomorphism R → S (called a gonflement) that lifts the inclusion k → ; see [2, Appendice] or [16, Chapter 10, Sect. 3]. Outline 6.3 Let (R, m, k) be a split local Dedekind-like ring. Step 1: Show that k has a field extension k →  such that Br() = 0, where  is a separable extension (possibly trivial) followed by an extension adjoining two variables. Step 2: Let (R, m, k) → (S, n, ) be a gonflement lifting the field extension k → . Show ascent of split local Dedekind-like for gonflements. That is, show that S is or can be chosen to be split local Dedekind-like. Step 3: Assuming that steps 1 and 2 go through, put them together and apply Theorem 5.1 to (S, n, ) to show that S is not wild. Step 4: Show descent of non-wildness for gonflements. That is, show that S is not wild implies that R is not wild. Remarks 6.4 Assume the notation of Outline 6.3. (1) We believe that step 1 is straightforward. In fact, we believe that every field k has an extension k →  such that Br() = 0, where  is a separable extension (possibly trivial) followed by an extension adjoining two variables. To support this claim, we show in Proposition 6.9 that, for every field k of characteristic = 2, the extension k →  := k(x, y) is such that Br() = 0, where x and y are indeterminates. In the case where the characteristic of k is 2, we suggest an adjustment, Example 6.12, to Example 6.8, and we assign Problem 6.13 to check the details. This would complete step 1. (2) Regarding step 1, Tsen’s theorem [7, Corollary 11.1.9] says that C(x) has trivial Brauer group. Therefore we really need to adjoin two algebraically independent elements over k to get an extension with non-trivial Brauer group. (3) The fact that, even in the case of characteristic two, there is no inseparable component in the extension k →  is essential for following Outline 6.3; see Example 6.5. Thus, in the case of characteristic 2, the algebraic part of the extension for step 1 is required to be separable. (3) Steps 2 and 4, respectively, are Problem 6.6 and 6.7 below. For step 3, by Theorem 5.1, mod-S does not have wild representation type.

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Example 6.5 Let k be an imperfect field of characteristic 2, and let α be an element 2 2 of k with no square root in k. The ring R := k[[x, y]]/(x √ + αy ) of [16, Example 10.17] is an exceptional Dedekind-like ring. Let  = k( α). The gonflement R → S := R ⊗k  = [[x, y]]/(x +

√ α y)2

lifts the purely inseparable field extension√ k → ; but S is not Dedekind-like, since it has the non-zero nilpotent element x + α y. Problem 6.6 Let (R, m, k) → (S, n, ) be a gonflement lifting the field extension k → , where R is a split local Dedekind-like ring. Assume k →  is either a finite separable extension or a finite separable extension (possibly trivial) followed by the adjunction of two independent indeterminates. Prove that the local ring S is split Dedekind-like. Problem 6.7 Let k →  be a field extension that is either a finite separable extension or a finite separable extension (possibly trivial) followed by the adjunction of two independent indeterminates. Let (R, m, k) → (S, n, ) be a gonflement lifting k → , where R and S are non-exceptional split Dedekind-like rings. If Br() = 0, prove that R is not wild. Informally, to solve Problem 6.7, one might proceed as follows: By Theorem 5.1, the ring S is not wild. By faithfully flat descent [8, (2.5.8)], the faithful functor M → S ⊗ R M from mod-R to mod-S reflects isomorphism. This means that the module category mod-R embeds in the category mod-S. Informally speaking, this ought to mean that the mod-R is no more complicated than mod-S, and hence that mod-R does not have wild representation type. But, alas, that’s not a proof.

6.1 Field Extensions and Division Algebras Example 6.8 is useful for step 1 of Outline 6.3. Example 6.8 Let k be a field with characteristic not equal to 2, and let  = k(x, y) be the function field in two variables over k. Let D be the -algebra [a, b], where a 2 = x, b2 = y, and ba = −ab. This is a four-dimensional vector space over , with basis {1, a, b, ab}. In fact, D is a well-defined ring called a symbol algebra [7, Lemma 13.4.6]. Clearly D is non-commutative, because the characteristic is not 2. Proposition 6.9 Let k be a field with characteristic not equal to 2 and let  := k(x, y), as in Example 6.8. Then  is an extension field of k with non-trivial Brauer group, i.e., Br() = 0. In particular, in the notation of Example 6.8, the non-commutative ring D is a division algebra over . Proof To prove the proposition, it suffices to prove the “In particular …” statement. To see that D is a division algebra, we imitate what one does to show that the

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quaternions form a division algebra. Let g be a nonzero element of D, and find a common denominator v ∈ k[x, y] so that gv = r + sa + tb + uab, for some r, s, t, u ∈ k[x, y], not all zero. Let h = r − sa − tb − uab, and check that gvh = hgv = r 2 − s 2 x − t 2 y + u 2 x y. By Lemma 6.11 below, since at least one of r, s, t, u is nonzero, gvh = hgv is a non-zero element of k[x, y], and therefore is a unit of . Now gv and v are units of D, and hence g is a unit. Thus D is a division algebra.  The proof of Lemma 6.11 depends on some easy facts: Facts 6.10 Let k be a field, x, y indeterminates, T = k[x, y] m = (x, y), n ∈ N0 , and α, s, t ∈ T . (1) (2) (3) (4)

If xα ∈ mn+1 , then α ∈ mn . If α ∈ mn and α 2 ∈ m2n+1 , then α ∈ mn+1 . If s, t ∈ mn and s 2 x + t 2 y ∈ m2n+2 , then s, t ∈ mn+1 . If r ∈ mn+1 , u ∈ mn , and r 2 + u 2 x y ∈ m2n+3 , then u ∈ mn+1 .

Proof To prove (1) we use the equation mn+1 = xmn + (y n+1 ) to write xα = xa + y n+1 b, with a ∈ mn and b ∈ T . Now x(α − a) = y n+1 b, and x | b, by unique factorization. Thus α − a = y n+1 bx , and α = a + y n+1 bx ∈ mn . For the proofs of (2), (3), and (4), we use the usual grading on T = k[x, y], with x and y having degree 1. Thus Tn is the k-vector space with basis {x n , x n−1 y, . . . , x y n−1 , y n }. For (2), suppose α ∈ / mn+1 , and write α = αn + h, where 0 = αn ∈ Tn and h ∈ n+1 2 2 m . Then α = αn + 2αn h + h 2 , which has a non-zero component of degree 2n, / m2n+1 , a contradiction. namely αn2 . This means that α 2 ∈ Suppose (3) fails, and choose the least n for which it fails. Then s ∈ / mn+1 or n+1 n+1 t∈ / m . Clearly n ≥ 1. Write s = sn + u, where sn ∈ Tn and u ∈ m ; and write t = tn + v, where tn ∈ Tn and v ∈ mn+1 . By assumption, sn = 0 or tn = 0. The term of degree 2n + 1 in s 2 x + t 2 y (which, by assumption, is 0) is sn2 x + tn2 y. From the equation sn2 x + tn2 y = 0, we see, using unique factorization, that x | tn and y | sn . Write tn = x p and sn = yq, where p, q ∈ Tn−1 and at least one is nonzero. Then: y 2 q 2 x + x 2 p 2 y = 0 and hence p 2 x + q 2 y = 0 ∈ m2(n−1)+2 . Moreover p, q ∈ mn−1 , and, by minimality of n, we have p, q ∈ mn . But then sn and tn are both in mn+1 , and so s and t are both in mn+1 , a contradiction to the alleged failure. Finally, suppose (4) is false, that is, u ∈ mn \ mn+1 . Then u = u n + g, where u n ∈ Tn \ {0} and g ∈ mn+1 . Write r = rn+1 + h, with rn+1 ∈ Tn+1 and h ∈ mn+2 .

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By (1) and (2) we are done if vn = 0 or u n = 0. Thus assume vn = 0 or u n = 0. 2 + x yu 2n ∈ T2n+2 ∩ m2n+3 = 0. Putting a = rn+1 and b = u n , we have an Then rn+1 expression a 2 + x yb2 = 0 ,

6.10.i

with a ∈ Tn+1 and b ∈ Tn \ {0}. Assume that n is the smallest non-negative integer for which such an expression exists. Then x | a 2 and hence x | a. Write a = cx, where c ∈ Tn . By (6.10.i), c2 x + b2 y = 0 .

6.10.ii

From (6.10.ii), we see that y | c. Thus n ≥ 1 and c = yd, where d ∈ Tn \ {0}. Now we have, from (6.10.ii), the equation b2 + d 2 x y = 0, with b ∈ Tn \ {0} and d ∈  Tn−1 \ {0}. Since n ≥ 1, we have a contradiction to the minimality of n. Lemma 6.11 If r, s, t, u ∈ k[x.y] and r 2 − s 2 x − t 2 y + u 2 x y = 0,

6.11.0

then r = s = t = u = 0. Proof It will suffice to show that r, s, t, u ∈ mn , for every n ∈ N0 , where m = (x, y). This is true for n = 0. Assume, inductively, that n ∈ N0 and r, s, t, u ∈ mn . We show that r, s, t, u ∈ n+1 m . Equation 6.11.0 implies r 2 ∈ m2n+1 . By Fact 6.10(2) we have r ∈ mn+1 . Since u ∈ mn , we have s 2 x + t 2 y ∈ m2n+2 , by Eq. 6.11.0. By Fact 6.10(3), we have s, t ∈ mn+1 . Now r 2 + u 2 x y ∈ m2n+3 , by Eq. 6.11.0, and hence u ∈ mn+1 , by Fact 6.10(4).  Thus we have proved that every field k of characteristic different from two has an extension k →  such that Br() = 0, where  is a separable extension (here this part is trivial) followed by an extension adjoining two variables, as desired for step 1 of Outline 6.3. In Example 6.12, where the field k has characteristic two, we make a slight adjustment to Example 6.8 in order to obtain an extension k →  that fits step 1 of Outline 6.3. Example 6.12 For a field k of characteristic 2, we can adjoin, if necessary, a primitive cube root of unity ζ , and construct the symbol algebra D = [a, b] over the function field  = k(ζ )(x, y) in two variables over k(ζ ) (or over k if ζ ∈ k), with relations a 3 = x, b3 = y, and ba = ζ ab. A brave reader can attempt to show that this algebra D, a nine-dimensional vector space over , is in fact a division algebra. Problem 6.13 Fill in the details and proof for Example 6.12, the case of characteristic two. Modulo the solution of Problems 6.6, 6.7, and 6.13, we have a proof of Conjecture 6.1 in case R is split local Dedekind-like.

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6.2 Unsplit Dedekind-Like Rings What if (R, m, k) is unsplit? That is, R is local Dedekind-like and the normalization R is a discrete valuation domain with residue field K of degree 2 over k. Does Conjecture 6.1 hold? Assuming R is not exceptional, we know that the residue-field growth k → K is separable. Choose a gonflement (R, m, k) → (S, n, K ) lifting k → K . Problem 6.14 Prove that the gonflement ring (S, n, K ) above is split local Dedekindlike. A solution to Problem 6.14 and the descent assertion in Problem 6.7, together with the split case of Conjecture 6.1 will yield the general case of the conjecture.

References 1. K. Bongartz, Representation Embeddings and the Second Brauer-Thrall Conjecture. arXiv:1611.02017 2. N. Bourbaki, Éléments de mathématique. Algèbre commutative. Chapitres 8 et 9 (Springer, Berlin, 2006). Reprint of the 1983 original. MR2284892 3. S. Brenner, Decomposition properties of some small diagrams of modules. Symp. Math. 13, 127–141 (1974) 4. H.L. Byun, Endomorphism rings of modules over Dedekind-like rings. Kyungpook Math. J. 26(2), 89–112 (1986) 5. A. Crabbe, G.J. Leuschke, Wild hypersurfaces. J. Pure Appl. Algebra 215, 2884–2891 (2011) 6. W.W. Crawley-Boevey, On tame algebras and Bocses. Proc. London Math. Soc. 56, 451–483 (1988) 7. T.J. Ford, Separable algebras. Grad. Stud. Math. 183, AMS (2017) 8. A. Grothendieck, J. Dieudonné, Éléments de géometrie algébrique IV, Partie 2, Publ. Math. I.H.E.S. 24 (1967) 9. R.M. Guralnick, L.S. Levy, R.B. Warfield, Cancellation counterexamples in Krull dimension 1. Proc. AMS 109(2), 323–326 (1990) 10. W. Heinzer, L.S. Levy, Domains of dimension 1 with infinitely many singular maximal ideals. Rocky Mount. J. Math. 37, 203–214 (2007) 11. I. Kaplansky, Commutative Rings (Allyn and Bacon Inc, Boston, 1970) 12. L. Klingler, L.S. Levy, Representation type of commutative Noetherian rings. I. Local wildness. Pac. J. Math. 200, 345–386 (2001) 13. L. Klingler and L. S. Levy, Representation type of commutative Noetherian rings. II. Local tameness. Pac. J. Math. 200, 387–483 (2001) 14. L. Klingler, L. S. Levy, Representation type of commutative Noetherian rings. III. Global wildness and tameness. Mem. AMS 176(832), viii+170 pp (2005) 15. L. Klingler, L.S. Levy, Representation type of Commutative Noetherian rings (introduction), Algebras, Rings and their Representations, 113–151 (World Scientific Publishing, Hackensack, NJ, 2006) 16. G.L. Leuschke, R. Wiegand, Cohen-Macaulay representations. Math. Surv. Monogr. 181, AMS (2012) 17. L.S. Levy, Modules over Dedekind-like rings. J. Algebra 37, 1–116 (1985) 18. K.M. Ringel, The representation type of local algebras. Lecture Notes in Mathematics, vol. 488 (Springer, 1975), pp. 282–305

On the Characterization of τ(n) -Atoms A. Hernández-Espiet and R. M. Ortiz-Albino

Abstract In 2006, Anderson and Frazier define the concept of τ(n) -factorization, where τ(n) is a restriction of the modulo n equivalence relation. These relations have been worked mostly for small values of n. One of these problems is finding τ(n) -irreducible elements or τ(n) -atoms in order to characterize elements that have a τ(n) -factorization in τ(n) -atoms. The τ(n) -irreducible elements are well known for n = 0, 1, 2, 3, 4, 5, 6, 8, 10, 12. However, the problem of determining the τ(n) -atoms becomes much more difficult the larger n is. In this work, we present an algorithm to construct families of τ(n) -atoms. It is shown that the algorithm terminates in finitely many steps when n is the safe prime associated to a Sophie Germain prime. Keywords τ(n) -factorizations · Factorizations over integral domains · Generalized factorizations 2010 Mathematics Subject Classification 13A05 · 11A99 · 20K01

1 Introduction Anderson and Frazier developed the theory of τ -factorizations [1], or τ -products on integral domains, where τ is a symmetric relation that determines which elements are allowed to be multiplied. This concept can be visualized as the study of a restriction to the multiplicative operation. That is, two nonzero nonunit elements are allowed to be multiplied if and only if they are related with respect to the symmetric relation τ . Formally, let τ be a symmetric relation on the nonzero nonunit elements of an Department of Mathematics, University of Puerto Rico at Mayagüez. A. Hernández-Espiet HC20 Box 26292, San Lorenzo 00754, Puerto Rico e-mail: [email protected] R. M. Ortiz-Albino (B) 1011 Sonsire Chalets, Mayagüez 00682, Puerto Rico e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. Badawi and J. Coykendall (eds.), Rings, Monoids and Module Theory, Springer Proceedings in Mathematics & Statistics 382, https://doi.org/10.1007/978-981-16-8422-7_12

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integral domain D (denoted by D # ). We say that x ∈ D # has a τ -factorization if x = λx1 ∗ ∗ ∗ xn where λ is a unit and for any i = j, xi τ x j . In such case, we also say that each xi is a τ -factor of x (denoted by xi |τ x). Notice that, x = x and x = λ(λ−1 x) are both (vacuously) τ -factorizations, called the trivial ones. In order to distinguish a usual product from a τ -product we will denote by a1 · a2 · · · an (respectively, a1 ∗ a2 ∗ ∗ ∗ an ) the usual product (respectively, τ -product) of the elements a1 , a2 , . . . , an . An element x ∈ D # whose only τ -factorizations are the trivial ones is called a τ -atom or τ -irreducible element. The theory of τ -factorization was also called the theory of generalized factorizations. This is because one can consider the relation τ = S × S, where S is any specific subset of D # . If τ = S × S where S is a subset of the nonzero nonunit elements, then the τ -products are the usual products of elements in S. As an example, take S to be the set of irreducible elements (resp. primes, primal elements, etc.), hence the τ -factorizations would be factorizations into irreducible elements (resp. into primes, primal elements, etc.). Also, if S = D # , then the τ -products and the usual products of D # coincide. In [1], the authors presented results based on a classification of types of relations. Two of these types are divisive (if xτ y and x  |x, then x  τ y) and associated-preserving (if xτ y and x  ∼ x, x  τ y) relations. Associated-preserving relations are well behaved compared to other relations, as can be seen in [2]. These relations allow us to omit the unit multiple in front, that is, if λx1 ∗ ∗ ∗ xn is a τ -factorization, so is x1 ∗ ∗ ∗ xi−1 ∗ (λxi ) ∗ xi+1 ∗ ∗ ∗ xn . Divisive relations allow us to admit τ -refinements; that is, whenever x = λx1 ∗ ∗ ∗ xn and xi = y1 ∗ ∗ ∗ ym are τ -factorizations, so is x = λx1 ∗ ∗ ∗ xi−1 ∗ y1 ∗ ∗ ∗ ym ∗ xi+1 ∗ ∗ ∗ xn . It is important to note that if a relation is divisive, then it is associated-preserving. Most of the relevant results with respect to τ -factorizations in [1] assumed relations to be divisive. On the other hand, to avoid studying the usual product structure, the relations to consider cannot be both reflexive and divisive. If this were the case, then for any x, y ∈ D # , (x y)τ (x y), by reflexivity. On the other hand, by the divisivity of τ , xτ y. This motivated Ortiz and Serna [10] to study the behavior of the τ -factorizations when τ is an equivalence relation (aside from their historical importance). Their main results were based on unital equivalence relations (that is, equivalence relations with the following property: if xτ y, then for any unit λ, (λx)τ (λy)). If τ  is the associated-preserving closure of τ (the smallest associate-preserving equivalence relation that contains τ ), then x has a τ -factorization (respectively, x|τ y, x is a τ -atom) if and only if x has a τ  -factorization (respectively, x|τ  y, x is a τ  -atom). Theorem 4.17 of [10] says that: x1 ∗ ∗ ∗ xn is a τ  -factorization if and only if there are units λ, λ1 , . . . , λn , such that λ(λ1 x1 ) ∗ ∗ ∗ (λn xn ) is a τ -factorization. For example, “=” is not an associate-preserving unital equivalence relation, but its associate-preserving closure is. Moreover, (−1)(2) ∗ (−2) ∗ (2) ∗ (2) is a = -factorization that induces a =-factorization given by 2 ∗ 2 ∗ 2 ∗ 2. So every time one can find a τ  -factorization there will be a τ -factorization. With respect to finding τ -factorizations, if τ is a unital equivalence relation, one may assume that the unital equivalence relation is also associated-preserving. See [10] to see other properties that are preserved with respect to τ and τ  .

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As a special case, the authors of [10] consider the relation τ(n) defined on Z# by Anderson and Frazier [1], and studied further by Hamon [5]. The relation τ(n) = {(x, y)|x − y ∈ (n)}, can be seen as the equivalence relation modulo n restricted to Z# . Hamon gave a characterization of n for which every element of Z# has a τ(n) factorization into τ(n) -atoms. Hamon’s results include {0, 1, 2, 3, 4, 5, 6, 8, 10, 12}. In 2013, Juett (Example 4.14(5) in [6]) found out that 12 does not satisfy such property. It is important to note that these results did not require knowing all elements which are τ(n) -atoms. An attempt to find τ(n) -atoms for any n was given by Lanterman [8] in 2012. In his paper, there is no information about the technique or method used to reduce the problem of finding the τ(n) -atoms to a finite number of cases by hand. It must be  . A second attempt noted that he used a relation called μn which coincides with τ(n) was done by Molina in [9], by using the formula of the number of τ(n) -factors for any positive integer. He gave the explicit forms of the τ(n) -atoms for n = 8, 10, 12. This research shows that the technique we think is used in [8] only works when n are both prime. We also formally define algorithms to find these τ(n) -atoms. and n−1 2 The authors recognize that this paper may seem technical an number theory oriented, but the results are needed in order to continue the notions of τ(n) -factorization into τ(n) -irreducible elements.

2 Preliminaries By Serna’s result (Theorem 4.17 of [10]), in order to determine whether or not a  -factorizations. In general, number has a τ(n) -factorization, it suffices to work with τ(n)  . We shall now define the results obtained in this paper use the equivalence relation τ(n)  a group that is compatible with τ(n) . For n ≥ 2, let U (n) = {[m] ∈ Z/nZ| gcd(n, m) = 1}, the multiplicative group of units in Z/nZ with order ϕ(n), where ϕ(n) is Euler’s totient function, as in [3]. We define U  (n) for n ≥ 3 to be the set {k|[k] ∈ U (n)}, where k¯ = [k] ∪ [−k] = {x ∈ Z|x ≡ k(modn) or .x ≡ −k(modn)} . Since U  (2) = U (2), we focus on n ≥ 3. We shall see that U  (n) is a group with a multiplicative operation. Proposition 2.1 (U  (n), ·) is an abelian group of order U (n)/{±1}.

ϕ(n) , 2

isomorphic to

Proof It is straightforward that this is an abelian group with operation x · y = x y. Define the map f from U (n) to U  (n) as f ([k]) = k. We shall prove that f is a homomorphism. If k ≡ k  (mod n), then k = {[x] ∈ U (n)|x ≡ ±k (mod n)} = {[x] ∈ U (n)|x ≡ ±k  (mod n)} = k  . Therefore, f is well defined. If k ∈ U  (n), gcd(k, n) = 1. Hence, [k] ∈ f −1 (k), making f surjective. Also, if [a], [b] ∈ U (n),

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then f ([ab]) = ab = a · b = f ([a]) f ([b]), so that f is a homomorphism. If [x] ∈ ker( f ), then x = 1. So, ker( f ) = {±[1]}. By the first isomorphism theorem, ϕ(n)  U (n)/{[±[1]} ∼ = U  (n), and |U  (n)| = 2 . Proposition 2.2 Let x be a composite number and p a prime such that p−1 is also a 2 (not necessarily distinct) prime divisors of x in some prime. If there are at least p−1 2 i = 1, then x has a τ( p) -factorization. Proof Let p1 , p2 , . . . , pk in i with i = 1 be the aforementioned prime divisors of x. Let x  = p1 ···x pk , or equivalently, x = p1 · · · pk x  . Since U  ( p) has prime order, U  ( p) is cyclic. Additionally, each nonidentity element in U  ( p) is a generator of the group. . Since In other words, i generates U  ( p). By assumption, we also have that k ≥ p−1 2  −1 j   U ( p) is a group, x has a multiplicative inverse. Hence, (x ) = (i) , for some . If j = p−1 , then x  = 1. This would imply that p1 ∗ ∗ ∗ pk−1 ∗ ( pk x  ) 0 < j ≤ p−1 2 2  − 1, then x  = i, so that p1 ∗ ∗ ∗ pk−1 ∗ pk ∗ x  is a τ( p) -factorization of x. If j = p−1 2 is a τ( p) -factorization of x. Otherwise, 0 < j ≤ p−1 − 2. For these cases it is true 2 that k − j ≥ 2. Then, p1 ∗ p2 ∗ ∗ ∗ ∗ ∗ pk− j−1 ∗ ( pk− j ( pk− j+1 · · · pk x  )) is a τ( p) factorization of x, because pk− j+1 · · · pk x  = 1. By Theorem 4.17 of [10], x has a τ( p) -factorization.  This result yields the first method for determining all the τ(n) -atoms for certain prime values of n. In these cases, for x to be a τ( p) -atom, it needs to have at most |U  ( p)| primes that are equivalent in U  ( p) in its factorization. Note that whether or not numbers have a τ( p) -factorization depends only on which elements of U  ( p) the dividing primes lie in, and not the actual value of the primes. By the last proposition, we only have to check finitely many numbers, the product of primes to powers lesser , before we determine all τ( p) -atoms for these values of p. It seems like than p−1 2 this is the technique used by Lanterman in [8]. In Sect. 4, we shall show that this is not prime. technique does not hold in general, in particular when p−1 2

3 Structure of U  (n) Recall that Proposition 2.2 depended on the group structure of U  (n). Thus, it will prove useful to understand the group structure of U  (n). First, we shall determine exactly when U  (n) is cyclic. In Proposition 2.2, we never used the fact that p has to be a prime. The condition was prime. A question is, for which n does this that really mattered was that ϕ(n) 2 happen? Well, the answer is for n = 9, 12, 18, p, 2 p for p any safe prime associated to a Sophie Germain prime q (that is, p = 2q + 1 is also prime). Proposition 3.1 prime p.

ϕ(n) 2

is prime if and only if n = 9, 12, 18, p, or n = 2 p for any safe

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Proof Let n = pa , p prime, a > 1. We have that ϕ(n) = 21 pa−1 ( p − 1). The only 2 p−1 way this is prime is if a = 2, and 2 = 1, so that p = 3, and n = 9. Now, let n be squarefree, but not prime or 2 times a prime. If n is a product of . With the exception of p = 3 or 2 odd primes, p and q, then 21 ϕ( pq) = ( p−1)(q−1) 2 q = 3, however, p − 1, q − 1 both have more than 1 factor. This would tell us that would not be prime. If p = 3, ( p − 1)(q − 1) has at least 4 factors so that ( p−1)(q−1) 2 it is still true that ( p − 1)(q − 1) has at least 3 factors so that, again, ( p−1)(q−1) is not 2 prime. This shows us that this case produces no numbers that satisfy the hypothesis. For the same reason, if n is odd, squarefree and more than two primes divide it, then we obtain no numbers that satisfy the hypothesis. By the same explanation, if n is even, squarefree, and more than three primes divide it, then such numbers do not satisfy the hypothesis. Let n be neither squarefree, nor a prime power. A similar analysis as in the first case shows that the only numbers that satisfy the conditions of the hypothesis are n = 12, 18. This only leaves us with n = p or n = 2 p, where p is a prime. This case follows directly from the definition of Sophie Germain primes. This concludes our proof.  Lemma 3.2 If U (n) is cyclic, then U  (n) is cyclic. ∼ U (n)/{±[1]}, the result follows from the fact that all quotients Proof Since U  (n) = of cyclic groups are cyclic.  By the Primitive Root Theorem in [7], Lemma 3.2 implies that n = 2, 4, p k , 2 p k (for any odd prime p) are values for which U  (n) is cyclic. It turns out that these are not the only values of n for which U  (n) is cyclic. The next propositions will help us develop a criterion to find the other values of n such that U  (n) is cyclic. Proposition 3.3 Let [x] ∈ U (n) with |[x]| = ϕ(n) . 4

ϕ(n) . 2

If x i ≡ −1 (mod n), then i =

Proof If x i ≡ −1 (mod n), then without loss of generality let 1 < i < ϕ(n) , due to 2 ϕ(n) 2i the order of x. So x ≡ 1 (mod n). By Lagrange’s Theorem in [3], 2 divides 2i. This means, 2i = ϕ(n) k for some k. Since 1 < i < ϕ(n) , 2 < 2i < ϕ(n). This forces 2 2 ϕ(n) ϕ(n) k = 1 and 2i = 2 . Hence i = 4 .  Note that x k = 1 if and only if x k ≡ ±1 (mod n). Hence, if x k ≡ ±1 (mod n) , then x is necessarily a generator of U  (n). The previous proposition for 1 ≤ k < ϕ(n) 2 ϕ(n) shows that if there is an [x] ∈ U (n) such that |[x]| = ϕ(n) and x 4 ≡ −1 (mod n), 2 then U  (n) = x . We shall now see how the group structure of U (n) gets restricted by having an . element of order ϕ(n) 2 Proposition 3.4 If U (n) is not cyclic and has an element of order (Z/n 1 Z) × (Z/n 2 Z), with gcd(n 1 , n 2 ) = 2.

ϕ(n) , then U (n) 2

∼ =

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Proof By Theorem 1 of [4], and the discussion after the theorem, U (n) is isomorphic to a finite product of cyclic groups with even order. If n = p1k1 · · · pmkm , then k k U (n) ∼ = U ( p11 ) × · · · × U ( pmkm ). If pi is odd, then U ( pi i ) is cyclic by the Primitive Root Theorem [7], and if pi = 2, and U (2ki ) is not cyclic, then it is isomorphic to Z/2ki −2 Z × Z/2Z. For simplicity, let the decomposition by cyclic factors be U (n) ∼ = (Z/n 1 Z) × · · · × (Z/n k Z), where Z/n i Z either corresponds uniquely kj to U ( p j ) for some j, or corresponds uniquely to one of the cyclic factors of U (2k j ), for some j, if it exists. This choice of decomposition for U (n) guarantees that 2 | n i for all i. By assumption, the product will have at least 2 multiplicands. Also, ϕ(n) = n 1 · · · n k . The largest possible order of an element in U (n) will be . However, since the n j ’s are lcm(n 1 , n 2 , . . . , n k ), so that the lcm is equal to ϕ(n) 2 ···n k . Thus, 1 = k − 1 or k = 2, as all even, we also have lcm(n 1 , n 2 , . . . , n k ) ≤ n21 k−1 claimed. If gcd(n 1 , n 2 ) > 2, then lcm(n 1 , n 2 ) = diction to the existence of an element of (Z/n 2 Z), where gcd(n 1 , n 2 ) = 2.

n1 n2 gcd(n 1 ,n 2 ) . order ϕ(n) 2


2. Additionally, b, c = 0, since otherwise α would not be an irreducible element form. Without loss of generality, let b > 1. All of these conditions would imply that pr2 = 1, meaning that pr is its own inverse. However, ps is the inverse of pr . This implies that r = s, a contradiction to α being an irreducible element form.  With this, we can eliminate all the cases in which we would attempt a double substitution in the algorithm (which in practice saves about half the run time). Modifying the algorithm, we obtain: Algorithm 5.3 Let Ła1 , . . . , Łak partition U  (n). To find all irreducible element forms in U  (n), (1) (2) (3) (4) (5) (6) (7) (8)

Fix a1 , . . . , ak ∈ U  (n). Set i = 1 Set the element form (x)1 to be equivalent to ai . Obtain all element forms obtainable by one single substitution on (x)1 . Keep the ones that are irreducible and discard that are not. Repeat this process with the resulting element forms. Return to step (5) until all irreducible element forms equivalent to ai are found. Set i = i + 1 and return to step (3) or move to next step if i = k. Apply all automorphisms of U  (n) to all the irreducibles, and keep one copy of each.

The same comments that were said for the previous algorithm apply for this algorithm, especially the observation that if n is not a safe prime, then there is no guarantee that this algorithm will terminate in a finite number of steps. We shall give an example illustrating this algorithm for n = 11. In this case, U  (n) has 5 elements, 1, 2, 3, 4, 5. 2

4·5

2

=⇒

2

2

3

=⇒

2 ·5

(1.3)

4·5

(1.2)

2

(1.1)

At each level, every product has the same number of factors as the number of the level it is in. So start with a single node, as in (1.1). We do all simple substitutions, which can be seen in (1.2), in the second step. Now, with this level, (1.2), the rightmost 2  number 3 turns out to τ(11) -factorize as 3 ∗ 3. Thus, we cross it out, as it does not produce any irreducible elements. The next level, (1.3), produces 4 terms given by

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2

2

3

2

2 · 5, 3 · 5 = 3 ∗ 5 , 2 · 3 · 4 = (2 · 4) ∗ 3, and 4 = 4 ∗ 4 ∗ 4. Only 2 · 5 is a τ(11) irreducible element. Finally, doing substitutions on the remaining term produces no irreducible elements. This concludes the process for a2 in Algorithm 5.3, by the previous discussion. When the correspondence is made from the previous graph to the element forms, and all the automorphisms ϕ are applied, we obtain the following graph: (ϕ(2))2 · (ϕ(5))1

=⇒

(ϕ(4))1 · (ϕ(5))1 (ϕ(3))2

(ϕ(2))1

=⇒

(ϕ(4))1 · (ϕ(5))1

(ϕ(2))1

(ϕ(2))1

There are three automorphisms other than the identity for U  (11): the one that sends 2 to 3, the one that sends 2 to 4 and the one that sends 2 to 5. Therefore, by Proposition 4.5, the previous graph contains all the element forms equivalent to a2 , a3 , a4 , and a5 (in Algorithm 5.3). Explicitly, the irreducible element forms are (1)k (5)1 , (1)k (4)1 , (1)k (3)1 , (1)k (2)1 , (1)k (2)1 (3)1 , (1)k (2)1 (4)1 , (1)k (3)1 (5)1 , (1)k (4)1 (5)1 , (1)k (2)2 (5)1 , (1)k (3)2 (4)1 , (1)k (3)1 (4)2 , and (1)k (2)1 (5)2 . Similarly, we can compute this graph for a1 = 1 in Algorithm 5.3. Again, discarding all the elements that factorize, (3 in the fourth level and 12 in the fifth), we get: 3

2 ·4

2

3·2

3

3

4 ·5

2·4

3 ·2

2

2

5·3

2·5

3·4

1

3

5 ·3

4·5

2

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Explicitly, the irreducible element forms for this graph are: (1)1 , (2)1 (5)1 , (3)1 (4)1 , (2)2 (3)1 , (2)1 (4)2 , (3)2 (5)1 , (4)1 (5)2 , (2)3 (4)1 , (4)3 (5)1 , (2)1 (3)3 , (3)1 (5)3 . The important thing to keep in mind is that the irreducible element forms are easily recoverable from this graph. We conclude that the numbers that remain in the graphs uniquely induce the element forms containing all irreducible elements in U  (11), for a total of 3 · 4 + 11 = 23 irreducible element forms. We give a list of all τ(11) -atoms lesser than 100, that are not divisible by 11: 2, 3, 5, 6, 7, 10, 12, 13, 14, 15, 17, 19, 20, 21, 23, 29, 31, 34, 35, 37, 38, 39, 41, 43, 45, 46, 47, 50, 51, 53, 54, 56, 58, 59, 61, 63, 65, 67, 68, 69, 71, 73, 74, 76, 78, 79, 82, 83, 86, 87, 89, 91, 93, 94, 95, 97, and 98. There are two final things worth taking note of, with respect to the implementation of Algorithm 5.3. First, for any given element x ∈ U  (n), how do we know the values of y, z such that x = y · z? Secondly, how do we know that a specific element form is not irreducible, so that it can be discarded as in steps (4) and (5) of the algorithm? Both of these questions are crucial to the algorithm in steps (4) and (5). Throughout the paper, the irreducibility of element forms has depended almost exclusively on the group structure of U  (n). For example, since U  (7) ∼ = U  (9), it is easy to see that the irreducible element forms for both n = 7 and n = 9 are in bijection. By considering all different values of n individually, there is bound to be a lot of redundancy. In order to avoid this, we could instead work with the more familiar finite abelian groups that U  (n) is isomorphic to. For example, since U  (7), U  (9) ∼ = Z/3Z, we could extend the rules for factorization to Z/3Z and work there alone. The elements of Z/3Z are 0, 1, 2. The rule for factorization is that if you have some sum involving the three elements of Z/3Z, and the sum can be simplified in some way which makes all new terms (there have to be at least 2 terms in the simplification) the same, then a number is said to factor. This same rule can be extended to any finite abelian group, including non cyclic groups like Z/2Z × Z/2Z, which are necessary for the study of U  (n) for values like n = 24. A formalization of this is seen as follows: Definition 5.4 Let (G, +) be an abelian group. A sum x1 + x2 + · · · + xk is said to have a τG -factorization if it can be simplified so that x1 + x2 + · · · + xk = l · x, for x ∈ G and l ∈ Z# . Analogous concepts for τG -atoms/τG -irreducible elements, τG -element forms, single substitutions, double substitutions, and the partition of G into sets Łx with x ∈ G are defined similarly. For the first question, a naïve implementation could involve making the Cayley table for each U  (n), and then making separate lists for each element, determine the possible single substitutions. However, if n is a value for which U  (n) ∼ = Z/mZ, then the previous discussion suggests that the algorithm can be run on Z/mZ instead. The process of generating numbers y, x such that z = x + y is less complicated, as the list 1 + z − 1, 2 + z − 2, . . . , m − 1 + z − m + 1

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contains all relevant values of x and y with multiplicity 2. So, this solves the first question, especially when U  (n) is cyclic. For the second question, consider the number x, which belongs to an element form generated by the algorithm. If one wanted to see whether or not this number has a τ(n) -factorization, a brute force method could be to generate all possible factorizations of x and then checking, one by one, if any of the factorizations is also a τ(n) -factorization. To illustrate why this is difficult and redundant, consider p k . The number of factorizations that p k has in Z is the number of partitions of k. Since the number of partitions grows incredibly fast as k is bigger, this would prove difficult to do for big values of k. The situation for a general number x is even worse. So, consider the following method. Assume that we are working in Z/mZ (this method can be extended to other groups, however). Consider an element form α in the m-th level of the tree produced by the algorithm, which may or may not be irreducible. Since α is in the l-th level, this means that α is a sum with exactly l terms. First consider the case where α is not equivalent to 0. Let α not be the multiple of some element of Z/mZ (i.e. α = ki, this is easy to detect). If α is not irreducible, then there exists some single or double simplification of α that is not irreducible either. For contradiction, assume this to be false. Since α is not a multiple of some fixed element, it would take at least one simplification on α before the resulting form is the multiple of some element. In particular, if any simplification of α is taken with respect to two element y, z ∈ Z/mZ which appear grouped together in any factorization of α, then such element form has to be not irreducible as well, a contradiction. This fails in the case that α is equivalent to 0 because in the definition of simplification, we discard factors of 0 that arise in any simplification. Thus, if α is a sum of length l ≥ 3 (for fixed l), the only other extra condition that has to be imposed for α to be irreducible is that no two terms in α sum to 0. To summarize, a method of determining, in steps (4) and (5) of the algorithm, whether or not an element form, α, is irreducible or not, we only have to check three conditions: if α is equivalent to 0 and is a sum of at least 3 terms, that there do not exist two terms that sum to 0, that any one simplification done on α is in the tree produced by the algorithm, and that α is not the multiple of some element in Z/mZ. While this seems more convoluted, in the worst of cases (when every term in α is different), if α has l terms, one would only have to check O(l 2 ) cases. The following presents an algorithm summarizing this process for the abelian group (G, +): Algorithm 5.5 Let Ła1 , . . . , Łak partition G. To find all irreducible element forms in G, Fix a1 , . . . , ak ∈ G. Set i = 1 Set the element form 1 · (x) to be equivalent to ai . Obtain all element forms obtainable by one single substitution on 1 · (x). Keep the ones that are irreducible and discard that are not. (5) Repeat this process with the resulting element forms.

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

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(6) Return to step (5) until all irreducible element forms equivalent to ai are found. (7) Set i = i + 1 and return to step (3) or move to next step if i = k. (8) Apply all automorphisms of G to all the irreducibles, and keep one copy of each. In the case where G = Z/mZ, we obtain the improved algorithm: Algorithm 5.6 To find all irreducible element forms in Z/mZ, (1) (2) (3) (4) (5) (6) (7) (8) (9)

Set d0 = 0 and d1 = 1. Let d2 , . . . , dk be the non trivial divisors of m. Set i = 0 Set the element form 1 · (x) to be equivalent to di . Obtain all element forms obtainable by one single substitution on 1 · (x), as described by the answer of the first question. For each of the element forms obtained, keep the ones that are irreducible and discard that are not, as described by the answer of the second question. Repeat this process with the resulting element forms. Return to step (6) until all irreducible element forms equivalent to di are found. Set i = i + 1 and return to step (3) or move to next step if i = k. Apply all automorphisms of Z/mZ to all the irreducibles, and keep one copy of each.

With all this in mind, the following table summarizes the number of irreducible element forms for Z/mZ, for the values of m that have been completely classified: Values with asterisks (∗) imply that the number represents the number of element forms in which any fixed term is repeated less times that the term’s order. In general, all irreducible elements that do not satisfy this condition can be determined from the element forms that satisfy the previous condition. However, the methods known to determine all the irreducible element forms for these values of n (i.e. m = 6) require checking cases by hand. It is for this reason that composite values of m have not been explored much. On the other hand, to see why the third column of the table is 0 sometimes, it is enough to use Proposition 3.1. This proposition tells us that almost all the primes for which there is a corresponding U  (n) are Sophie Germain primes. In particular, 7, 13, 17 and 19 are not Sophie Germain primes. These values were chosen because the algorithm terminates in a finite number of steps, and is able to be executed without any human intervention. As the size of the numbers in Table 1 shows, computing these element forms for large n requires computer aid. As a final note, if U  ( p) ∼ = Z/mZ and m appears on the table, then for those values of p we have found all τ( p) -atoms. It is not too difficult to check that, if x = pt, then x is a τ( p) -atom if and only if p  t. This would mean that all τ( p) -atoms for these cases would be the atoms referred to in the table along with numbers pt with p  t.

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Table 1 Algorithm results summary m Number of irreducible element Number of values n such that forms for Z/mZ U  (n) ∼ = Z/mZ 1 1 3 2 2 4 3 4 4 ∗ 4 7 (infinitely many) 4 5 23 2 6 124∗ (infinitely many) 6 7 108 0 11 1398 2 13 4367 0 17 33321 0 19 84544 0 23 465774 2

6 Conclusion We were able to find exactly when U  (n) is cyclic and explain the relevance of cyclic groups with respect to τ(n) -atoms. Additionally, we provided algorithms that help to determine τ(n) -atoms. Finally, due to the algorithm ending in finitely many steps for safe primes, there might be further applications for Sophie Germain primes.

References 1. D.D. Anderson, A. Frazier, On a general theory of factorization in integral domains. Rocky Mount. J. Math. 41(3), 663–705 (2011) 2. D.D. Anderson, R.M. Ortiz-Albino, Three frameworks for a general theory of factorization. Arab. J. Math. 1:1–16, 04 (2012) 3. J.A. Gallian. Contemporary Abstract Algebra (BROOKS/COLE, 2013) 4. J.A. Gallian, D.J. Rusin, Factoring groups of integers modulo n. Math. Mag. 53(1), 33–36 (1980) 5. S.M. Hamon. Some topics in τ -factorizations. Ph.D. thesis, The University of Iowa (2013) 6. J. Juett, Some Topics in Abstract Factorizations. Ph.D. thesis, The University of Iowa (2007) 7. R. Kumanduri, C. Romero, Number Theory with Computer Applications (Prentice Hall, 1997) 8. J. Lanterman, Irreducibles in the Integers Modulo n. In: arXiv:1210.2991, Oct. 2012 9. C. Molina, On the number of τ(n) -factors. Master’s thesis, University of Puerto Rico at Mayagüez (2016) 10. C. Serna, Factorizaciones donde cada factor de un elemento pertenece a solo una clase de equivalencia. Master’s thesis, University of Puerto Rico at Mayagüez (2014)

Bounded Periodic Rings D. D. Anderson and P. V. Danchev

Abstract We define a subclass of the class of periodic rings by calling a ring R an (n, m)-ring if for each x ∈ R there exist two fixed positive integers n, m with n > m such that x n = x m . We study when an (n, m)-ring is either Boolean, potent, reduced or commutative. Keywords Jacobson theorem · Jacobson radical · Periodic rings · Commutativity 2010 Mathematics Subject Classification 16D60 · 16S34 · 16U60

1 Introduction and Fundamentals Throughout this article all rings are associative with an identity element. Let us recall that a ring R is said to be Boolean if x 2 = x for each x ∈ R. These rings have a complete characterization as the subdirect product of a family of copies of the field Z2 of two elements (see, e.g., [6]). Hence Boolean rings are always commutative. We now present some generalizations of Boolean rings and give some historical facts concerning them. For a fixed prime p, a p-ring is a ring R in which a p = a and pa = 0 for all a ∈ R. Thus any Boolean ring is simply a 2-ring. It is well known [13] that a ring is a p-ring if and only if it is a subdirect product of fields of order p. More generally, for a prime p and a positive integer k, a p k -ring is a ring R in which k a p = a and pa = 0 for all a ∈ R. The structure of p k -rings has been completely described in [3]. All of them are commutative too.

D. D. Anderson (B) Department of Mathematics, The University of Iowa, Iowa City, IA 52242-1419, USA e-mail: [email protected] P. V. Danchev Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, “Acad. G. Bonchev” str., bl. 8, 1113 Sofia, Bulgaria e-mail: [email protected]; © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. Badawi and J. Coykendall (eds.), Rings, Monoids and Module Theory, Springer Proceedings in Mathematics & Statistics 382, https://doi.org/10.1007/978-981-16-8422-7_13

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A ring R is said to be potent if for each a ∈ R there is an integer n(a) > 1 such that a n(a) = a. Every potent ring is commutative by the fundamental result due to Jacobson [10]. If the natural number n(a) does not depend on the choice of the element a, that is, x n = x for some n > 1 and for all x, then the ring R is called a J (n)-ring or in the terminology we introduce R just satisfies the (n, 1)-property. It was proved in [12] that a ring is a J(n)-ring for some n ∈ N if and only if it is the direct product of finitely many p k -rings. However, the complete description of J(n)rings was given in [14] (see [9] as well). It was proved in [14] that R is a J(n)-ring if and only if R is a subdirect product of the finite fields GF( p k ), where p is a prime and k is an integer such that p k − 1 divides n − 1. We refer also to [7, 11] for more detailed information on this topic. On the other hand, a ring R is called periodic if for every r ∈ R there exist two distinct natural numbers m, n depending on r such that the equality r m = r n holds. When the existing pair (n, m) is just fixed for all elements of R, we shall say that R satisfies the (n, m)-property or that R is an (n, m)-ring. It is not too hard to check that these rings are, in general, not commutative by looking at the equation r 2 = r 4 which as it can be checked directly is always satisfied by the upper 2 × 2 triangular matrix ring over Z2 . The purpose of this article is to study rings that satisfy the equation x n = x m for some fixed natural numbers n > m. We call such a periodic ring an (n, m)-ring. And we call an (n, m)-ring R a proper (n, m)-ring if R is not an (n  , m  )-ring where (n  , m  ) < (n, m) in the product order. Thus an (n, m)-ring is always periodic, but the converse is false as the algebraic closure of a finite field is periodic, and even potent, but there is no bound on the periodicity. These rings are closely related to the previously mentioned significant theorem of Jacobson and were investigated in [1]. We consider the question of when an (n, m)-ring is either Boolean, potent, reduced, or commutative. We are also interested in the question of when an (n, m)-ring is an (n  , m  )-ring where (n  , m  ) < (n, m). As we will see, there is a dichotomy as to whether n and m have the same or opposite parity. Suppose that n and m have opposite parity. Then (−1)n = (−1)m gives that R has characteristic 2. Moreover, R is an (n, m)-ring if and only if R is an (n − m + 1, 1)-ring (see our Theorem 2.4 stated below). Thus R is potent and hence commutative by virtue of a celebrated theorem of Jacobson [10]. For some other special relationships concerning Euler rings, potent rings and periodic rings, we suggest to the interested reader the articles [2, 4, 5, 8], respectively. Throughout, we will use the obvious fact that for any two fixed different positive integers n and m, the class of (n, m)-rings forms a variety, that is, it is closed under subrings, homomorphic images, and direct products. We denote the set of all natural numbers by N and the Jacobson radical of R by J (R).

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2 The Main Results We begin with the simple observation that an (n, m)-ring is just “a bounded periodic ring”. Proposition 2.1 For a ring R the following are equivalent. (1) R is an (n, m)-ring for some natural numbers n > m. (2) There exists a finite set of natural numbers n 1 , m 1 ; . . . ; n s , m s , s ≥ 1, with n i > m i for each i, 1 ≤ i ≤ s, such that for every x ∈ R, x ni = x m i for some index i. (3) R is a bounded periodic ring, that is, there exists a natural number n so that for each x ∈ R, x r = x s where r and s are natural numbers with s < r ≤ n. Proof Clearly (2) ⇐⇒ (3) and (1) ⇒ (2). As for the reverse implication (2) ⇒ (1), let x ∈ R, so there is an index i with x ni = mi x . Let s be the least positive integer with x s = x t where 1 < t ≤ s. So s ≤ n i . Now S = {x t , x t+1 , . . . , x s−1 } is a multiplicative cyclic group of order s − t ≤ n i . Thus the identity element of S is idempotent. So there exists a natural number l(x) ≤ n i with x 2l(x) = x l(x) . Hence there exists a finite set of natural numbers l1 , . . . , lk so that if x ∈ R, there exists an i with x 2li = x li . Put l = l1 · · · lk . Then x 2l = x l . Therefore, R is a (2l, l)-ring.  Let the set T = {(n, m) | n, m ∈ N, n > m} be partially ordered by the product order (n, m) ≤ (n  , m  ) ⇐⇒ n ≤ n  and m ≤ m  . So (2, 1) is the minimal element of T. Given (n, m), (n  , m  ) ∈ T, we write (n, m) ⇒ (n  , m  ) if every (n, m)-ring is an (n  , m  )-ring, and we write (n, m) ⇐⇒ (n  , m  ) when (n, m) ⇒ (n  , m  ) and (n  , m  ) ⇒ (n, m). We also say that (n, m) ∈ T is primitive if (n, m)  (n  , m  ) for any (n  , m  ) < (n, m). Thus the pair (n, m) is primitive exactly when there exists a proper (n, m)-ring. Given a property P, we say that the pair (n, m) ∈ T satisfies P if every (n, m)-ring satisfies P. So (n, m) is Boolean (resp., potent, etc.) if each (n, m)ring is Boolean (resp., potent, etc.). So a (2, 1)-ring is just a Boolean ring. Certainly (2, 1) is primitive and (2, 1) ⇒ (n, m) for any (n, m) ∈ T. Note that Proposition 2.1 shows that (n, m) ⇒ (2k, k) for some natural number k. However, as the next example shows, we can have (n, m) ⇒ (n  , m  ) where (n  , m  ) < (n, m). Example 2.2 (2, 1) ⇐⇒ (3, 2). So (3, 2) is not primitive. Evidently (2, 1) ⇒ (3, 2). To prove the converse implication, suppose R satisfies (3, 2), that is, x 3 = x 2 . Then −1 = (−1)3 = (−1)2 = 1, so char(R) = 2. For an arbitrary x ∈ R, 1 + x + x 2 + x 3 = (1 + x)3 = (1 + x)2 = 1 + x 2 . Hence x = x 3 , so that x 2 = x 3 = x, as required. It is worth noticing that this is a quite special case of Theorem 2.4 stated below. Moreover, it is worthwhile to notice that there is a ring R of characteristic 2 in which x 2 = x 4 , ∀ x ∈ R, but neither x 2 = x 3 nor x = x 3 . In fact, let us consider the group ring R = BG, where B  Z2 is a Boolean ring and G is a group consisting

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only of elements of order at most 2. Clearly, the equality x 4 = x 2 is valid for each element x ∈ R as char(R) = 2. However, taking into account that B contains nontrivial idempotents, simple calculations show that both inequalities x 3 = x 2 and x 3 = x are fulfilled for some element x ∈ R. There is a natural dichotomy for (n, m)-rings as to whether n and m have the same parity or opposite parity. First suppose R is an (n, m)-ring where n and m have opposite parity. Then (−1)n = (−1)m gives that R has characteristic 2. Also, in this case (n, m) ⇐⇒ (n − m + 1, 1) (see Theorem 2.4 below). So R is potent and thus is a commutative von Neumann regular ring. Thus R is also reduced, i.e., it does not possess nonzero nilpotent elements. However, the case where n and m have the same parity is much more different and more complicated as the next examples illustrate: Consider the three rings (1) Z2 [X ]/(X 2 ), (2) Z4 , and (3) T2 (Z2 ), the ring of all 2 × 2 upper triangular matrices over Z2 . It is not too hard to check that these three rings satisfy the (4, 2)-property and hence the (n, m)-property for any even natural numbers n > m. However, the ring in (3) is definitely not commutative, the ring in (2) does not have characteristic 2, all three rings have nontrivial nilpotent elements, and even though the ring in (1) is commutative having characteristic 2, it is obviously not potent. We recall that a commutative ring R is said to be von Neumann regular if for each r ∈ R there is b ∈ R such that r = r 2 b. We next give a characterization of rings that satisfy the (n, 1)-property for some fixed n ∈ N greater than one. The following assertion is probably well-known, but the authors have not seen it in the existing literature, so being relevant to the current subject we include it only for the sake of completeness and clarity of the exposition. Theorem 2.3 For a ring R the following conditions are equivalent. (1) R satisfies (n, 1) for some natural number n > 1. (2) R is a commutative von Neumann regular ring with {|R/M| | M ∈ Max(R)} bounded. (3) R is a commutative ring with J (R) = 0 and {|R/M| | M ∈ Max(R)} bounded. Proof (1) ⇒ (2). Suppose R is an (n, 1)-ring. Thus x n = x for all elements x ∈ R and some n > 1. Obviously, R is commutative since it is potent. Furthermore, x = x x n−2 x implying that R is von Neumann regular. For a maximal ideal M of R, the quotient ring R/M satisfies (n, 1) as well. So for 0 = x ∈ R/M, x n = x gives x n−1 = 1. So |R/M| ≤ n. (2) ⇒ (3). It is well known that a commutative von Neumann regular ring R has J (R) = 0. (3) ⇒ (1). Suppose that {|R/M| | M ∈ Max(R)} is bounded. So there exists a natural number m such that for x ∈ R and each maximal ideal M of R, either x = 0 or x m = 1 in R = R/M. Thus x m+1 = x in R, or x m+1 − x ∈ M. Consequently, x m+1 − x ∈ ∩ M∈Max(R) M = J (R) = 0. So R satisfies (m + 1, 1), as required. 

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We also establish the following useful statement. Theorem 2.4 Let (n, m) ∈ T where n and m have opposite parity. Then (n, m) ⇐⇒ (n − m + 1, 1), that is, a ring satisfies the (n, m)-property if and only if it satisfies the (n − m + 1, 1)-property. In addition, if R is an (n, m)-ring for these numbers n, m, then R has characteristic 2 and is a potent ring (whence a commutative von Neumann regular ring). Proof By Corollary 2.2 in [1], one has that (n, m) ⇒ (n − m + 1, 1). However, for any (n, m) ∈ T, we have the implication (n − m + 1, 1) ⇒ (n, m) as seen by multiplying both sides of the equation x n−m+1 = x by x m−1 . The last part follows immediately from Theorem 2.3.  We next determine those pairs (n, m) for which an (n, m)-ring is Boolean. Theorem 2.5 Let n > m ≥ 1 be integers. Then an (n, m)-ring is Boolean if and only if n and m have opposite parity and n − m is not divisible by 2k − 1 for any natural number k > 1. Proof “⇒”. Suppose that R is an (n, m)-ring which is Boolean. Note that the nonBoolean ring Z3 always satisfies (n, m) whenever n and m have the same parity. Thus n and m must have opposite parity. By using Theorem 2.4, the (n, m)-property is equivalent to the (n − m + 1, 1)-property. Hence (n − m + 1, 1) is a Boolean pair. Suppose there is a k > 1 with 2k − 1 / n − m. Now consider the finite field k GF(2k ). For any 0 = x from this field we have that x 2 −1 = 1, whence x n−m = 1 and x n−m+1 = x. Consequently, GF(2k ) satisfies (n − m + 1, 1) and so it has to be Boolean. This contradicts the inequality k > 1, so that we are done. “⇐”. Suppose that the natural numbers n and m have opposite parity and 2k − 1 does not divide n − m for any k > 1. Let R be an (n, m)-ring. We need to show that R is, in fact, Boolean. By the preceding Theorem 2.4, R is an (n − m + 1, 1)ring. Hence char(R) = 2 and R is commutative von Neumann regular according to Theorem 2.3. Let M be a maximal ideal of R. Then the localization R M of R at M is a finite field of characteristic 2 with R M ∼ = R/M. Suppose that R M ∼ = GF(2k ). Then n−m+1 n−m = x, as x = 1 for every for each element x from this field we have that x nonzero x. Hence 2k − 1 divides n − m and thus we must have k = 1. So for each maximal ideal M of R, R M ∼  = GF(2). Thus R is a Boolean ring, as claimed. As an interesting consequence, we have the following result. Corollary 2.6 Let p be an odd prime. Then the pair ( p + 1, 1) is Boolean if and only if p is not a Mersenne prime. Proof Since p is odd, p + 1 and 1 have opposite parity. Note that 2k − 1 divides ( p + 1) − 1, i.e., 2k − 1 divides p for k > 1, exactly when 2k − 1 = p, that is, p is a Mersenne prime.  We next consider the more general question of when an arbitrary pair (n, m) is potent.

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Theorem 2.7 Let n > m be natural numbers. Then the following are equivalent. (1) The pair (n, m) is potent, that is, every (n, m)-ring R is potent. (2) (n, m) ⇒ (n − m + 1, 1) (or, equivalently, (n, m) ⇐⇒ (n − m + 1, 1)). (3) (n, m) ⇒ (l, 1) for some natural number l > 1. (4) Either n and m have opposite parity or m = 1. Proof (1) ⇒ (2). Let R be an (n, m)-ring. By hypothesis, R is also potent. So, by what we have seen above in Theorem 2.3, R is a commutative von Neumann regular ring. Let M be a maximal ideal of R. Then R M , the localization of R at M, is a field that satisfies (n, m). So for 0 = x ∈ R M , x n = x m and hence x n−m = 1. Then for any x ∈ R M , x n−m+1 = x. Thus for x ∈ R, x n−m+1 = x locally and hence globally. So R is an (n − m + 1, 1)-ring. Thus (n, m) forces (n − m + 1, 1). Note that both implications (2) ⇒ (3) ⇒ (1) are clear. (4) ⇒ (1). If n and m have opposite parity, Theorem 2.4 gives that the pair (n, m) is potent and certainly the pair (n, 1) is potent as well. (1) ⇒ (4). Suppose that the pair (n, m) is potent and n and m have the same parity with m > 1. Note that the ring Z4 satisfies (n, m) which is the desired contradiction since this ring is obviously not potent.  We next show that (n, m) is commutative or reduced if and only if (n, m) is potent, that is, by the previous theorem either n and m have opposite parity or m = 1. Theorem 2.8 Let n > m be natural numbers. Then the following four conditions are equivalent. (1) n and m have opposite parity or m = 1. (2) (n, m) is potent. (3) (n, m) is commutative. (4) (n, m) is reduced. Proof By Theorem 2.7, (1) and (2) are equivalent. Now a potent ring is both commutative and reduced, so (2) ⇒ (3) and (2) ⇒ (4). We now concentrate on (3),(4) ⇒ (1). To that aim, suppose that (n, m) is commutative or reduced. Assume also that n and m have the same parity. First suppose that n and m are both even. Now T2 (Z2 ) satisfies (4, 2) and hence it satisfies (n, m). But this matrix ring is not commutative and has nonzero nilpotents, a contradiction. So suppose that n and m are both odd. Letting n > m > 1, we then again look at T2 (Z2 ) to get the desired contradiction. Finally, m = 1, as asserted.  We next have the following result. Theorem 2.9 Suppose that n > m ≥ 1 have opposite parity. Then (n, m) is primitive if and only if m = 1 and n − 1 = LCM(2k1 − 1, . . . , 2kl − 1) where 1 ≤ k1 < · · · < kl with 2ki − 1/n − 1 for i ∈ [1, l] and l ∈ N.

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Proof “⇐”. We just need to show that (n, 1) is primitive where n − 1 = LCM(2k1 − 1, . . . , 2kl − 1). For this purpose, consider the ring R = GF(2k1 ) × · · · × GF(2kl ). Then it can be easily verified that R is an (n, 1)-ring but not a ( j, 1)-ring for any 2 ≤ j < n. So (n, 1) is primitive. “⇒”. Suppose that (n, m) is primitive. Since n and m have opposite parity, by Theorem 2.4 we have that (n, m) ⇒ (n − m + 1, 1). So we must have m = 1. Suppose now that f := LCM(2k1 − 1, . . . , 2kl − 1) < n − 1. Let R be an (n, 1)-ring. So by Theorem 2.3, R is commutative von Neumann regular having characteristic 2. Let M be an arbitrary maximal ideal of R. Therefore, the localization R M of R at M is isomorphic to GF(2k ) for some k ∈ N and obviously is an (n, 1)-ring. Thus 2k − 1 divides n − 1 and hence R M is an ( f, 1)-ring. Since this is true for all maximal ideals, we deduce that (n, 1) ⇒ ( f, 1) for f < n − 1, which contradicts the primitivity of the pair (n, m). Thus f = n − 1, as claimed.  A slight modification of the proof of the previous theorem gives when the pair (n, 1) is primitive for n odd, so we will only sketch the main arguments as we leave the details to the interested reader. Theorem 2.10 Let n ≥ 3 be an odd integer. Then (n, 1) is primitive if and only if n − 1 = LCM({ p k − 1 | p is an odd prime, k ≥ 1, and p k − 1 / n − 1}).  Proof “⇐”. Consider the ring R = GF( p k ) where the product ranges over all p k such that p is an odd prime, k ≥ 1, and p k − 1 divides n − 1. Then R is an (n, 1)-ring, but not a ( j, 1)-ring for any 2 ≤ j < n. “⇒”. Set f := LCM({ p k − 1 | p is an odd prime, k ≥ 1, and p k − 1 / n − 1}). Let M be a maximal ideal of R. So R M ∼ = GF( p k ) is an (n, 1)-ring, whence p is an odd prime. Now n − 1 is a multiple of all p k − 1, so p k − 1 divides f and hence R M is an ( f + 1, 1)-ring. Since this is true locally, it is true globally. Consequently, (n, 1) ⇒ ( f + 1, 1). However (n, 1) is primitive, which allows us to conclude that n = f + 1, as asserted.  It remains to determine when the pair (n, m) is primitive for n > m ≥ 2 where n and m have the same parity. Note that (n, m) is primitive if and only if there exists a proper (n, m)-ring. The first such case is (4, 2). It is easily checked that Z4 is a proper (4, 2)-ring, or in other words that the pair (4, 2) is primitive. The next such case is (5, 3). Again, it is easy to check that Z8 is a proper (5, 3)-ring, i.e., (5, 3) is primitive. Note that in these two cases and in the cases where (n, m) is primitive with n and m of opposite parity (so m = 1), or when n is odd and m = 1, there is a commutative proper (n, m)-ring. In that aspect, we end with the following questions. Question 1. For which natural numbers n and m of the same parity with n > m ≥ 2 is the pair (n, m) primitive? Question 2. Suppose that the pair (n, m) is primitive for some two positive integers n, m with n > m, that is, there exists a proper (n, m)-ring for two such natural numbers n and m. Then does there exist a commutative proper (n, m)-ring?

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Acknowledgements The authors are grateful to the referee for his/her helpful comments as well as for suggesting to us references [2, 4, 5]. Funding The work of the second named author P. V. Danchev is partially supported by the Bulgarian National Science Fund under Grant KP-06 N 32/1 of Dec. 07, 2019.

References 1. D.D. Anderson, P.V. Danchev, A note on a theorem of Jacobson related to periodic rings. Proc. Am. Math. Soc. 148(12) 5087–5089 (2020) 2. D.F. Anderson, A. Badawi, Von Neumann regular and related elements in commutative rings. Algebra Colloq. 19(Spec Issue 1), 1017–1040 (2012) 3. R.F. Arens, I. Kaplansky, Topological representation of algebras. Trans. Am. Math. Soc. 63, 457–481 (1948) 4. A. Badawi, A.Y.M. Chin, H.V. Chen, On rings with near idempotent elements. Internat. J. Pure & Appl. Math. 1(3), 253–259 (2002) 5. H.E. Bell, A commutativity study for periodic rings. Pac. J. Math. 70, 29–36 (1977) 6. G. Birkhoff, M. Ward, A characterization of Boolean algebras. Ann. Math. 40, 609–610 (1939) 7. T. Chinburg, M. Henriksen, Multiplicatively periodic rings. Am. Math. Mon. 83, 547–549 (1976) 8. J. Cui, P. Danchev, A new characterization of periodic rings. J. Algebra & Appl. 19(12) (2020) 9. P.V. Danchev, A characterization of weakly J (n)- rings. J. Math. Appl. 41, 53–61 (2018) 10. N. Jacobson, Structure theory for algebraic algebras of bounded degree. Ann. Math. 46, 695– 707 (1945) 11. T.-K. Lee, Y. Zhou, From Boolean rings to clean rings. Contemp. Math. 609, 223–232 (2014) 12. J. Luh, On the structure of J-rings. Am. Math. Mon. 74, 164–166 (1967) 13. N.H. McCoy, D. Montgomery, A representation of generalized Boolean rings. Duke Math. J. 3, 455–459 (1937) 14. V. Peri´c, On rings with polynomial identity x n − x = 0, Publ. Inst. Math. (Beograd) (N.S.) 34(48), 165–168 (1983)

On Gracefully and Harmoniously Labeling Zero-Divisor Graphs Christopher P. Mooney

Abstract In this article, we study two popular types of vertex labelings of the zerodivisor graph which arises naturally from a commutative ring R with zero-divisors. As a continuation of the early study of zero-divisor graphs about the coloring number, we are instead interested in the graceful labeling of Rosa and the harmonious labeling of Graham and Sloane. We prove several results towards determining which commutative rings have zero-divisor graphs which are graceful or harmonious. We demonstrate infinite classes of rings that are graceful and harmonious, as well as infinite classes of rings that are neither graceful nor harmonious. While we are unable to completely answer the question for all commutative rings, we are able to provide the answer to this question for all of the zero-divisor graphs on up to 14 vertices provided by Redmond. Keywords Zero-divisor graphs · Graceful labeling · Harmonious labeling

1 Introduction Undoubtedly the biggest mathematical mark left by Dan Anderson is in the world of commutative algebra and factorization theory, but we wanted to be sure to point out another area of mathematics where Dan Anderson has made a large and lasting impact. This other area involves the intersection of commutative algebra and graph theory. We will begin with some historical context and try to keep the definitions to a minimum initially. In 1988, I. Beck introduced the concept of a zero-divisor graph for a commutative ring R (with 1 = 0) with zero-divisors, [11]. The original graph, as defined by Beck, has vertex set R and edges between distinct a and b if ab = 0. The modern treatment of the zero-divisor graph, due to Dan’s twin brother David Anderson and Phil Livingston in [4], involves using the set Z (R)∗ , the collection C. P. Mooney (B) Department of Mathematics, Statistics and Computer Science, University of Wisconsin - Stout, Menomonie, WI 54751, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. Badawi and J. Coykendall (eds.), Rings, Monoids and Module Theory, Springer Proceedings in Mathematics & Statistics 382, https://doi.org/10.1007/978-981-16-8422-7_14

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of nonzero zero-divisors as the vertex set and the same edge relation. This is what we will mean then by a zero-divisor graph of a commutative ring R and we will write (R). The vertex set, V ((R)) = Z (R)∗ , and edge set defined by distinct a, b ∈ Z (R)∗ are adjacent if ab = 0. This has since been studied and developed by many authors in numerous important articles including, but not limited to [2–9, 16, 23–25]. Associating a graph with a given algebraic structure has proven to be a fruitful field of research and has been generalized and studied in numerous ways in the last thirty some years. To give the reader a sense, a search for ‘divisor graphs’ in Math Reviews returns around 1700 articles at the present moment. In the initial paper, [11], Beck was interested in studying the chromatic (coloring) number of these graphs, which is the minimum distinct colors one would need to color the vertices so that no two adjacent vertices have the same color. The main conjecture coming out of this initial article was whether the clique number (the size of the biggest complete subgraph) of zero-divisor graphs was always the same as the chromatic number (such graphs where these numbers coincide are called perfect graphs). While there have been numerous fascinating results proven about properties of these graphs, it was Dan Anderson and Muhammad Naseer who were able to resolve this conjecture by providing the first counter-example of a ring which had a properly higher coloring number than its clique number in [6], in 1993, thus showing zero-divisor graphs were not all perfect. This counterexample was further shown to be the minimal such example where this happens. One might think with this initial conjecture resolved some five years later, this would end the study of zero-divisor graphs. On the contrary, this area of research has exploded over the last thirty years in numerous directions: irreducible divisor graphs [15], total graphs [10], comaximal graphs [22], and regular graphs [1], to name a few. While the initial focus on zero-divisor graphs was on coloring, there were numerous graph theoretic properties. While the initial focus on zero-divisor graphs was on coloring, there were numerous graph theoretic properties proven about zero-divisor graphs which make them quite nice for studying other labeling properties as well which are discussed below. While most of my interest in zero-divisor graphs came about for studying factorization properties in rings with zero-divisors through irreducible divisor graphs, as a graduate student, I found myself intrigued by the coloring problem of these graphs. I found Dan Anderson’s interest in studying the interplay between commutative rings and a highly combinatorial problem of graph coloring to be a beautiful symbiotic relationship with lots of potential to advance both areas of mathematics. This opened up a new world of graph theory and connections to algebra to which I had not previously been exposed much. Thus, in this contribution, we turn our attention to a similar idea of studying vertex labelings of a graph associated to a commutative ring with zero-divisors. We are interested in graceful labeling which was introduced by A. Rosa in [29] and is probably the most well studied type of labeling perhaps with the exception of coloring. Due to the strong connection between graceful and harmonious labeling introduced by Graham and Sloane in [20], we also study harmonious labeling here. These are two of the most famous vertex labelings in graph theory after coloring, so it seems a natural place to continue the investigation.

On Gracefully and Harmoniously Labeling Zero-Divisor Graphs

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In Sect. 2, we provide the requisite definitions and notation as well as provide several known and useful results from the theory of zero-divisor graphs, graceful, and harmonious labeling which will be used throughout the paper. In Sect. 3, we are able to prove several results that demonstrate infinite classes of commutative rings with zero-divisor graphs which are and are not graceful and harmonious. This seems to indicate that this is an interesting and deep question for a general commutative ring. We conclude the paper, in Sect. 4, with the process of actually finding graceful and harmonious labelings of zero-divisor graphs of small order or proving that none exist. We use the zero-divisor graphs provided by Redmond in [26, 27] where the author is able to find all rings up to isomorphism with zero-divisor graphs with up to 14 vertices. In the final section, we are able to determine which graphs are harmonious or graceful in all cases for the zero-divisor graphs up to 14 vertices from [26, 27]. We conclude by posing a few open questions to suggest some possible results from the patterns noticed in these small cases that could be interesting for future research.

2 Background In this section, we provide formal definitions and results from the three main areas of interest for our purposes: zero-divisor graphs, graceful labeling, and harmonious labeling.

2.1 Zero-Divisor Graphs As in [4], we let R be a commutative ring with unity and zero-divisors. We define (R) to be the zero-divisor graph of R. The vertex set is V ((R)) = Z (R)∗ , i.e., the nonzero zero-divisors. The edge set is defined by the relation: Let a, b ∈ Z (R)∗ be distinct, then there is an edge between a and b if and only if ab = 0. This is a simple graph, so there are no loops even if a 2 = 0. Example 2.1 We draw (R) for two small rings to give the reader a sense of the zero-divisor graph in Fig. 1 below. We now collect a few results on zero-divisor graphs which will be useful for determining which rings have zero-divisor graphs that admit a graceful labeling (defined soon). Theorem 2.2 ([4, Theorem 2.2]) Let R be a commutative ring with 1 = 0. Then V ((R)) is finite if and only if R is finite or R is an integral domain. Theorem 2.3 ([4, Theorem 2.3]) Let R be a commutative ring. Then (R) is connected and diam((R)) ≤ 3.

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Fig. 1 a (Z6 )

C. P. Mooney

b) (Z2 × Z4 )

These two theorems both guarantee that the question of gracefully or harmoniously labeling a zero-divisor graph is even possible and greatly restricts the class of rings we need to consider. For any commutative ring R, the zero-divisor graph is connected. Moreover, for a connected graph to have a finite edge set, there must be a finite number of vertices. While some work has been done on infinite analogues of graceful and harmonious labelings, we will not be considering the infinite case here. Thus, we may restrict to finite commutative rings which are not integral domains.

2.2 Graceful Labeling Graceful labeling was introduced by Rosa in 1967 in [29]. A graceful labeling of a graph G with n edges is an injective function f : V (G) → {0, 1, . . . , n} such that when each edge x—y ∈ E(G) is assigned the edge label, | f (x) − f (y)|, all the edge labels are distinct. A graph is said to be graceful if it admits a graceful labeling. While there are some natural extensions for infinite graphs, we insist that our graphs have a finite number of edges to be graceful. There is a substantial amount of research on which graphs admit a graceful labeling. The dynamic survey article by J.A. Gallian, [18], collects many of these results very nicely. It also includes an incredibly extensive bibliography. We direct the interested reader there for more information. Below we list several graceful labeling results that we will find useful. Theorem 2.4 The following types of graphs are graceful. (1) Complete bipartite graphs K n,m for any n, m ∈ N [19]. (2) Complete graphs on n ∈ N vertices, K n , if and only if 1 ≤ n ≤ 4 [19]. (3) Trees on n vertices if n ≤ 34 [17]. Example 2.5 We gracefully label the zero-divisor graphs provided above here to illustrate what a graceful labeling of a simple connected graph should look like. We use Pn to denote a path on n vertices and Ss,t to denote the spire graph on s vertices

On Gracefully and Harmoniously Labeling Zero-Divisor Graphs

Fig. 2 Graceful labelings of a (Z6 ) ∼ = P3

243

b (Z2 × Z4 ) ∼ = S5,2

with the spire on the t-th vertex of the main path. There are many more examples provided in Sect. 4. It is perhaps worth noting that these examples above are also both trees. There is a long standing conjecture as to whether all trees are graceful, and as of writing this paper it remains open. It has been verified that all trees up to 34 vertices are indeed graceful in [17].

2.3 Harmonious Labeling Harmonious labeling was introduced by R.L. Graham and N.J.A. Sloane in the 1980 article [20] as an analogue of graceful labeling which works under modular arithmetic. A harmonious labeling of a graph G with n edges is an injective function f : V (G) → {0, 1, . . . , n − 1} such that when each edge x—y ∈ E(G) is assigned the edge label, f (x) + f (y) (mod n), all the edge labels are distinct. If G is a tree, then exactly one label may be used on two vertices. A graph is said to be harmonious if it admits a harmonious labeling. Again, we insist that our graphs have a finite number of edges to be considered harmonious. Below we list several harmonious labeling results which we will find useful. Theorem 2.6 The following types of graph are harmonious. (1) Complete bipartite graphs K n,m if and only if m = 1 or n = 1 [20]. (2) Complete graphs on n ∈ N vertices, K n , if and only if 1 ≤ n ≤ 4 [20]. (3) Trees on n vertices if n ≤ 31 [17]. Example 2.7 We provide harmonious labelings of the previous two graphs in Fig. 3 below. Again, both of these examples are trees and it is conjectured that all trees would be harmonious as well and has been verified up to trees on 31 vertices [17].

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Fig. 3 Harmonious labeling of a (Z6 ) ∼ = P3 (mod 2) b (Z2 × Z4 ) ∼ = S5,2 (mod 4)

3 Labeling Results on Zero-Divisor Graphs of Commutative Rings In this section, we answer the question of whether a ring has a graceful or harmonious zero-divisor graph for some infinite classes of rings. Theorem 3.1 If R is an infinite ring which is not an integral domain, then (R) is neither graceful nor harmonious. Proof By Theorem 2.2, if R is an infinite ring which is not an integral domain, then V ((R)) is an infinite set. Since (R) is connected by Theorem 2.3, there are infinitely many edges. To be graceful or harmonious by our definition, there must be a finite number of edges. Thus, the zero-divisor graph of any infinite commutative ring with non-trivial zero-divisors cannot be graceful or harmonious.  Theorem 3.2 ([4, Theorem 2.10]) Let R be a finite commutative ring with unity and (R) be the zero-divisor graph of R. The following are equivalent. (1) (R) is complete. (2) R ∼ = Z2 × Z2 or x y = 0 for all x, y ∈ Z (R). (3) The relation ∼ on Z (R) defined by x ∼ y for x, y ∈ Z (R) if and only if x y = 0 or x = y is transitive. (4) The relation ∼ as above is an equivalence relation. (5) The relation ∼∗ on Z (R) defined by x ∼∗ y for x, y ∈ Z (R) if and only if x y = 0 is transitive and R  Z2 × Z2 . (6) The relation ∼∗ as above is an equivalence relation and R  Z2 × Z2 . (7) R ∼ = Z2 × Z2 or R is local with maximal ideal M and M 2 = 0. Furthermore, if (R) is complete, then R ∼ = Z2 × Z2 or R is local with char(R) = p or p 2 , and | (R) |= p n − 1, where p is prime and n ≥ 1. Theorem 3.3 The only rings R (up to isomorphism) for which (R) is both complete and graceful (resp. harmonious) are listed below. (1) Z4 or Z2 [X ]/(X 2 ), in which case (R) ∼ = K1. (2) Z9 , Z2 × Z2 , or Z3 [X ]/(X 2 ), in which case (R) ∼ = K2. (3) Z2 [X, Y ]/(X, Y )2 , Z4 [X ]/(2, X )2 , F4 [X ]/(X 2 ) or Z4 [X ]/(X 2 + X + 1), in which case (R) ∼ = K3.

On Gracefully and Harmoniously Labeling Zero-Divisor Graphs

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(4) Z25 or Z5 [X ]/(X 2 ), in which case (R) ∼ = K4. Proof If R is a ring such that (R) is complete and n = |V ((R))| ≥ 5, then (R) ∼ = K n and cannot be graceful by Theorem 2.4 (2). It also cannot be harmonious by Theorem 2.6 (2). On the other hand, if R has a zero-divisor graph which is complete and n = |V ((R))| < 5, then we know from the work of [5] all rings up to isomorphism with zero-divisor graph K n . They are listed above and again by Theorem 2.4 these are graceful and by Theorem 2.6 these are also harmonious.  We now may use the construction from [4, Example 2.11] to find an infinite number of commutative rings with zero-divisors R such that (R) is complete and neither graceful nor harmonious. Proposition 3.4 Let p be a prime with p > 5 then (Z/ p 2 Z) and (Z p [X ]/(X 2 )) are neither graceful nor harmonious. Proof By [4, Example 2.11], both (Z/ p 2 Z) and (Z p [X ]/(X 2 )) are isomorphic to K p−1 . Thus, since p > 5, K p−1 is not graceful. Thus, neither (Z/ p 2 Z) nor  (Z p [X ]/(X 2 )) are graceful or harmonious. This shows not only are not all zero-divisor graphs graceful or harmonious, but there are infinitely many such rings with zero-divisor graphs which are neither graceful nor harmonious. We now demonstrate an infinite class of rings which do have graceful zero-divisor graphs. Proposition 3.5 We have the following. (1) If A and B are integral domains, then (A × B) is a complete bipartite graph. (2) If R = F1 × F2 where F1 and F2 are finite fields, then (R) ∼ = K |F1 |−1,|F2 |−1 . (3) If R ∼ = Z2 × F where F is a finite field with |F| ≥ 2, then (R) is a star graph. Proof (1) This result is found in [4], but is clear as one can simply make V1 = {(a, 0) | a ∈ A∗ } and V2 = {(0, b) | b ∈ B ∗ }. Then every vertex in V1 is adjacent to each vertex in V2 , yet since A and B are integral domains, none are adjacent within V1 and V2 , respectively. (2) For (R) to be finite R must be finite, thus both A and B must be finite. Any finite integral domain is a field. (3) A star graph is simply a complete bipartite graph of the form K 1,n .  Corollary 3.6 Let R ∼ = F1 × F2 where F1 and F2 are finite fields. Then (R) is graceful. Proof By Proposition 3.5, (R) ∼ = K |F1 |−1,|F2 |−1 . Then by Theorem 2.4 (1), complete bipartite graphs are graceful. 

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On the other hand, since the only complete bipartite graphs which are harmonious are star graphs, we have the following additional corollary. Corollary 3.7 Let R ∼ = Z2 × F2 where F2 is a finite field. Then (R) is both graceful and harmonious. Proof By Proposition 3.5, (R) ∼ = K 1,|F2 |−1 which is a star graph. Then by Theorem 2.4 (1) and Theorem 2.6 (1), star graphs are both graceful and harmonious.  Example 3.8 For the sake of completeness, we show examples of how to gracefully label star graphs and complete bipartite graphs in Fig. 4. We use the zero-divisor graph of Z2 × F, where F is a finite field of order p m which has a zero-divisor graph a star graph K 1, pm −1 . We also use the zero-divisor graph of F1 × F2 where F1 is a finite field of order p m and F2 is a finite field of order q n which has zero-divisor graph K pm −1,q n −1 , a complete bipartite graph.

Fig. 4 Graceful labeling of a (Z2 × F) ∼ = K 1, pm −1 and b (F1 × F2 ) ∼ = K pm −1,q n −1

As indicated above, among these complete bipartite graphs, it is only the star graphs which are harmonious. We conclude this section by providing an example, in the figure below, to show how one can harmoniously label the star graph K 1, pm −1 which would come as the zero-divisor graph of R = Z2 × F where F is a finite field of order p m (Fig. 5).

On Gracefully and Harmoniously Labeling Zero-Divisor Graphs

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Fig. 5 Harmonious labeling of (Z2 × F) ∼ = K 1, pm −1

4 Small Zero-Divisor Graphs In this section, we present a series of tables which resolve the question of which zero-divisor graphs are graceful and/or harmonious for rings with a small number of zero-divisors. In [26, 27], all rings which have zero-divisor graphs with up to 14 vertices are classified up to isomorphism. We use the tables provided there to answer this question for all graphs up to 14 vertices. We know from the previous section when a zero-divisor graph is harmonious or graceful when the graph is either complete or complete bipartite. Thus, we omit the diagrams illustrating the graceful or harmonious labeling of all complete or complete bipartite graphs. On the other hand, more work is required for the graphs which are neither complete nor complete bipartite graphs. We follow [26, 27] for all rings which have a particular zero-divisor graphs on n ≤ 14 vertices up to isomorphism to begin classifying the graceful and harmonious zero-divisor graphs. In most cases, we actually provide the graceful or harmonious labeling of the given graph. In some cases, we simply cite a known result from the world of graph theory which proves this particular graph is graceful or harmonious. The author has done his best to give credit to those who proved it first, but he would like to acknowledge that many graphs are called different names by different authors and may well have mistakenly not been attributed to the first person to prove it. There are some graphs in which an argument is required to indicate why no harmonious labeling exists since they do not seem to fall into categories of graphs which have already been determined. Since the author is, as of this moment, unaware of a result already in the literature, we go ahead and prove that no harmonious labeling would be possible for these particular cases. We give these results their own theorems, although we suspect these results are likely already known somewhere in

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the literature. We apologize for not citing the appropriate person if this is indeed the case. For these arguments, a result in [20] will come in handy. Theorem 4.1 ([20, Theorem 8]) Let G be a graph with n edges and f : V (G) → Zn be a harmonious labeling. Then the map g(v) = a · f (v) + b is harmonious where a is a unit in Zn and b is any element of Zn . In particular, any translation g(v) = 1 · f (v) + b, of a harmonious labeling is harmonious. Because of this above result, we can assume if a graph is harmonious, then we could find a harmonious labeling with any particular vertex being labeled as 0. Corollary 4.2 Let G be a graph with n edges and f : V (G) → Zn be a harmonious labeling. Then for any vertex v0 ∈ V (G), we can find a harmonious labeling g : V (G) → Zn with g(v0 ) = 0. Proof Suppose f (v0 ) = λ where f is the assumed harmonious labeling. Then g : V (G) → Zn can be defined by g(v) = 1 · f (v) − λ and this will be harmonious by  Theorem 4.1. Moreover g(v0 ) = 1 · f (v0 ) − λ = λ − λ = 0 Vertices 1

R Z4 Z2 [X ]/(X 2 )

|R| 4 4

Graph K1 K1

Graceful? Yes - Theorem 2.4 Yes - Theorem 2.4

Harmonious? Yes - Theorem 2.6 Yes - Theorem 2.6

Vertices 2

R Z9 Z2 × Z2 Z3 [X ]/(X 2 )

|R| 9 4 9

Graph K2 K2 K2

Graceful? Yes - Theorem 2.4 Yes - Theorem 2.4 Yes - Theorem 2.4

Harmonious? Yes - Theorem 2.6 Yes - Theorem 2.6 Yes - Theorem 2.6

Vertices 3

R Z6 Z8 Z2 [X ]/(X 3 ) Z4 [X ]/(2X, X 2 − 2) Z2 [X, Y ]/(X, Y )2 Z4 [X ]/(2, X )2 F4 [X ]/(X 2 ) Z4 [X ]/(X 2 + X + 1)

|R| 6 8 8 8 8 8 16 16

Graph K 1,2 K 1,2 K 1,2 K 1,2 K3 K3 K3 K3

Graceful? Yes - Theorem 2.4 Yes - Theorem 2.4 Yes - Theorem 2.4 Yes - Theorem 2.4 Yes - Theorem 2.4 Yes - Theorem 2.4 Yes - Theorem 2.4 Yes - Theorem 2.4

Harmonious? Yes - Theorem 2.6 Yes - Theorem 2.6 Yes - Theorem 2.6 Yes - Theorem 2.6 Yes - Theorem 2.6 Yes - Theorem 2.6 Yes - Theorem 2.6 Yes - Theorem 2.6

On Gracefully and Harmoniously Labeling Zero-Divisor Graphs

Vertices 4

Vertices 5

Vertices 6

R Z2 × F4 Z3 × Z3 Z25 Z5 [X ]/(X 2 )

R Z2 × Z5 Z3 × F4 Z2 × Z4 Z2 × Z2 [X ]/(X 2 )

R Z3 × Z5 F4 × F4 Z49 Z7 [X ]/(X 2 ) Z2 × Z2 × Z2

|R| 8 9 25 25

Graph K 1,3 K 2,2 K4 K4

|R| 10 12 8 8

|R| 15 16 49 49 8

Graceful? Yes - Theorem 2.4 Yes - Theorem 2.4 Yes - Theorem 2.4 Yes - Theorem 2.4

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Harmonious? Yes - Theorem 2.6 No - Theorem 2.6 Yes - Theorem 2.6 Yes - Theorem 2.6

Graph K 1,4 K 2,3 S5,2 S5,2

Graceful? Yes - Theorem 2.4 Yes - Theorem 2.4 Yes - Theorem 2.4 Yes - Fig. 2

Harmonious? Yes - Theorem 2.6 No - Theorem 2.6 Yes - Theorem 2.6 Yes - Fig. 3

Graph K 2,4 K 3,3 K6 K6 Fig. 6a, b

Graceful? Yes - Theorem 2.4 Yes - Theorem 2.4 No - Theorem 2.4 No - Theorem 2.4 Yes - Fig. 6a

Harmonious? No - Theorem 2.6 No - Theorem 2.6 No - Theorem 2.6 No - Theorem 2.6 Yes - Fig. 6b

250 Vertices 7

C. P. Mooney R Z2 × Z7 F4 × Z5 Z3 × Z4 Z3 × Z2 [X ]/(X 2 ) Z16 Z2 [X ]/(X 4 ) Z4 [X ]/(X 2 + 2) Z4 [X ]/(X 2 + 2X + 2) Z4 [X ]/(X 3 − 2, 2X 2 , 2X ) Z2 [X, Y ]/(X 3 , X Y, Y 2 ) Z8 [X ]/(2X, X 2 ) Z4 [X ]/(X 3 , 2X 2 , 2X )

|R| 14 20 12 12 16 16 16 16 16 16 16 16 16

Graph K 1,6 K 3,4 Fig. 7a, b Fig. 7a, b Fig. 8a, b Fig. 8a, b Fig. 8a, b Fig. 8a, b Fig. 8a, b Fig. 8c, d (B(4, 3, 4)) Fig. 8c, d (B(4, 3, 4)) Fig. 8c, d (B(4, 3, 4)) Fig. 8c, d B(4, 3, 4)

Graceful? Yes - Theorem 2.4 Yes - Theorem 2.4 Yes - Fig. 7a Yes - Fig. 7a Yes - Fig. 8a Yes - Fig. 8a Yes - Fig. 8a Yes - Fig. 8a Yes - Fig. 8a Yes [14, 21] Yes [14, 21] Yes [14, 21] Yes [14, 21]

Harmonious? Yes - Theorem 2.6 No - Theorem 2.6 No - Theorem 4.3 No - Theorem 4.3 Yes - Fig. 8b Yes - Fig. 8b Yes - Fig. 8b Yes - Fig. 8b Yes - Fig. 8b Yes - Fig. 8d Yes - Fig. 8d Yes - Fig. 8d Yes - Fig. 8d

Z4 [X ]/(X 2 + 2X ) Z8 [X ]/(2X, X 2 + 4) Z2 [X, Y ]/(X 2 , Y 2 − X Y )

16 16 16 16

Fig. 9a, b Fig. 9a, b Fig. 9a, b Fig. 9a, b

Yes - Fig. 8a Yes - Fig. 8a Yes - Fig. 8a Yes - Fig. 8a

Yes - Fig. 8b Yes - Fig. 8b Yes - Fig. 8b Yes - Fig. 8b

Z4 [X,Y ] (X 2 −2,X Y,Y 2 ,2X,2Y )

Z4 [X,Y ] (X 2 ,Y 2 −X Y,X Y −2,2X,2Y ) Z4 [X,Y ] 2 2 (X ,Y ,X Y −2,2X,2Y ) Z2 [X, Y ]/(X 2 , Y 2 ) Z4 [X ]/(X 2 ) Z2 [X, Y, Z ]/(X, Y, Z )2 Z4 [X,Y ] (X 2 ,Y 2 ,X Y,2X,2Y ) F8 [X ]/(X 2 ) Z4 [X ]/(X 3 + X + 1)

16

Fig. 9c, d

No - [12],[13]

Yes - Fig. 8d

16 16 16 16

Fig. 9c, d Fig. 9c, d K7 K7

No - [12],[13] No - [12],[13] No - Theorem 2.4 No - Theorem 2.4

Yes - Fig. 8d Yes - Fig. 8d No - Theorem 2.6 No - Theorem 2.6

64 64

K7 K7

No - Theorem 2.4 No - Theorem 2.4

No - Theorem 2.6 No - Theorem 2.6

Fig. 6 (a) Graceful (b) Harmonious labeling of (Z2 × Z2 × Z2 )

Fig. 7 (a) Graceful labeling of (Z3 × Z4 ) (b) Figure for Theorem 4.3

Theorem 4.3 The graph in Fig. 7b is not harmonious.

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Proof We will refer to the above figure throughout the proof to help guide the reader. Suppose the graph were harmonious, then let f : V (G) → Z8 be a harmonious labeling. We may use Corollary 4.2 to assume one vertex of our choosing is labeled 0 as above. Then, we observe that this vertex is adjacent to four other vertices, with labels which we have called a, b, c, d as in the above figure. These will induce edges with the labels a, b, c, d, respectively. There are two remaining vertices which we have called their labels x and y. We see by symmetry that these are interchangeable, as are a and b. The edges which remain, adjacent to the vertices labeled x and y, must receive the labels of 0, x, and y in some order since these values remain unassigned otherwise. We may as well assume (by symmetry) the edge f −1 (a)— f −1 (y) receives the label 0, which means y = −a. Since the edges adjacent to the vertex labeled x cannot receive the label x, or else this forces a or b to be 0 which contradicts the injectivity of f , this means the edge f −1 (b)— f −1 (y) must get the label x, so x = b + y = b + (−a). This leaves two cases which remain for which edge will get labeled y, either (1) f −1 (a)— f −1 (x) or (2) f −1 (b)— f −1 (x). In the first case, if a + x = y, then substituting x = b − a yields a + (b − a) = y which implies b = y which is a contradiction of the injectivity of the labeling. Thus, we turn our attention to the second case where b + x = y. After substituting again, we see this means b + (b − a) = −a or 2b = 0. Since b cannot be 0, the only solution in Z8 is that b = 4. If b = 4 and x + b = −a, then x + 4 = −a. This means our final edge will receive, x + a = x + (−x − b) = −b = −4 = 4 ( mod 8). This is the same label as edge f −1 (0)— f −1 (b) receives which means our harmonious labeling f is not actually harmonious, a contradiction in both cases. Thus, there is no harmonious labeling of this graph (Fig. 7b). 

Fig. 8 (a) Graceful (b) Harmonious labeling of (Z16 ) (c) No graceful labeling of (Z8 [x]/(2X, X 2 )) exists (d) Harmonious labeling of (Z8 [x]/(2X, X 2 ))

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Fig. 9 (a) Graceful (b) Harmonious labeling of (Z4 [X ]/(X 2 + 2X )) (c) No graceful labeling of (Z4 [x]/(X 2 )) exists (d) Harmonious labeling of (Z4 [x]/(X 2 ))

Vertices 8

R Z2 × F8 Z3 × Z7 Z5 × Z5 Z27 Z9 [X ]/(3X, X 2 − 3) Z9 [X ]/(3X, X 2 − 6) Z3 [X ]/(X 3 ) Z3 [X, Y ]/(X, Y )2 Z9 [X, Y ]/(3, X )2 F9 [X ]/(X 2 ) Z9 [X ]/(X 2 + 1)

|R| 16 21 25 27 27 27 27 27 27 81 81

Graph K 1,7 K 2,6 K 4,4 Fig. 10a, b Fig. 10a, b Fig. 10a, b Fig. 10a, b K8 K8 K8 K8

Graceful? Yes Yes Yes Yes - Fig. 10a Yes - Fig. 10a Yes - Fig. 10a Yes - Fig. 10a No - Theorem 2.4 No - Theorem 2.4 No - Theorem 2.4 No - Theorem 2.4

Harmonious? Yes No No Yes - Fig. 10b Yes - Fig. 10b Yes - Fig. 10b Yes - Fig. 10b No - Theorem 2.6 No - Theorem 2.6 No - Theorem 2.6 No - Theorem 2.6

On Gracefully and Harmoniously Labeling Zero-Divisor Graphs

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Fig. 10 (a) Graceful (b) Harmonious labeling of (Z27 )

Vertices 9

R Z2 × F9 Z3 × F8 F4 × Z7 Z2 × Z2 × Z3 Z4 × F4 Z2 [X ]/(X 2 ) × F4

|R| 18 24 28 12 16 16

Graph K 1,8 K 2,7 K 3,6 Fig. 11a, b Fig. 11c, d Fig. 11c, d

Graceful? Yes - Theorem 2.4 Yes - Theorem 2.4 Yes - Theorem 2.4 Yes - Fig. 11a Yes - Fig. 11c Yes - Fig. 11c

Harmonious? Yes - Theorem 2.6 No - Theorem 2.6 No - Theorem 2.6 Yes - Fig. 11b No - Theorem 4.4 No - Theorem 4.4

Theorem 4.4 The graph in Fig. 11(d) is not harmonious. Proof We will refer to Fig. 11(d). Suppose there were in fact a harmonious labeling of this graph. Then by Corollary 4.2, we may choose any vertex we wish to be labeled 0 under the labeling f : V (G) → Z12 possibly after translating the harmonious labeling. This vertex which we chose be to labeled 0 is adjacent to 6 vertices labeled a, b, c, d, e, f , and thus, will induce an edge label which is also a, b, c, d, e, f , respectively. The remaining two vertices will receive labels x and y which must be distinct from 0, a, b, c, d, e, f since harmonious labelings must be injective. One edge must receive a labeling of 0, which means x or y must be the additive inverse of a, b, or c. By symmetry, we may as well assume x = −a. We will also need edges to receive the labeling of x and y. Clearly, edge f −1 (a)— f −1 (x), f −1 (b)— f −1 (x), f −1 (c)— f −1 (x) cannot receive x or else a, b, c would have to be 0 contradicting the injectivity of the labeling. Similarly, f −1 (a)— f −1 (y), f −1 (b)— f −1 (y), f −1 (c)— f −1 (y) cannot receive a label of y or else a, b, c would have to be 0, again a contradiction. This means either f −1 (b)— f −1 (x) or f −1 (c)— f −1 (x) must be labeled y. We may assume it is f −1 (b)— f −1 (x) that receives y. But this means b + (−a) = y, hence b = y + a. But then the edge f −1 (a)— f −1 (y) must receive edge label a + y, which is the same label that f −1 (0)— f −1 (b) receives, i.e.,

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Fig. 11 (a) Graceful (b) Harmonious labeling of (Z2 × Z2 × Z3 ) (c) Graceful labeling of (Z4 × F4 ) (d) Graph for (Z4 × F4 ) with no harmonious labeling for Theorem 4.4

0 + (y + a) = a + y. Thus, the labeling f is not harmonious as the edges do not get unique labels, a contradiction. Hence, no such harmonious labeling can exist.  Vertices 10

Vertices 11

R Z3 × F9 F 4 × F8 Z5 × Z7 Z121 Z11 [X ]/(X 2 )

|R| 27 32 35 121 121

R Z2 × Z11 F 4 × F9 Z5 × F8 Z2 × Z9 Z2 × Z3 [X ]/(X 2 ) Z5 × Z4 Z5 × Z2 [X ]/(X 2 ) Z2 × Z8 Z2 × Z2 [X ]/(X 3 ) Z2 × Z4 [X ]/(2X, X 2 − 2) Z2 × Z2 [X, Y ]/(X, Y )2 Z2 × Z4 [X ]/(2, X )2 Z4 × Z4 Z4 × Z2 [X ]/(X 2 ) Z2 [X ]/(X 2 ) × Z2 [X ]/(X 2 )

Graph K 2,8 K 3,7 K 4,6 K 10 K 10

|R| 22 36 40 18 18 20 20 16 16 16 16 16 16 16 16

Graceful? Yes - Theorem 2.4 Yes - Theorem 2.4 Yes - Theorem 2.4 No - Theorem 2.4 No - Theorem 2.4

Graph K 1,10 K 3,8 K 4,7 Fig. 12a, b Fig. 12a, b Fig. 12c, d Fig. 12c, d Fig. 13a, b Fig. 13a, b Fig. 13a, b Fig. 13c, d Fig. 13c, d Fig. 14a, b Fig. 14a, b Fig. 14a, b

Graceful? Yes - Theorem 2.4 Yes - Theorem 2.4 Yes - Theorem 2.4 Yes - Fig. 12a Yes - Fig. 12a Yes - Fig. 12c Yes - Fig. 12c Yes - Fig. 13a Yes - Fig. 13a Yes - Fig. 13a Yes - Fig. 13c Yes - Fig. 13c Yes - Fig. 14a Yes - Fig. 14a Yes - Fig. 14a

Harmonious? No - Theorem 2.6 No - Theorem 2.6 No - Theorem 2.6 No - Theorem 2.6 No - Theorem 2.6

Harmonious? Yes - Theorem 2.6 No - Theorem 2.6 No - Theorem 2.6 Yes - Fig. 12b Yes - Fig. 12b No - Theorem 4.5 No - Theorem 4.5 Yes - Fig. 13b Yes - Fig. 13b Yes - Fig. 13b Yes - Fig. 13d Yes - Fig. 13d Yes - Fig. 14b Yes - Fig. 14b Yes - Fig. 14b

On Gracefully and Harmoniously Labeling Zero-Divisor Graphs

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Fig. 12 (a) Graceful (b) Harmonious labeling of (Z2 × Z9 ) (c) Graceful labeling of (Z5 × Z4 ) (d) Graph for (Z5 × Z4 ) with no harmonious labeling for Theorem 4.5

Theorem 4.5 There is no harmonious labeling of the graph in Fig. 12(d). Proof We argue this in much the same way as the previous results. If we assume there were a harmonious labeling, f : V (G) → Z16 , we can assume one vertex of our choosing is labeled 0 by translation as in Corollary 4.2. We choose the most central one, again, as we see this is the most useful for our argument. This vertex is adjacent to all vertices except two, so we label these a, b, c, d, e, f, g, h, as above and then note that these will induce the edge labels a, b, c, d, e, f, g, h respectively. The two remaining vertices, will be called x and y. Again, we notice by symmetry, x and y are completely interchangeable. One edge must receive a label of 0, so we go ahead and assume by symmetry this will be edge f −1 (x)— f −1 (a), which implies that a = −x. We will need an edge to receive the label of y as well. It is immediate that the edge which receives the edge label of y cannot be adjacent to y or else it would force a, b, c or d to be 0, contradicting the injectivity of f . This means edge f −1 (x)— f −1 (b), f −1 (x)— f −1 (c), or f −1 (x)— f −1 (d) must be labeled y. Again, by symmetry we may assume it is f −1 (x)— f −1 (b), which implies x + b = y. Hence, b = y − x. We then can observe that since f −1 (b) is adjacent to the vertex labeled 0, we have edge f −1 (0)— f −1 (b) receiving 0 + b = y − x. We then note that edge f −1 (y)— f −1 (a) also receives y + a = y + (−x), which means this labeling is not harmonious since two edges receive the same label. This shows it is not possible to harmoniously label this graph. 

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Fig. 13 (a) Graceful (b) Harmonious labeling of (Z2 × Z8 ) (c) Graceful (d) Harmonious labeling of (Z2 × Z2 [X, Y ]/(X, Y 2 ))

Fig. 14 (a) Graceful (b) Harmonious labeling of (Z4 × Z4 )

Vertices 12

R Z3 × Z11 Z5 × F9 Z7 × Z7 Z2 × Z2 × F4 Z169 Z13 [X ]/(X 2 )

|R| 33 45 49 16 169 169

Graph K 2,10 K 4,8 K 6,6 Fig. 16a, b K 12 K 12

Graceful? Yes - Theorem 2.4 Yes - Theorem 2.4 Yes - Theorem 2.4 Yes - Fig. 16a No - Theorem 2.4 No - Theorem 2.4

Harmonious? No - Theorem 2.6 No - Theorem 2.6 No - Theorem 2.6 Yes - Fig. 16b No - Theorem 2.6 No - Theorem 2.6

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Fig. 15 (a) Graceful (b) Harmonious labeling of (Z2 × Z3 × Z3 )

Fig. 16 (a) Graceful (b) Harmonious labeling of (Z2 × Z2 × F4 )

Vertices 13

R Z2 × Z13 F4 × Z11 Z7 × F8 Z2 × Z3 × Z3 Z2 × Z2 × Z4 Z2 × Z2 × Z2 [X ]/(X 2 )

|R| 26 44 56 18 16 16

Graph K 1,12 K 3,10 K 6,7 Fig. 15a, b Fig. 17a, b Fig. 17a, b

Graceful? Yes - Theorem 2.4 Yes - Theorem 2.4 Yes - Theorem 2.4 Yes - Fig. 15a Yes - Fig. 17a Yes - Fig. 17a

Fig. 17 (a) Graceful (b) Harmonious labeling of (Z2 × Z2 × Z4 )

Harmonious? Yes - Theorem 2.6 No - Theorem 2.6 No - Theorem 2.6 Yes - Fig. 15b Yes - Fig. 17b Yes - Fig. 17b

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Fig. 18 (a) Graceful (b) Harmonious labeling of (Z2 × Z2 × Z2 × Z2 )

Fig. 19 (a) Graceful (b) Harmonious labeling of (Z3 × Z9 )

Vertices 14

R Z3 × Z13 Z5 × Z11 Z7 × F9 F8 × F8 Z2 × Z2 × Z2 × Z2 Z3 × Z9 Z3 × Z3 [X ]/(X 2 )

|R| 39 55 63 64 16 27 27

Graph K 2,12 K 4,10 K 6,8 K 7,7 Fig. 18a, b Fig. 19a, b Fig. 19a, b

Graceful? Yes - Theorem 2.4 Yes - Theorem 2.4 Yes - Theorem 2.4 Yes - Theorem 2.4 Yes - Fig. 18a Yes - Fig. 19a Yes - Fig. 19a

Harmonious? No - Theorem 2.6 No - Theorem 2.6 No - Theorem 2.6 No - Theorem 2.6 Yes - Fig. 18b Yes - Fig. 19b Yes - Fig. 19b

Questions 4.6 We conclude with a few open questions that seem interesting for potential future research based on patterns observed in these tables of small examples so far. (1) Are there an infinite number of integers n such that every commutative ring R that has |V ((R))| = n is graceful (or harmonious)? What about for primes? (2) Along these lines, it seems that 1, 4, and 9 all have the property that every ring has a graceful zero-divisor graph. Does this pattern continue for n 2 ? Perhaps for p 2 where p is a prime?

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(3) Are there other necessary or sufficient algebraic properties that determine whether a zero-divisor graph is graceful or harmonious besides the ones described in this paper? (4) There is some progress on which numbers of vertices admit a zero-divisor graph. For example, Redmond has show there is no zero-divisor graph with 1209, 3341, or 5465 vertices [28]. Could we compile a similar list of numbers for which there is no graceful or harmonious zero-divisor graphs with that number of vertices? Acknowledgements The author would like to thank the referee for their careful reading of the article and helpful suggestions for improving the quality of the article. The bibliography is significantly improved thanks to their insight.

References 1. S. Akbari, F. Heydari, The regular graph of a commutative ring. Period. Math. Hung. 67, 211–220 (2013) 2. S. Akbari, A. Mohammadian, On the zero-divisor graph of a commutative ring. J. Algebra 274, 847–855 (2004) 3. D.F. Anderson, A. Badawi, The zero-divisor graph of a commutative semigroup: a survey, in Groups, Modules, and Model Theory–surveys and Recent Developments. (Springer, Cham, 2017), pp. 23–39 4. D.F. Anderson, P.S. Livingston, The zero-divisor graphs of a commutative ring. J. Algebra 217, 434–447 (1999) 5. D.F. Anderson, A. Frazier, A. Lauve, P.S. Livingston, The zero-divisor graphs of a commutative ring, II, in Ideal theoretic methods in commutative algebra (Columbia, MO, 1999), Lecture Notes in Pure and Applied Mathematics, vol. 220 (Dekker, New York, 2001), pp. 61–72 6. D.D. Anderson, M. Naseer, Beck’s coloring of a commutative ring. J. Algebra 159, 500–514 (1993) 7. D.F. Anderson, M.C. Axtell, J.A. Stickles Jr., Zero-divisor graphs in commutative rings, in Commutative Algebra Noetherian and Non-Noetherian Perspectives. (Springer, New York, 2011), pp. 23–45 8. D.F. Anderson, A. Badawi, On the zero-divisor graph of a ring. Comm. Algebra 36(8), 3073– 3092 (2008) 9. D.F. Anderson, R. Levy, J. Shapiro, Zero-divisor graphs, von Neumann regular rings, and Boolean algebras. J. Pure Appl. Algebra 180(3), 221–241 (2003) 10. A. Badawi, On the total graph of a ring and its related graphs: a survey, in Commutative Algebra. ed. by M. Fontana, S. Frisch, S. Glaz (Springer, New York, NY, 2014) 11. I. Beck, Coloring of commutative rings. J. Algebra 116, 208–226 (1988) 12. J.C. Bermond, A.E. Brouwer, A. Germa, Systèmes de triplets et différences associèes, Problèmes Combinatoires et Théorie des Graphes, Colloq. Intern. du CNRS, vol. 260, Editions du Centre Nationale de la Recherche Scientifique, Paris (1978), pp. 35–38 13. J.C. Bermond, A. Kotzig, J. Turgeon, On a Combinatorial Problem of Antennas in Radioastronomy, Combinatorics (North- Holland, Amsterdam, 1978), pp. 135–149 14. J.C. Bermond, Graceful Graphs, Radio Antennae and French Windmills, Graph Theory and Combinatorics (Pitman, London, 1979), pp. 18–37 15. J. Coykendall, J. Maney, Irreducible divisor graphs. Comm. Algebra 35(3), 885–895 (2007) 16. F. DeMeyer, K. Schneider, Automorphisms and zero-divisor graphs of commutative rings. Int. J. Commut. Rings 1, 93–106 (2002) 17. W. Fang, A computational approach to the graceful tree conjecture. arXiv:1003.3045

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18. J. Gallian, A dynamic survey of graph labeling. Electron. J. Comb. DS06, 1–389 (1996) 19. S.W. Golomb, How to number a graph, in Graph Theory and Computing. ed. by R.C. Read (Academic Press, New York, 1972), pp. 23–37 20. R.L. Graham, N.J.A. Sloane, On additive bases and harmonious graphs. SIAM J. Alg. Discrete Methods 1, 382–404 (1980) 21. K.M. Koh, D.G. Rogers, C.K. Lim, On graceful graphs: sum of graphs, Research Report 78. College of Graduate Studies. (Nanyang University, 1979) 22. H.R. Maimani, M. Salimi, A. Sattari, S. Yassemi, Comaximal graph of commutative rings. J. Algebra 319(4), 1801–1808 (2008) 23. C.P. Mooney, Generalized factorization in commutative rings with zero-divisors. Houston J. Math. 41(1), 15–32 (2015) 24. C.P. Mooney, Generalized factorization in commutative rings with zero-divisors, Ph.D. thesis, The University of Iowa (2013) 25. S.B. Mulay, Cycles and symmetries of zero-divisors. Comm. Algebra 30(7), 3533–3558 (2002) 26. S.P. Redmond, On zero-divisor graphs of small finite commutative rings. Discrete Math. 307(9– 10), 1155–1166 (2007) 27. S.P. Redmond, Corrigendum to: “On zero-divisor graphs of small finite commutative rings” [Discrete Math. 307 (2007), no. 9–10, 1155-1166. Discrete Math. 307(21), 2449–2452 (2007) 28. S.P. Redmond, Counting zero-divisors, in Commutative Rings: New Research. ed. by J. Lee (Nova Science Publishers, Hauppauge, NY, 2009) 29. A. Rosa, On certain valuations of the vertices of a graph, Theory of Graphs (Internat. Symposium, Rome, July, 1966) Gordon and Breach. N. Y. and Dunod Paris (1967), pp. 349–355

A Survey on Genus of Selected Graphs from Commutative Rings T. Asir, K. Mano, and M. Subathra

Abstract In this paper, we survey recent results on the orientable and nonorientable genus of graphs arising from commutative rings. Specifically, we restrict our attention to the genus of zero-divisor graphs, Jacobson graphs, annihilator graphs, unit graphs and unitary Cayley graphs. Keywords Genus number · Crosscap number · Zero-divisor graph · Jacobson graph · Annihilator graph · Unit graph · Unitary cayley graph 2000 Mathematics Subject Classification Primary 13A15 · 13M05 · Secondary 05C75 · 05C25

1 Introduction In this article, we review the topological concept called genus of graphs constructed out of ring structure. The prime objective of topological graph theory is to draw a graph on a surface so that no two edges cross, an intuitive geometric problem that can be enriched by specifying symmetries or combinatorial side-conditions. Graphs on surfaces form a natural link between discrete and continuous mathematics. They enable us to understand both graphs and surfaces better. The theoretical importance of minimum graph embedding topics has been enhanced by impressive connections The research of T. Asir was in part supported by a grant from Science and Engineering Research Board (SERB-MATRICS Project-Ref. MTR/2017/000830). T. Asir (B) Department of Mathematics-DDE, Madurai Kamaraj University, Madurai 625021, TN, India K. Mano Department of Mathematics, Fatima College, Madurai 625021, TN, India M. Subathra Department of Mathematics, Lady Doak College, Madurai 625021, TN, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. Badawi and J. Coykendall (eds.), Rings, Monoids and Module Theory, Springer Proceedings in Mathematics & Statistics 382, https://doi.org/10.1007/978-981-16-8422-7_15

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to areas such as VLSI design, computer algorithms and complexity, and computer graphics. On the other hand, the study of algebraic structures, using the properties of graph theory, tends to be an exciting research topic in the last decade. There are so many ways to construct graphs from commutative ring structures. Some of the popular constructions are zero-divisor graphs, total graphs, Jacobson graphs, annihilator graphs, unit graphs and unitary Cayley graphs. Through these constructions, the interplay between algebraic structures and graphs are studied. Over the past several years, the topological structures of these graphs are widely investigated; more specifically, embeddings of graphs arising from ring structure. The aim of this survey article is to present results obtained with regard to embeddings of graphs constructed out of commutative rings. For non-negative integers  and k, let S denote the sphere with  handles and Nk denote a sphere with k crosscaps attached to it. Note that any orientable surface is homeomorphic to one of the surfaces S and any nonorientable surface is homeomorphic to one of the surfaces Nk . The genus of a graph G, denoted by γ(G), is the minimal integer  such that the graph can be embedded in S . Intuitively, G is embedded in a surface if it can be drawn in the surface so that its edges intersect only at their common vertices. We say that a graph G is planar if γ(G) = 0, and toroidal if γ(G) = 1. Therefore a planar graph G has an embedding in the plane. The nonorientable genus of a graph G, denoted γ(G), ˜ is the minimal integer k such that the graph can be embedded in Nk . For example, projective space is of crosscap one, and the Klein bottle is of crosscap two. A nonplanar graph is said to be projective if it can be embedded into the projective plane. Determining the genus γ(G) of a graph G is one of the fundamental problems in topological graph theory. Note that Thomassen [48] announced that the graph genus problem is NP-complete. Throughout the paper, let R be a commutative ring. Let Z (R) be the set of zerodivisors of R and U (R) be the group of units of R. We denote the ring of integers module n by Zn , the finite field with q elements by Fq and the cardinality of the set A by |A|. Let G = (V, E) be a simple graph. We denote the complete graph with n vertices by K n and the complete bipartite graph with parts of size m and n by K m,n . Further, the degree of a vertex v in G and the minimum degree of G is denoted by deg(v) and δ(G), respectively. For terminology and notation that are not defined in this article can be seen in [37] for ring theory and in [26] for graph theory.

2 Genus Properties of Graphs In this section, we gather some facts and bounds related to the genus of a simple graph. Observe that if H is a subgraph of a graph G, then γ(H ) ≤ γ(G) and γ(H ˜ ) ≤ γ(G). ˜ Let us display the genus and crosscap formula for complete and complete bipartite graphs. Proposition 2.1 ([50, Theorem 6.37,6.38,6.39,6.42]) Let m and n ≥ 3 be integers. Then

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(i) γ(K n ) =



(ii) γ(K m,n ) =

(n−3)(n−4) 12



263

 ;



(m−2)(n−2) ; 4 (mn−2)(n−1) = ; 2 2

(iii) γ(K mn,n,n ) (iv) γ(K n,n,n,n ) = (n − 1) for n = 3 and γ(K 3,3,3,3 ) = 5. Proposition 2.2 ([41, Theorem 4.4.5,4.4.6,4.4.7]) Let m ≥ 3 and n ≥ 3 be integers. Then   (n−3)(n−4) if n ≥ 3 and n = 7; 6 (i) γ(K ˜ n) = 3 if n = 7.   (m−2)(n−2) . (ii) γ(K ˜ m,n ) = 2 According to the crosscap formula, the nonplanar graphs K 5 , K 6 , K 3,3 and K 3,4 are all projective. Next result is the famous Euler Formula for graph embedding. Proposition 2.3 ([50]) (Euler Formula) If G is a finite connected graph with n vertices, m edges, and genus γ, then n − m + f = 2 − 2γ, where f is the number of faces created when G is minimally embedded on a surface of genus γ. Euler’s formula for nonorientable surfaces says that if G is a graph with n vertices, m edges and nonorientable genus γ, ˜ then n − m + f = 2 − γ˜ where f is the number of regions created when G is embedded on a nonorientable surface of genus γ. ˜   |E|−|V |−1 . The genus of any embedding of a graph G is an integer between 0 and 2 The crosscap number is an integer between 0 and |E| − |V | − 1. We close the section by mentioning a relation between the orientable and nonorientable genus of graph. An important relation between the orientable and nonorientable genus of graph is due to Gross et al. [33, Theorem 3.4.5], which says that γ(G) ˜ ≤ 2γ(G) + 1. Whereas, there are some graphs having crosscap one but arbitrarily large genus, see [20]. For further details on the notion of embedding of a graph in a surface, see White [50, Chapter 6].

3 Zero-Divisor Graphs The graph denoted by 0 (R) associated to zero-divisors of a commutative ring R was introduced by Beck [23] in 1988. He defined 0 (R) to be the graph whose vertices are the elements of R and in which two distinct vertices x and y are adjacent if and only if the product of x and y is zero. The definition along with the name for zero-divisor graph was first introduced by Anderson and Livingston [6] in 1999. Actually they defined the subgraph (R) of 0 (R). Definition 3.1 ([6]) Let R be a commutative ring with identity and Z (R) be its set of zero-divisors. The zero-divisor graph of R, denoted by (R), is the simple graph with vertex set Z (R)∗ = Z (R) \ {0}, and two distinct vertices x and y are adjacent if x y = 0.

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Fig. 1 Zero-divisor graphs of order less than four

Fig. 2 Zero-divisor graphs on 5 and 6 vertices

Example 3.2 In Figs. 1 and 2, several planar zero-divisor graphs are given. First, we focus on the planar zero-divisor graphs. Recall that a graph is planar if it can be drawn in the plane in such a way that no edges intersect, except of course at a common end-vertex. The first work towards planar zero-divisor graphs was given in [7] where they gave necessary and sufficient condition for zero-divisor graph to be planar for some specific rings. Theorem 3.3 ([7, Theorem 5.1]) (i) Let R = Zn where n ≥ 2 is not prime. Then (R) is planar if and only if n ∈ {8, 12, 16, 18, 25, 27} ∪ {2 p, 3 p : p is prime}. (ii) Let R = Zn 1 × · · · × Znr where r ≥ 2 and 2 ≤ n 1 ≤ · · · ≤ n r . Then (R) is planar if and only if R is one of Z2 × Z4 , Z2 × Z6 , Z2 × Z8 , Z2 × Z9 , Z2 × Z p , Z3 × Z4 , Z3 × Z9 , Z3 × Zq , Z2 × Z2 × Z2 , Z2 × Z2 × Z3 where p ≥ 2 and q ≥ 3 are primes. Theorem 3.4 ([7, Theorem 5.2]) Let Rn,m = Zn [x]/(x m ) where m, n ≥ 2; (i) (Rn,2 ) is planar if and only if n ≤ 5; (ii) (Rn,3 ) is planar if and only if n ≤ 3; (iii) (Rn,4 ) is planar if and only if n = 2; (iv) (Rn,m ) is never planar when m ≥ 5. After establishing the above two results, Anderson et al. [7] considered the problem of which finite rings in general determine a planar zero-divisor graph. This was

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partially answered by Akbari et al. [2] who were able to refine the question to local rings of cardinality at most thirty-two. Theorem 3.5 ([2, Theorems 1.2, 1.4]) Let R be a finite commutative local ring with maximal ideal m = 0. Then (R) is nonplanar if one of the following holds. (i) |R/m| ≥ 4 and |R| ≥ 26; (ii) |R/m| = 3 and |R| ≥ 28; (iii) |R/m| = 2 and |R| ≥ 33. Akbari et al. [2] also proved the following result regarding the planarity of zerodivisor graphs. Theorem 3.6 ([2, Theorem 1.6]) Let R be a local commutative Artinian ring such that (R) is a planar graph. Then R is quasi-Frobenius, or R ∼ = Z4 [x]/(2, x)2 or 2 ∼ R = Z2 [x, y]/(x, y) . Note that, if R is a finite commutative local ring that is not a field and contains at least thirty-three elements, then (R) is not planar. Also note that (Z27 ) is planar. In view of these, Akbari et al. [2] posed the question: For any local ring R (not a field) of cardinality 32, “Is (R) nonplanar?”. This question has also been answered independently by Wang [49], Belshoff and Chapman [24]. Theorem 3.7 ([49, Theorem 3.2], [24, Proposition 5]) If R is a commutative local ring of cardinality 32 and R is not a field, then (R) is not planar. In 2003, Smith [45] provided the classification of finite commutative rings R for which (R) is planar listing forty-four isomorphism classes in all. The corresponding results are given in Theorem 3.8 and Theorem 3.9. Belshoff and Chapman [24] also gave all finite commutative local rings with planar zero-divisor graphs using slightly different techniques than [45]. Theorem 3.8 ([45, Lemma 2.1, Lemma 2.2, Lemma 2.3, Theorem 3.7]) Let R be a finite commutative local ring which is not a field. Then (R) is a planar graph if and only if R is isomorphic to one of the following 29 rings. Z4 , Z2 [x]/(x 2 ), Z9 , Z3 [x]/(x 2 ), Z8 , Z2 [x]/(x 3 ), Z4 [x]/(2x, x 2 − 2), Z16 , Z2 [x]/(x 4 ), Z4 [x]/(2x, x 3 − 2), Z4 [x]/(x 2 − 2), Z4 [x]/(x 2 + 2x + 2), F4 [x]/(x 2 ), Z4 [x]/(x 2 + x + 1), Z2 [x, y]/(x, y)2 , Z4 [x]/(2, x)2 , Z27 , Z3 [x]/(x 3 ), Z9 [x]/(x 2 − 3, 3x), Z9 [x]/(x 2 − 6, 3x), Z2 [x, y]/(x 2 , y 2 − x y), Z2 [x, y]/(x 2 , y 2 ), Z8 [x]/(2x − 4, x 2 ), Z4 [x]/(x 2 ), Z4 [x]/(x 2 − 2x), Z4 [x, y]/(x 2 , x y − 2, y 2 − x y, 2x, 2y), Z4 [x, y]/(x 2 , x y − 2, y 2 , 2x, 2y), Z25 , Z5 [x]/(x 2 ).

Theorem 3.9 ([45, Theorem 2.12]) Let R be a finite commutative nonlocal ring and F be a finite field. Then (R) is a planar graph if and only if R is isomorphic to one of the following 15 rings:

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Z2 × F, Z3 × F, Z2 × Z4 , Z2 × Z2 [x]/(x 2 ), Z3 × Z4 , Z3 × Z2 [x]/(x 2 ), Z2 × Z2 × Z2 , Z2 × Z2 × Z3 , Z2 × Z8 , Z2 × Z2 [x]/(x 3 ), Z2 × Z4 [x]/(2x, x 2 − 2), Z2 × Z9 , Z2 × Z3 [x]/(x 2 ), Z3 × Z9 , Z3 × Z3 [x]/(x 2 ).

For results regarding the planarity of zero-divisor graph of infinite commutative rings, we refer the readers to Smith [44] and Yao [53]. Next, we are interested in the commutative rings that have genus one zero-divisor graphs. This study was first initiated by Wang in [49] and continued by Wickham [51]. Actually, he found all rings of the form Z p1α1 × · · · × Z pnαn and Zn [x]/(x m ) that have genus one zero-divisor graph. The next theorem characterizes all rings of the form Zn whose (R) has genus at most one. Theorem 3.10 ([49, Theorem 3.5]) Let R be a homomorphic image of Zn and p be a prime number. Then γ((R)) ≤ 1 if and only if R is one of the following rings. Z p , Z4 , Z9 , Z25 , Z49 , Z8 , Z27 , Z16 , Z32 , Z2 × Z p , Z3 × Z p , Z5 × Z5 , Z2 × Z4 , Z3 × Z4 , Z2 × Z9 , Z3 × Z9 , Z4 × Z5 , Z4 × Z7 , Z4 × Z4 , Z2 × Z8 , Z3 × Z8 , Z2 × Z2 × Z2 , Z2 × Z2 × Z3 , Z2 × Z2 × Z5 , Z2 × Z2 × Z7 , Z2 × Z3 × Z3 , Z3 × Z3 × Z3 , Z2 × Z2 × Z4 , Z2 × Z2 × Z2 × Z2 .

The next theorem characterizes all rings of the form Zn [x]/(x m ) whose (R) has genus at most one. Theorem 3.11 ([49, Theorem 3.11]) Let Rm,n = Zn [x]/(x m ) where m, n ≥ 2. Then (i) (ii) (iii) (iv) (v)

γ((R2,n )) ≤ 1 if and only if n = 2, 3, 4, 5, 7. γ((R3,n )) ≤ 1 if and only if n = 2, 3. γ((R4,n )) ≤ 1 if and only if n = 2. γ((R5,n )) ≤ 1 if and only if n = 2. γ((Rm,n )) ≥ 2 whenever m ≥ 6.

After the work of Wang [49], Cameron Wickham [52] and Hung-Jen ChiangHsieh et al. [28] independently characterized all the finite commutative rings whose zero-divisor graph has genus one. The following theorem completely characterizes all finite local commutative rings whose zero-divisor graph has genus one. Theorem 3.12 ([28, Theorem 3.5.2],[52, Theorem 4.1]) Let R be a finite commutative local ring which is not a field. Then γ((R)) = 1 if and only if R is isomorphic to one of the following 17 rings: Z32 , Z49 , Z2 [x]/(x 5 ), F8 [x]/(x 2 ), Z2 [x, y]/(x 3 , x y, y 2 ), Z2 [x, y, z]/(x, y, z)2 , Z4 [x]/(x 3 + x + 1), Z4 [x]/(x 3 − 2, x 5 ), Z4 [x]/(x 4 − 2, x 5 ), Z4 [x, y]/(x 3 , x 2 − 2, x y, y 2 ), Z4 [x]/(x 3 , 2x), Z4 [x, y]/(2x, 2y, x 2 , x y, y 2 ), Z7 [x]/(x 2 ), Z8 [x]/(x 2 , 2x), Z8 [x]/(x 2 − 2, x 5 ), Z8 [x]/(x 2 + 2x − 2, x 5 ), Z8 [x]/(x 2 − 2x + 2, x 5 ).

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The following result provides a complete list of nonlocal commutative rings whose zero-divisor graphs has genus one. Theorem 3.13 ([28, Theorem 3.6.2], [52, Theorem 3.1]) Let R be a finite commutative ring which is not local. Then γ((R)) = 1 if and only if R is isomorphic to one of the following 29 rings: F4 × F4 , F4 × Z5 , F4 × Z7 , Z5 × Z5 , Z2 × F4 [x]/(x 2 ), Z2 × Z4 [x]/(x 2 + x + 1), Z2 × Z2 [x, y]/(x 2 , x y, y 2 ), Z2 × Z4 [x]/(2x, x 2 ), Z3 × Z2 [x]/(x 3 ), Z3 × Z4 [x]/(x 2 − 2, x 3 ), Z3 × Z8 , Z4 × F4 , F4 × Z2 [x]/(x 2 ), Z4 × Z4 , Z4 × Z2 [x]/(x 2 ), Z2 [x]/(x 2 ) × Z2 [x]/(x 2 ), Z4 × Z5 , Z5 × Z2 [x]/(x 2 ), Z4 × Z7 , Z7 × Z2 [x]/(x 2 ), Z2 × Z3 × Z3 , Z3 × Z3 × Z3 , Z2 × Z3 × F4 , Z2 × Z2 × Z5 , Z2 × Z2 × Z7 , Z2 × Z2 × F4 , Z2 × Z2 × Z4 , Z2 × Z2 × Z2 [x]/(x 2 ), Z2 × Z2 × Z2 × Z2 ,

where Fn is the field with n elements. Now, we discuss the genus two (double toroidal) zero-divisor graphs. This work was first initiated by Bloomfield and Wickham in [25]. Actually, they enumerated all finite commutative local rings whose zero-divisor graphs have genus two. Theorem 3.14 ([25, Theorem 1]) Let R be a finite commutative local ring. Then γ((R)) = 2 if and only if R is isomorphic to one of the following 14 rings: Z81 , F3 [x]/(x 4 ), Z9 [x]/(3x, x 3 − 3), Z9 [x]/(x 2 − 3), Z9 [x]/(x 2 − 6), Z9 [x, y]/(x, y)2 , Z9 [x]/(3x, x 2 ), F9 [x]/(x 2 ), Z9 [x]/(x 2 + 1), F2 [x, y]/(x 4 , x y, y 2 − x 3 ), Z4 [x, y]/(x 2 − 2, x y, y 2 − 2x, 2y), Z4 [x, y]/(x 3 − 2, x y, 2x, y 2 − 2), Z8 [x]/(2x, x 3 − 4), Z16 [x]/(2x, x 2 − 8).

Then in [34], Huadong Su and Pailing Li determined when γ((Zn )) = 2 or 3. Theorem 3.15 ([34, Theorem 9]) The following are true. (i) γ((Zn )) = 2 if and only if n ∈ {30, 35, 36, 44, 50, 81}. (ii) γ((Zn )) = 3 if and only if n ∈ {42, 45, 52, 54, 64}. After 10 years of commutative local rings with genus two zero-divisor graph characterization given by Bloomfield and Wickham [25], recently, Asir and Mano [11] determined precisely all non-local commutative rings whose zero-divisor graphs have genus two, see [11, Theorem 1]. But there are some flaws in the statement and proof of [11, Theorem 1], which has been corrected in [16, Theorem 1]. The revised statement is given below. Theorem 3.16 ([11, Theorem 1], [16, Theorem 1]) Let R be a non-local commutative ring. Then the genus of the zero-divisor graph is 2 if and only if R is isomorphic to one of the following 31 rings:

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Z4 × Z9 , Z2 × Z25 , Z2 × Z5 [x]/(x 2 ), Z2 × Z16 , Z2 × Z2 [x]/(x 4 ), Z2 × Z4 [x]/(x 2 − 2, x 4 ), Z2 × Z4 [x]/(x 3 − 2, x 4 ), Z2 × Z4 [x]/(x 3 + x 2 − 2, x 4 ), F4 × Z8 , F4 × Z2 [x]/(x 3 ), F4 × Z9 , F4 × Z3 [x]/(x 2 ), Z2 × Z2 × F8 , Z2 × Z2 × F9 , Z2 × Z2 × Z11 , Z2 × Z3 × Z4 , Z2 × Z3 × Z2 [x]/(x 2 ), Z2 × Z3 × Z5 , Z3 × Z3 × F4 , Z2 × Z2 × Z2 × Z3 .

Next we move on to the nonorientable genus of zero-divisor graphs. The embedding of the zero-divisor graph in nonorientable compact surfaces was first studied by Hung-Jen Chiang-Hsieh [27]. First of all, he determined all finite commutative local rings whose zero-divisor graphs are of crosscap one. The corresponding result is stated below. Theorem 3.17 ([27, Theorem 2.8]) Let (R, m) be a finite commutative local ring which is not a field. Then γ((R)) ˜ = 1 if and only if R is isomorphic to one of the following 13 rings. Z49 , Z7 [x]/(x 2 ), Z32 , Z2 [x]/(x 5 ), Z4 [x]/(x 3 − 2, x 5 ), Z4 [x]/(x 4 − 2, x 5 ), Z8 [x]/(x 2 − 2, x 5 ), Z8 [x]/(x 2 − 2x + 2, x 5 ), Z8 [x]/(x 2 + 2x − 2, x 5 ), Z2 [x, y]/(x 3 , x y, y 2 ), Z4 [x]/(x 3 , 2x), Z4 [x, y]/(x 3 , x 2 − 2, x y, y 2 ), Z8 [x]/(x 2 , 2x).

The next three results, classified all finite nonlocal commutative rings R with γ((R)) ˜ = 1. Theorem 3.18 ([27, Theorem 3.8]) Let R be a finite commutative ring with |Spec(R)| = 2. Then γ((R)) ˜ = 1 if and only if R is isomorphic to one of the following 16 rings. F4 × F4 , F4 × Z5 , Z2 × F4 [x]/(x 2 ), Z2 × Z4 [x]/(x 2 + x + 1), Z2 × Z2 [x, y]/(x 2 , x y, y 2 ), Z2 × Z4 [x]/(x 2 , 2x), Z3 × Z8 , Z3 × Z2 [x]/(x 3 ), Z3 × Z4 [x]/(x 3 , x 2 − 2), Z4 × Z5 , Z5 × Z2 [x]/(x 2 ), F4 × Z4 , F4 × Z2 [x]/(x 2 ), Z4 × Z4 , Z4 × Z2 [x]/(x 2 ), Z2 [x]/(x 2 ) × Z2 [x]/(x 2 ).

Theorem 3.19 ([27, Theorem 3.10]) Let R be a finite commutative ring with |Spec(R)| = 3. Then γ((R)) ˜ = 1 if and only if R is isomorphic to one of the following 6 rings: Z2 × Z2 × F4 , Z2 × Z2 × Z4 , Z2 × Z2 × Z2 [x]/(x 2 ), Z2 × Z2 × Z5 , Z2 × Z3 × Z3 , Z3 × Z3 × Z3 . Theorem 3.20 ([27, Theorem 3.11]) Let R be a finite commutative ring such that |Spec(R)| = 4. Then γ((R)) ˜ = 1 if and only if R is isomorphic to Z2 × Z2 × Z2 × Z2 .

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We close this section by discussing an interesting open problem related to the orientable and nonorientable genus of zero-divisor graphs. Note that γ(G) ˜ ≤ 2γ(G) + 1 for any graph G, see [33, Theorem 3.4.5]. At the same time, there are some graphs having crosscap one but arbitrarily large genus, see [20]. But in case of zero-divisor graphs, when comparing the list of finite commutative rings with toroidal zerodivisor graphs and the ones with projective zero-divisor graphs, one can find that if a zero-divisor graph is projective, then it must be toroidal. That is γ((R)) ˜ ≥ γ((R)) when γ((R)) = 1. However, since a zero-divisor graph usually contains some complete subgraphs or complete bi-partite subgraphs, Hsieh [27] thought that the above inequality holds not merely by accident. Hence, he proposed the following conjecture. Conjecture: γ((R)) ˜ ≥ γ((R)) for every finite commutative ring R. Towards this direction, in 2019, Asir and Mano [12] enumerated all isomorphism classes of commutative rings with identity whose zero-divisor graph has crosscap two. Theorem 3.21 ([12, Theorem 3.11]) Let R be a finite commutative ring. Then γ((R)) ˜ = 2 if and only if R is isomorphic to one of the following 4 rings F4 × Z7 , Z5 × Z5 , Z2 × Z2 × Z7 , Z2 × Z3 × F4 . By comparing Theorem 3.21 with Theorem 3.13, the orientable genus of all the four rings in Theorem 3.21 is one. So the authors of [12] arrived at the following result, which answers a case of a conjecture proposed by Chiang-Hsieh in [27]. Theorem 3.22 ([12, Theorem 3.12]) Let R be a commutative ring with crosscap number of the zero-divisor graph (R) less than or equal to 2. Then the inequality γ((R)) ˜ ≥ γ((R)) holds. The conjecture is still open all the commutative ring with γ((R)) ˜ ≥ 3.

4 Jacobson Graphs In this section, we deal with what is known as the Jacobson graph of a ring. Definition 4.1 The Jacobson graph of R, denoted by J R , is defined as the graph with vertex set R \ J (R) such that two distinct vertices x and y are adjacent if and only if 1 − x y ∈ / U (R) where J (R) is the Jacobson radical of R and U (R) is the set of all unit elements of R. The concept of Jacobson graph is due to Azimi et al. [21]. The graphs in Fig. 3 are the Jacobson graphs of the rings indicated. In this section, we list the classification of the orientable and nonorientable genus of the Jacobson graph. First, let us discuss some bounds for the genus of Jacobson graphs. These bounds are due to Asir et al. [17]. The following result provides a lower bound for the genus of J R through the cardinality of a maximal ideal of R.

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Fig. 3 The Jacobson graphs of some specific rings

Theorem 4.2 ([17, Theorem 2.4]) Let R be a ring with maximal ideal m such that |m| = λ ≥ 3 and |R/m| = μ (λ and μ may be infinite cardinals also). Then    ⎧  μ−3 (λ−3)(λ−4) (λ−2)2 ⎪ 2 + ( if μ is odd, ) ⎪ 12 2 4 ⎨ γ(J R ) ≥

    ⎪ ⎪ ⎩ (λ−3)(λ−4) + ( μ−2 ) (λ−2)2 12

2

4

if μ is even.

Moreover, if R is local, then the equality holds true. The next result exhibits a lower bound for the genus of J R , where R is isomorphic to the direct product of local rings, in terms of the cardinalities of the local rings. Theorem 4.3 ([17, Theorem 2.5]) Let R1 , . . . , Rn be rings and R ∼ = R1 × · · · × Rn (n ≥ 2). Then the following statements hold true. (i) Suppose that for every 1 ≤ i ≤ n, Ri is finite. Let min{|R1 |, . . . , |Rn |} = |R j | and set r = |R1 | · · ·  |R j |· · · |Rn |, where the notation  |R j | means that |R j | is 

−4) if r ≥ 3 and γ(J R ) ≥ 0 if r = 2. removed. Then γ(J R ) ≥ (r −3)(r 12 (ii) Suppose that at least for one 1 ≤ i ≤ n, Ri is infinite. Then γ(J R ) = ∞.

By part(ii) of Theorem 4.3, we have the following result. Corollary 4.4 ([10, Corollary 2.7]) Let R be an Artinian ring. Then γ(J R ) = ∞ if and only if R is infinite. One more lower bound for genus of Jacobson graphs is provided in the following theorem. Theorem 4.5 ([17, Theorem 2.8]) Let R be a finite ring. Let R ∼ = R1 × · · · × Rn , where every Ri is local. Set ki = γ(J Ri ), i = |Ri \ J (Ri )|, m 1 = max{1 k2 + k1 , 2 k1 + k2 } and m j = max{1 · · ·  j k j+1 + m j−1 ,  j+1 m j−1 + k j+1 } for j = 2, . . . , n − 1. Then γ(J R ) ≥ m n−1 . The following theorem provides an upper bound for γ(J R ).

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Theorem 4.6 ([17, Theorem 2.10]) Let R be a finite ring such that R \ U (R) =  m R i=1 mi , where mi ’s are maximal ideals of R. Let |mi | = λi ≥ 3 and | mi | = μi for i = 1, . . . , m. Rearrange mi ’s such that each μi is odd for 1 ≤ i ≤ j and each μi is even for j + 1 ≤ i ≤ m. Then γ(J R ) ≤ (m − 1)(|R \ J (R)| − 1)

     j (λi − 3)(λi − 4) μi − 3 (λi − 2)2 2 + + 12 2 4 i=1,|λi |>2    

 m μi − 1 (λi − 2)2 (λi − 3)(λi − 4) + . + 12 2 4 i= j+1,|λi |>2

The following theorem provides an effective tool to compare the genus of Jacobson graphs of same order rings. Theorem 4.7 ([17, Theorem 3.3]) Let R ∼ = R1 × · · · × Rn and S ∼ = S1 × · · · × Sn be two finite rings, where Ri ’s and Si ’s are local. Then the following statements hold true. (i) If for every 1 ≤ i ≤ n, |Ri | = |Si | and J Ri ∼ = J Si , then J R ∼ = JS . (ii) If for every 1 ≤ i ≤ n, |Ri | = |Si | and |K Ri | = |K Si |, then J R ∼ = JS . The next result is due to Azimi [21] which characterized all finite commutative rings with planar Jacobson graphs. Theorem 4.8 ([21, Theorem 4.3]) Let R be a finite commutative ring. Then γ(J R ) = 0 if and only if either R is a field or R is isomorphic to one of the following rings: Z4 , Z2 × Z2 , Z2 [x]/(x 2 ), Z6 , Z8 , Z2 × Z4 , Z2 × Z2 [x]/(x 2 ), Z2 × Z2 × Z2 , Z2 [x]/(x 3 ), Z4 [x]/(2x, x 2 ), Z2 × Z2 [x]/(x 2 + x + 1), Z4 [x]/(x 2 − 2, 2x), Z2 [x, y]/(x, y)2 , Z9 , Z3 × Z3 , Z3 [x]/(x 2 ). In 2018, Amraei et al. [4] characterized all finite commutative rings whose J R is toroidal. The same result is also proved in [17]. Theorem 4.9 ([4, Theorem 1.1], [17, Theorem 2.11]) Let R be a finite commutative ring. Then γ(J R ) = 1 if and only if R is isomorphic to one of the following rings. Z2 × Z5 , Z2 × Z7 , Z3 × Z4 , Z3 × F4 , Z3 × Z2 [x]/(x 2 ), Z4 [x]/(x 2 + x + 1), F4 [x]/(x 2 ).

Subsequently, Asir [10] determined all isomorphism classes of commutative rings whose Jacobson graph has genus two. Theorem 4.10 ([10, Theorem 3.4]) Let R be a commutative Artinian ring. Then γ(J R ) = 2 if and only if R is isomorphic to one of the following rings.

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Z2 [x, y, z]/(x, y, z)2 , Z4 [x]/(x 2 ), Z4 [x]/(x 3 , 2x), Z4 [x]/(x 3 , x 2 − 2x), Z4 [x]/(x 2 − 2, x 4 ), Z4 [x]/(x 3 − 2, x 4 ), Z4 [x]/(x 3 + x 2 − 2, x 4 ), Z4 [x, y]/(x 2 , x y − 2, y 2 ), 2 Z4 [x, y]/(x , x y, y 2 , 2x, 2y), Z4 [x, y]/(x 3 , x 2 − 2, x y, y 2 ), Z4 [x, y]/(x 3 , x 2 − 2, x y, y 2 − 2, y 3 ), Z8 [x]/(x 2 , 2x), Z8 [x]/(x 2 − 4, 2x), Z3 × Z5 , Z2 × F8 , Z2 × Z2 × Z3 .

Next, we discuss the nonorientable genus of the Jacobson graphs arising from finite commutative rings R with nonzero identity. The following table illustrate the nonorientable genus of the Jacobson graphs of special rings. These values are due to Amraei et al. [5]. R Z4 , Z8 , Z9 , Z2 [x]/(x 2 ), Z2 [x]/(x 3 ), Z3 [x]/(x 2 ), Z4 [x]/(2x, x 2 ), Z4 [x]/(2x, x 2 − 2), Z2 [x, y]/(x, y)2 , F Z2 × Z2 , Z2 × Z3 , Z2 × Z4 , Z2 × Z2 [x]/(x 2 ), Z2 × F4 , Z3 × Z3 Z2 × Z2 × Z2 Z2 × Z5 , Z3 × Z4 , Z3 × F4 , Z3 × Z2 [x]/(x 2 ) Z4 [x]/(x 2 + x + 1), F4 [x]/(x 2 ) Z25 , Z5 [x]/(x 2 ) Z3 × Z5 F4 × F4 Z2 × Z2 × Z3 F4 × Z5 , Z5 × Z5

γ(J ˜ R) 0

A type of R A local ring

0

A product of two local rings

0 1 2 7 ≥2 ≥3 ≥3 ≥5

A product of three local rings A product of two local rings A local ring A local ring Product of two local rings Product of two local rings A product of three local rings A product of two local rings

Moreover, Amraei et al. [5] classified all the commutative rings with projective Jacobson graphs. The relevant result is stated below. Theorem 4.11 ([5, Main Theorem]) Let R be a finite commutative ring with nonzero identity. Then the Jacobson graph J R is a projective graph if and only if R is isomorphic to one of Z2 × Z5 , Z3 × Z4 , Z3 × Z2 [x]/(x 2 + x + 1) or Z3 × Z2 [x]/(x 2 ).

5 Annihilator Graphs In this section, we discuss the orientable and nonorientable genus of annihilator graph. The annihilator graph is a variant to the zero-divisor graph, which was introduced by Badawi [22] in 2014. Definition 5.1 ([22]) The annihilator graph of a commutative ring R, denoted by AG(R), is the simple graph AG(R) with vertices Z (R)∗ , and two distinct vertices x and y are adjacent if and only if Ann R (x y) = Ann R (x) ∪ Ann R (y) (equivalently Ann R (x) ∪ Ann R (y)  Ann R (x y)).

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Fig. 4 The annihilator graph of Z12

Clearly the zero-divisor graph (R) of a commutative ring R is a subgraph of AG(R). Example 5.2 (a) If R ∼ = F1 × F2 where F1 and F2 are fields, then AG(R) ∼ = K |F1∗ |,|F2∗ | is a complete bipartite graph. (b) If R ∼ = K 3 is a complete graph with = Z8 = {0, 1, . . . , 7}, then AG(R) ∼ V (AG(R)) = {2, 4, 6}. (c) Consider the ring Z12 = {0, . . . , 11}. Then the corresponding annihilator graph AG(Z12 ) is given in Fig. 4. The following theorem answers the question of when the annihilator graph AG(R) is identical to the zero-divisor graph (R). Theorem 5.3 ([22, Theorem 3.6]) Let R be a reduced commutative ring that is not an integral domain. Then AG(R) = (R) if and only if R has exactly two distinct minimal prime ideals. Tamizh Chelvam et al. in [46] and Nikmehr et al. in [42] have independently characterized all commutative rings whose annihilator graph has genus either zero or one. The corresponding results are stated below. Let us start with the characterization of all commutative rings with planar annihilator graphs. Theorem 5.4 ([42, Theorem 5]) Let R be a ring such that R ∼ = R1 × . . . × Rn , where n is a positive integer and Ri is a ring, for every 1 ≤ i ≤ n. Then the following statements holds: (i) If n ≥ 4, then AG(R) is not planar. (ii) If n = 3 and AG(R) is planar, then R ∼ = Z2 × Z2 × Z2 . The next theorem deals with reduced rings whose annihilator graphs are planar. Theorem 5.5 ([42, Theorem 6]) Let R be a reduced ring. Then AG(R) is planar if and only if one of the following statements hold. (i) R ∼ = Z2 × Z2 × Z2 . (ii) |Min(R)| = 2 and one of the minimal prime ideals of R has at most three distinct elements. The next theorem deals with non-reduced rings whose annihilator graphs are planar.

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Theorem 5.6 ([42, Theorem 9]) Let R be a non-reduced ring. Then AG(R) is planar if and only if one of the following statements holds. (i) R is ring-isomorphic to either Z2 × Z4 or Z2 × Z2 [x]/(x 2 ). (ii) Ann(Z (R)) is a prime ideal of R and 2 ≤ |N il(R)| ≤ 3. (iii) Z (R) = N il(R) and 4 ≤ |N il(R)| ≤ 5. Now, we state the result which completely characterized all finite commutative rings with planar annihilator graphs. Theorem 5.7 ([42, Corollary 10], [46, Theorems 14,15]) Let R be a finite ring. If AG(R) is planar, then R is isomorphic to one of the following rings. Z4 , Z2 [x]/(x 2 ), Z9 , Z3 [x]/(x 2 ), Z8 , Z2 [x]/(x 3 ), Z4 [x]/(x 2 − 2, 2x), Z2 [x, y]/(x 2 , x y, y 2 ), Z4 [x]/(2x, x 2 ), F4 [x]/(x 2 ), Z4 [x]/(x 2 + x + 1), Z25 , Z5 [x]/(x 2 ), Z2 × F pn , Z3 × F pn , Z2 × Z4 , Z2 × Z2 [x]/(x 2 ), Z2 × Z2 × Z2 .

Next we move on to the toroidal annihilator graphs. Theorem 5.8 ([42, Theorem 12]) Let R be a reduced ring. If AG(R) is toroidal, then R∼ = R1 × · · · × Rn , where 2 ≤ n ≤ 3. Moreover, one of the following statements holds. (i) If n = 3, then R ∼ = Z2 × Z2 × Z3 . (ii) If n = 2, then R is isomorphic to one of the following rings. Z5 × Z5 , F4 × Z7 , F4 × F4 , F4 × Z5 . The next two results provide a complete characterization of all finite commutative rings whose annihilator graphs are toroidal. Theorem 5.9 ([46, Theorems 16],[42, Lemma 14]) Let (R, m) be a finite commutative local ring with identity that is not a field. Then γ(AG(R)) = 1 if and only if R is isomorphic to one of the following 22 rings: Z49 , Z7 [x]/(x 2 ), Z16 , Z2 [x]/(x 4 ), Z4 [x]/(x 2 − 2, x 4 ), Z2 [x]/(x 3 − 2, x 4 ), Z4 [x]/(x 3 + x 2 − 2, x 4 ), Z2 [x, y]/(x 3 , x y, y 2 − x 2 ), Z4 [x]/(x 3 , x 2 − 2x), Z8 [x]/(x 2 − 4, 2x), Z4 [x, y]/(x 3 , x 2 − 2, x y, y 2 − 2, y 3 ), Z4 [x, y]/(x 2 , y 2 , x y − 2), Z4 [x]/(x 2 ), Z2 [x, y]/(x 2 , y 2 ), Z2 [x, y]/(x 2 , y 2 , x y), Z4 [x]/(x 3 , 2x), Z4 [x, y]/(x 3 , x 2 − 2, x y, y 2 ), Z8 [x]/(x 2 , 2x), F8 [x]/(x 2 ), Z4 [x]/(x 3 + x + 1), Z4 [x, y]/(2x, 2y, x 2 , x y, y 2 ), Z2 [x, y, z]/(x, y, z)2 .

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Theorem 5.10 ([46, Theorems 17],[42, Theorem 15]) Let R = R1 × . . . × Rn be a finite commutative nonlocal ring, where each Ri is a local ring and n ≥ 2. Then γ(AG(R)) = 1 if and only if R is isomorphic to one of the following 7 rings. Z5 × Z5 , F4 × Z7 , F4 × F4 , F4 × Z5 , Z4 × Z3 , Z2 [x]/(x 2 ) × Z3 , Z2 × Z2 × Z3 .

Recently, Asir et al. [14] characterized all commutative rings with double toroidal annihilator graphs. The relevant result is stated below. Theorem 5.11 ([14, Theorem 3.1]) Let R be a commutative ring with identity. Then the genus of annihilator graph is two if and only if the following conditions hold. (i) If R is a local ring, then R is isomorphic to one of the following 8 rings Z27 , Z9 [x]/(3x, x 2 − 3), Z9 [x]/(3x, x 2 − 6), Z9 [x]/(3, x)2 , Z9 [x]/(x 2 + 1), F9 [x]/(x 2 ), Z3 [x]/(x 3 ), Z3 [x, y]/(x, y)2 . (ii) If R is not a local ring, then R is isomorphic to one of the following 10 rings Z2 × Z9 , Z2 × Z3 [x]/(x 2 ), Z4 × F4 , Z2 [x]/(x 2 ) × F4 , Z4 × Z5 , Z2 [x]/(x 2 ) × Z5 , F4 × F8 , F4 × F9 , F4 × Z11 , Z5 × Z7 . We close this section, by mentioning a result which shows that the genus of the annihilator graph associated with an infinite ring is either zero or infinite. Theorem 5.12 ([42, Theorem 11]) Let R be an infinite ring. Then either γ(AG(R)) = 0 or γ(AG(R)) = ∞.

6 Unit Graphs The unit graph was first investigated by Grimaldi for Zn in [32]. In 2010, Ashrafi et al. [9] generalized the unit graph G(Zn ) to G(R) for an arbitrary ring R. A survey of the study of unit graphs can be found in [40]. Definition 6.1 Let R be a commutative ring with nonzero identity and U (R) be the set of all units in R. The unit graph of R, denoted by G(R), has vertex set as the set of all elements of R, for distinct vertices x and y are adjacent if and only if x + y ∈ U (R).

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Fig. 5 The unit graph of Z5

Example 6.2 Consider the ring Z5 . The corresponding unit graph is given in Fig. 5. In this section, we discuss about the embedding of unit graphs. The following theorem completely characterizes all commutative rings with planar unit graphs. Theorem 6.3 ([9, Theorem 5.14]) Let R be a finite commutative ring. Then the unit graph G(R) is planar if and only if R is isomorphic to one of the following rings. (i) Z5 , (ii) Z3 × Z3 , (iii) Z2 × . . . × Z2 ×S, where  ≥ 0 and S ∼ = Z2 , S ∼ = Z3 , S ∼ = Z4 , S ∼ = F4 or    −

times   ab S∼ | a, b ∈ Z2 . = 0a The next result gives all finite commutative rings with identity whose unit graph is toroidal. Theorem 6.4 ([29, Theorem 4.13], [19, Corollary 4.3]) Let R be a finite commutative ring with nonzero identity. Then the unit graph G(R) is a toroidal graph if and only if R is isomorphic to one of the following rings. Z7 , Z8 , Z2 [x]/(x 3 ), Z2 [x]/(2x, x 2 − 2), Z2 [x, y]/(x, y)2 , Z4 [x]/(2, x)2 , Z2 × Z5 , Z3 × Z4 , Z3 × Z2 [x]/(x 2 ) or Z2 × Z3 × Z3 .

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The following table, display some unit graphs with their genus number. S

(S)

(Z2 × S)

Z5 , Z3 × Z3 Z3 × Z4 , Z3 × Z2 [x]/(x 2 ), Z8 , Z2 [x]/(x 3 ), Z2 [x]/(2x, x 2 − 2), Z2 [x, y]/(x, y)2 , Z4 [x]/(2, x)2 F8 Z7 Z3 × F4 Z9 , Z3 [x]/(x 2 ) Z4 × Z4 , Z4 × Z2 [x]/(x 2 ), Z2 [x]/(x 2 ) × Z2 [x]/(x 2 ) Z4 × F4 , Z2 [x]/(x 2 ) × F4

0

1

1 1 2 ≥2 ≥2 2 ≥2

2 ≥5 ≥7 ≥7 ≥6 4 ≥9

Huadong Su et al. [35] determined all isomorphism classes of finite commutative rings whose unit graphs have genus at most three. In order to do this, they considered several special rings one by one in a series of lemma. Lemma 6.5 ([35, Lemma 2.5]) The following statements hold. (i) Let R = Z2 × Fq . Then γ(G(R)) ≥ 5 for q ≥ 7, γ(G(R)) = 1 for q = 5, and γ(G(R)) = 0 for q ≤ 4. (ii) Let R = Z2 × S, where S is a local ring of order eight which is not a field. Then γ(G(R)) = 2. (iii) Let R = Z2 × S, where S is a local ring of order nine which is not a field. Then γ(G(R)) ≥ 6. Lemma 6.6 ([35, Lemma 2.6]) The following statements hold. (i) Let R = Z3 × S, where S is a local ring of order four which is not a field. Then γ(G(R)) = 1. (ii) Let R = Z3 × F4 . Then γ(G(R)) = 3. (iii) Let R = Z3 × Z5 . Then γ(G(R)) ≥ 6. Lemma 6.7 ([35, Lemma 2.7]) The following statements hold. (i) Let R = S × T , where S and T are local rings of order four which are not fields. Then γ(G(R)) = 2. (ii) Let R = F4 × S, where S is a local ring of order four which is not a field. Then γ(G(R)) ≥ 5. Lemma 6.8 ([35, Lemma 2.8]) The following statements hold: (i) Let R be a local ring of order nine which is not a field. Then γ(G(R)) = 2. (ii) Let R = Z2 × Z3 × Z3 . Then γ(G(R)) = 1. Lemma 6.9 ([35, Lemma 2.9]) Let R = Z2 × Z2 × S, where S is a ring. Then G(R) is two copies of G(Z2 × S). The following theorem characterizes all commutative rings whose unit graph has genus two.

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Theorem 6.10 ([35, Theorem 2.11],[18, Corollary 3.6]) Let R be a commutative Artinian ring. Then the unit graph G(R) has genus two if and only if R is isomorphic to one of the following rings: F8 , Z9 , Z3 [x]/(x 2 ), Z2 × Z10 , Z2 × Z8 , Z2 × Z2 [x]/(x 3 ), Z2 × Z2 [x, y]/(x 2 , x y, y 2 ), Z2 × Z4 [x]/(x 2 , 2x), Z2 × Z4 [x]/(x 2 − 2, 2x), Z2 × Z3 × Z3 , Z2 × Z12 , Z2 × Z3 × Z2 [x]/(x 2 ), Z3 × F4 , Z4 × Z4 , Z2 [x]/(x 2 ) × Z2 [x]/(x 2 ), Z4 × Z2 [x]/(x 2 ).

Finally, we state below the classification of genus three unit graphs. Theorem 6.11 ([35, Theorem 2.11]) Let R be a finite ring. Then γ(G(R)) = 3 if and only if R ∼ = F9 or R ∼ = Z3 × F4 . Recently, Khorsandi et al. [39] studied the nonorientable genus of unit graph of a commutative ring and the main result is displayed as follows. Theorem 6.12 ([39, Corollary 4.12]) There is no finite ring R such that the unit graph G(R) is projective.

7 Unitary Cayley Graphs and Cayley Sum Graphs The definition of Cayley graph was introduced by Arthur Cayley in 1878 to explain the concept of abstract groups which are described by a set of generators. In the last 50 years, the theory of Cayley graphs has grown into a substantial branch of algebraic graph theory. The unitary Cayley graph of a ring was initially investigated for Zn by Dejter and Giudici in [30] where some properties of G Zn are presented. In 2009, Akhtar et al. [3] generalized the unitary Cayley graph G Zn to G R for a finite ring R and obtained various characterization results regarding connectedness, chromatic index, diameter, girth and planarity of G R . Definition 7.1 [3] Let R be a commutative ring and U (R) be the group of unit of R. The unitary Cayley graph of R, denoted by G R , is a simple graph whose vertex set is R and two distinct vertices x and y are adjacent if x − y ∈ U (R). Note that G R is the special case of Cayley graph, in fact G R = Cay(R, U (R)). Example 7.2 (a) If n is a prime, then G Zn is the complete graph on n vertices. (b) If n is a power of 2, then G Zn is the complete bipartite K n2 , n2 . (c) Consider the ring R = Z2 × Z4 = {(0, 0), (0, 1), (0, 2), (0, 3), (1, 0), (1, 1), (1, 2), (1, 3)}. Here U (R) = {(1, 1), (1, 3)}. The corresponding unitary Cayley graph G (Z2 ×Z4 ) is given in Fig. 6. First, let us consider the unitary Cayley graph of Zn . Note that U (Zn ) = {x ∈ Zn : gcd(x, n) = 1}. Thus G Zn has the vertex set V (G Zn ) = Zn and edge set E(G Zn ) = {(x, y) : x, y ∈ Zn and gcd(a − b, n) = 1}.

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Fig. 6 The unitary Cayley graph of Z2 × Z4

First, we discuss the orientable and nonorientable genus of unitary Cayley graph. Akhtar et al. [3] determined all finite commutative rings whose unitary Cayley graph Cay(R, U (R)) is planar. Theorem 7.3 ([3, Theorem 8.2]) Let R be a finite ring. Then Cay(R, U (R)) is planar if and only if R is one of the following rings. (Z2 )s , Z3 × (Z2 )s , Z4 × (Z2 )s , F4 × (Z2 )s , where F4 is the field with four elements and s > 0 is any integer value. Asir et al. in [19] characterized all commutative rings whose unitary Cayley graph is toroidal. Theorem 7.4 ([19, Corollary 4.5]) Let R be a commutative Artinian ring. Then the unitary Cayley graph Cay(R, U (R)) is toroidal if and only if R is isomorphic to one of the rings: Z5 , Z7 , Z8 , Z2 [x]/(x 3 ), Z2 [x, y]/(x 2 , x y, y 2 ), Z4 [x]/(x 2 , 2x), Z4 [x]/(x 2 − 2, 2x), Z9 , Z3 [x]/(x 2 ), Z2 × F4 , Z3 × Z3 , Z10 , Z12 , Z3 × Z2 [x]/(x 2 ).

Furthermore, the next result characterized all commutative rings whose unitary Cayley graph is double toroidal. Theorem 7.5 ([18, Corollary 3.7]) Let R be a commutative Artinian ring. Then the unitary Cayley graph Cay(R, U (R)) has genus two if and only if R is isomorphic to one of the following rings. F8 , Z2 × Z2 × F4 , Z2 × Z10 , Z2 × Z8 , Z2 × Z2 [x]/(x 3 ), Z2 × Z2 [x, y]/(x 2 , x y, y 2 ), Z2 × Z4 [x]/(x 2 , 2x), Z2 × Z4 [x]/(x 2 − 2, 2x), Z2 × Z3 × Z3 , Z2 × Z12 , Z2 × Z3 × Z2 [x]/(x 2 ), Z3 × F4 , Z4 × Z4 , Z2 [x]/(x 2 ) × Z2 [x]/(x 2 ), Z4 × Z2 [x]/(x 2 ).

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Recently, the authors of [39] proved the following result regarding the projective unitary Cayley graphs. Theorem 7.6 ([39, Corollary 4.13]) Let R be a finite ring. Then the unitary Cayley graph Cay(R, U (R)) is projective if and only if R is isomorphic to Z5 or Z3 × Z3 . Next, we discuss about the embedding of Cayley sum graph of ideals. In 2014, Afkhami et al. [1] introduced concept of Cayley sum graph of ideals of a commutative ring. Let R be a commutative ring, I (R) be the set of all ideals of R and S be a subset of I (R)∗ = I (R) \ {0}. The Cayley sum graph, denoted by Cay(I (R), S), is an undirected graph whose vertex set is the set I (R) and two distinct vertices I and J are adjacent whenever I + K = J or J + K = I , for some ideal K in S. Here, we consider S = I (R)∗ and we denote the Cayley sum graph Cay(I (R), S) by Cay(I (R), I ∗ ). The results related to genus of Cay(I (R), I ∗ ) are listed below. Theorem 7.7 ([1, Theorem 3.2]) If the Cayley sum graph Cay(I (R), I ∗ ) is planar, then R is an Artinian ring. Theorem 7.8 ([1, Theorem 3.3]) If the Cayley sum graph Cay(I (R), I ∗ ) is planar, then |Max(R)| ≤ 3. Theorem 7.9 ([1, Theorem 3.5]) The Cayley sum graph Cay(I (R), I ∗ ) is a planar graph if and only if R ∼ = F1 × F2 , where Fi ’s are fields, or (R, m) is a local ring which satisfies one of the following conditions. (i) dim mR ( mm2 ) = 2 and I (R) = {0, (x), (y), (x, y), R}, where x, y ∈ m (ii) dim mR ( mm2 ) = 1, m 2 = 0 and I (R) = {0, (x 2 ), (x), R}, where x ∈ m; (iii) dim mR ( mm2 ) = 1, m 2 = 0 and I (R) = {0, (x), R}, where x ∈ m. Theorem 7.10 ([1, Proposition 4.4]) If the genus of the Cayley sum graph Cay(I (R), I ∗ ) is finite, then R is an Artinian ring. Theorem 7.11 ([1, Proposition 4.5]) If |Max(R)| = t and γ(Cay(I (R), I ∗ )) = g √ is finite, then 2t < 5(1 + 2 + 2g). The next result characterized all commutative rings whose Cay(I (R), I ∗ ) has genus one. Theorem 7.12 ([1, Theorem 4.6]) Let R be a ring with at least three maximal ideals. Then γ(Cay(I (R), I ∗ )) = 1 if and only if R ∼ = F1 × F2 × F3 where Fi are fields. In 2018, Tamizh Chelvam et al. [47] studied the toroidal nature of particular Cayley sum graphs Cay(I (R), Max(R)). The main characterization result is stated below. Theorem 7.13 ([47, Theorem 5.6]) Let R be a commutative Artinian ring. Then Cay(I (R), Max(R)) is toroidal if and only if R ∼ = R1 × F1 × F2 where R1 has exactly one nonzero proper ideal and F1 and F2 are fields.

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8 Generalized Unit and Unitary Cayley Graphs In 2012, Khashyarmanesh et al. [38] defined the graph (R, G, S) where G is a multiplicative subgroup of U (R) and S is a non-empty subset of G such that S −1 = {s −1 : s ∈ S} ⊆ S, which generalizes both unit and unitary Cayley graphs of R. Definition 8.1 ([38]) Let R be a commutative ring, U (R) be the set of all unit elements of R, G be a multiplicative subgroup of U (R) and S be a non-empty subset of G such that S −1 = {s −1 : s ∈ S} ⊆ S. The graph (R, G, S) is a simple graph with vertex set R and two distinct vertices x and y are adjacent if there exists s ∈ S such that x + sy ∈ G. For simplicity of notation, (R, U (R), S) is denoted by (R, S). Note that if S = {1}, then (R, S) is same as the unit graph G(R) and if S = {−1}, then (R, S) is the unitary Cayley graph Cay(R, U (R)) of R. Example 8.2 Let R = Z3 × Z3 . Then U (R) = {(1, 1), (1, 2), (2, 1)}. The graph (Z3 × Z3 , U (Z3 × Z3 ), {1}) is given in Fig. 7. K. Khashyarmanesh et al. [38] determined all commutative Artinian rings R for which (R, S) is planar. The result is stated below for ready reference. Lemma 8.3 ([38, Theorem 3.7]) Let R be a commutative Artinian ring. Then (R, S) is planar if and only if one of the following conditions hold. (i) R ∼ = Z2 × T where  ≥ 0 and T is isomorphic to one of the following rings: Z2 , Z3 , Z4 or Z2 [x]/(x 2 ), (ii) R ∼ = F4 ,

Fig. 7 (Z3 × Z3 , {(1, 1)}) = G(Z3 × Z3 )

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∼ Z × F4 , where  > 0 with S = {1}, (iii) R = 2 ∼ Z5 with S = {1}, (iv) R = (v) R ∼ = Z3 × Z3 with S = {(1, 1)}, S = {(1, −1)}, or S = {(−1, 1)}. The following result is due to Asir et al. [19], which characterize all commutative Artinian rings R whose (R, S) has genus one. Theorem 8.4 ([19, Theorem 4.2]) Let R be a commutative Artinian ring. Then (R, S) is toroidal if and only if one of the following conditions hold. (i) R ∼ = Z5 with S = {1}, (ii) R ∼ = Z7 , Z2 [x,y] Z4 [x] Z4 [x] (iii) R ∼ = Z8 , Z(x2 [x] 3 ) , (x 2 ,x y,y 2 ) , (x 2 ,2x) , or (x 2 −2,2x) , ∼ (iv) R = Z2 × F4 with S = {(1, 1)}, (v) R ∼ = Z9 with S = {−1}, S = {2, 5}, or S = {−1, 2, 5}, 2 2 2 (vi) R ∼ = Z(x3 [x] 2 ) with S = {−1 + (x )}, S = {−1 + x + (x ), −1 − x + (x )}, or 2 2 2 S = {−1 + (x ), −1 + x + (x ), −1 − x + (x )}, (vii) R ∼ = Z3 × Z3 with S = {(−1, −1)}, S = {(1, 1), (1, −1)}, S = {(1, 1), (−1, 1)}, S = {(−1, −1), (1, −1)}, or S = {(−1, −1), (−1, 1)}, (viii) R ∼ = Z10 with S = {1} or S = {−1}, (ix) R ∼ = Z12 with S = {1}, S = {−1}, S = {5}, S = {7}, S = {1, 7}, or S = {5, −1}, with S = {(1, 1 + (x 2 ))}, S = {(−1, 1 + x + (x 2 ))}, (x) R ∼ = Z3 × Z(x2 [x] 2) S = {(1, 1 + x + (x 2 ))}, S = {(−1, 1 + (x 2 ))}, S = {(1, 1 + (x 2 )), (1, 1 + x + (x 2 ))}, or S = {(−1, 1 + (x 2 )), (−1, 1 + x + (x 2 ))}. As a continuation to the previous result, Asir et al. [18] characterized all commutative Artinian rings R whose (R, S) has genus two. Theorem 8.5 ([18, Theorem 3.5]) Let R be a commutative Artinian ring. Then γ((R, S)) = 2 if and only if one of the following holds. (i) R ∼ = F8 , (ii) R ∼ = Z9 with S = {1}, S = {4, 7} or S = {1, 4, 7}, with S = {1 + (x 2 )}, S = {1 + x + (x 2 ), 1 − x + (x 2 )} or S = (iii) R ∼ = Z(x3 [x] 2) 2 {1 + (x ), 1 + x + (x 2 ), 1 − x + (x 2 )}, (iv) R ∼ = Z3 × Z3 with S = {(1, 1), (−1, −1)}, S = {(1, −1), (−1, 1)} or |S| ≥ 3, (v) R ∼ = Z10 with S = {1} and S = {−1}, (vi) R ∼ = Z3 × F4 with S = {(1, 1)} or S = {(−1, −1)}, (vii) R ∼ = Z2 × Z2 × F4 with S = {(1, 1, 1)}, Z2 [x,y] Z4 [x] Z4 [x] (viii) R ∼ = Z2 × Z8 , Z2 × Z(x2 [x] 3 ) , Z2 × (x 2 ,x y,y 2 ) , Z2 × (x 2 ,2x) or Z2 × (x 2 −2,2x) , Z2 [x] Z2 [x] (ix) R ∼ = Z4 × Z4 , Z(x2 [x] 2 ) × (x 2 ) or Z4 × (x 2 ) , ∼ (x) R = Z2 × Z3 × Z3 with S = (1, 1, 1), S = (1, 1, −1), S = (1, −1, 1) and S = (1, −1, −1), (xi) R ∼ = Z2 × Z10 with S = {(1, 1)} or S = {(1, −1)}, (xii) R ∼ = Z2 × Z12 with S = {(1, 1)}, S = {(1, −1)}, S = {(1, 5)}, S = {(1, 7)}, S = {(1, 1), (1, 7)} or S = {(1, 5), (1, −1)},

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(xiii) R ∼ with S = {(1, 1, 1 + (x 2 ))}, S = {(1, −1, 1 + x + = Z2 × Z3 × Z(x2 [x] 2) (x 2 ))}, S={(1, 1, 1 + x + (x 2 ))}, S = {(1, −1, 1 + (x 2 ))}, S={(1, 1, 1 + (x 2 )), (1, 1, 1 + x + (x 2 ))} or S={(1, −1, 1 + (x 2 )), (1, −1, 1 + x + (x 2 ))}. Recently, Khorsandi et al. [39] determined all Artinian rings R whose (R, S) is projective. Theorem 8.6 ([39, Corollary 4.11]) Let R be a finite ring. Then (R, S) is projective if and only if one of the following conditions holds: (i) R ∼ = Z5 with S = {1}. (ii) R ∼ = Z3 × Z3 with S = {(−1, −1)}, {(−1, 1), (−1, −1)}.

S = {(1, −1), (−1, −1)}

or

S=

Further, Khorsandi et al. [39] proved that for an Artinian ring R, (R, S) has finite nonorientable genus if and only if R is finite. Moreover, they proved that for a given positive integer k, the number of finite rings R whose (R, S) has nonorientable genus k is finite.

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A Computational Approach to Shephard Groups Dong-Il Lee

Abstract For some of the primitive complex reflection groups, that is, several extensions of the binary tetrahedral and octahedral groups, we show the construction of their Gröbner-Shirshov bases and the corresponding sets of standard monomials. In addition, the freeness theorem and the symmetrizing trace conjecture for the Hecke algebras of complex reflection groups are briefly surveyed. Keywords Complex reflection group · Shephard group · Gröbner-Shirshov basis · Hecke algebra 2010 Mathematics Subject Classification Primary 20F55 · Secondary 05E15 · 16Z05 · 68W30

1 Introduction The objects in this paper are the complex reflection groups, which are crucial in invariant theory, representation theory, algebraic topology, mathematical physics and many others. It’s known that the invariant ring of a finite group possesses a set of fundamental invariants such that every invariant form is expressed as a polynomial in fundamental invariants if and only if the group is generated by reflections [16, 33]. The irreducible complex reflection groups are classified into infinite families including the Coxeter groups of types A, B, D, I , and the other 34 exceptional ones which are primitive. Our approach to understanding the structure of complex reflection groups is the noncommutative Gröbner basis theory, which is called the Gröbner-Shirshov basis theory. The theory of Gröbner-Shirshov bases provides a powerful tool for understanding the structure of (non)associative algebras and their representations, especially in computational aspects. With the ever-growing power of computers, it is D.-I. Lee (B) Department of Mathematics, Seoul Women’s University, Seoul 01797, Korea e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. Badawi and J. Coykendall (eds.), Rings, Monoids and Module Theory, Springer Proceedings in Mathematics & Statistics 382, https://doi.org/10.1007/978-981-16-8422-7_16

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now viewed as a universal engine behind algebraic or symbolic computation. The effective notion stems from Shirshov’s Composition Lemma and his algorithm [34] for Lie algebras and independently from Buchberger’s algorithm [13] of computing Gröbner bases for commutative algebras in 1960’s. In [4], Bokut applied Shirshov’s method to associative algebras, and Bergman mentioned the diamond lemma for ring theory [3]. In particular, if we are given a finitely presented group, that is, a group G generated by a set X equipped with the defining relations R, then we consider its group algebra F[G] over a field F, which can be viewed as an associative algebra generated by the X quotient by the two-sided ideal generated by R. One of the applications of Gröbner-Shirshov bases in group theory is the word problem [31]. It is solvable in a finite Coxeter group since any element has a unique normal form with respect to a Gröbner-Shirshov basis. For finite Coxeter groups and the imprimitive complex reflection groups, their Gröbner-Shirshov bases were constructed in [5, 6, 18, 19, 21–24, 35]. In this paper, we focus on the class of Shephard groups, that is, the collection of complex reflection groups with unbranched Coxeter diagrams. They correspond to the symmetry groups of regular complex polytopes [32]. Note that the Shephard groups are well-generated, which means that the Shephard groups of rank n are generated by n reflections. After brief preliminaries on the theory of Gröbner-Shirshov bases and the classification of irreducible finite complex reflection groups, we calculate, in Sects. 3 and 4, a Gröbner-Shirshov basis for each primitive Shephard group of tetrahedral and octahedral type. Starting from a presentation of any reflection group [12, 17], we complete the relations to find its Gröbner-Shirshov basis, thus obtaining the set of standard monomials. In the subsequent sections, we introduce the Hecke algebras of complex reflection groups, followed by a survey on the freeness theorems with the symmetrizing trace conjecture [11].

2 Preliminaries 2.1 Gröbner-Shirshov Bases In this subsection, we recall a basic theory of Gröbner-Shirshov bases for associative algebras to make the paper self-contained. Let X be a set and let X  be the free monoid of associative words on X . We denote the empty word by 1 and the length (or degree) of a word u by l(u). A well-ordering < on X  is called a monomial order if x < y implies axb < ayb for all a, b ∈ X . Fix a monomial order < on X  and let FX  be the free associative algebra generated by X over a field F. Given a nonzero element p ∈ FX , we denote by p the maximal monomial (called  the leading monomial) appearing in p under the ordering 1 **

Order n! d n en−1 n!

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

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

Cyclic group Zd = G(d, 1, 1) W (L2 ) = Z2 .T = 3[3]3 Z6 .T = 3[4]3 Z4 .T = 3[6]2 Z12 .T Z4 .O = 4[3]4 Z8 .O = 4[6]2 Z12 .O = 4[4]3 Z24 .O Z2 .O Z4 .O = 2, 3, 42 Z6 .O = 3[8]2 Z12 .O = 2, 3, 46 Z10 .I = 5[3]5 Z20 .I = 5[6]2 Z30 .I = 5[4]3 Z60 .I Z6 .I = 3[5]3 Z12 .I = 3[10]2 Z4 .I W (H3 ) = [5, 3] W (J3 (4)) W (L3 ) = 3[3]3[3]3 W (M3 ) = 3[3]3[4]2 W (J3 (5)) W (F4 ) = [3, 4, 3] W (N4 ) W (H4 ) = [5, 3, 3] W (EN4 ) = W (O4 ) W (L4 ) = 3[3]3[3]3[3]3 W (K5 ) W (K6 ) W (E6 ) = [32,2,1 ] W (E7 ) = [33,2,1 ] W (E8 ) = [34,2,1 ]

d 24 72 48 144 96 192 288 576 48 96 144 288 600 1200 1800 3600 360 720 240 120 336 648 1296 2160 1152 7680 14400 46080 155520 51840 39191040 51840 2903040 696729600

*The symbol a n denotes that the number of a-reflections is n. **G(2, 2, 2) is reducible.

Reflections* 2n(n−1)/2 2(de)n(n−1)/2 , δ nφ(δ) (δ|d, δ > 1) δ φ(δ) (δ|d, δ > 1) 38 316 26 38 26 316 26 412 218 412 26 316 412 218 316 412 212 218 212 316 218 316 548 230 548 340 548 230 340 548 340 230 340 230 215 221 324 29 324 245 224 240 260 260 380 245 2126 236 263 2120

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Note that G(2, 1, n) = W (Bn ) = [4, 3, . . . , 3] for n > 1, G(2, 2, n) = W (Dn ) for n > 3, G(6, 6, 2) = W (G 2 ) = [6], G(5, 5, 2) = W (H2 ) = [5], and G(e, e, 2) = W (I2 (e)) = [e] for e > 6.

3 The Shephard Groups of Tetrahedral Type From now on, the presentation of each primitive complex reflection group of rank 2 will be kept as usual in [12, 17]. We recall that a Shephard group is the symmetry group of a regular polytope, that is, the reflection group associated to a Coxeter diagram without branch. In this section, we give a Gröbner-Shirshov basis calculation for each of three Shephard groups of tetrahedral type, with writing a proof in detail only for the first subsection.

3.1 The Group G 4 Let G be the complex reflection group of type L 2 , which is G 4 following the Shephard-Todd notations in [33]. Note that it is also called the binary tetrahedral group. Take the complex field C as our base field, then the group algebra C[G] is generated by s, t with the defining relations s 3 = 1 = t 3 , tst = sts. The corresponding Dynkin-type diagram is given by 3

s

3

t

.

Set our monomial order to be the degree-lexicographic order with s < t. Denote by R the set of reflection relations and the braid relation, that is, R = {s 3 − 1, t 3 − 1, tst − sts}. Then C[G] is isomorphic to Cs, t/I , where I is the two-sided ideal generated by R in the free associative algebra Cs, t. From now on, we identify C[G] with Cs, t/I . We say that the algebra C[G] is defined by R. Lemma 3.1 The following relations hold in C[G] : (a) ts 2 ts − sts 2 t = 0, (b) ts 2 t 2 − s 2 t 2 s = 0, (c) t 2 s 2 t − st 2 s 2 = 0. Proof (a) Consider the composition between tst − sts and itself. Then we get a new polynomial:

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(tst − sts)st − ts(tst − sts) = ts 2 ts − sts 2 t. (b) The composition between (a) and tst − sts is (ts 2 ts − sts 2 t)t − ts 2 (tst − sts) = −sts 2 t 2 + t 2 s from s 3 = 1. We multiply this polynomial by −s 2 from the left to obtain our homogeneous polynomial. (c) We calculate the composition between (a) and itself to have (ts 2 ts − sts 2 t)sts − ts 2 (ts 2 ts − sts 2 t) = t 2 s 2 t − sts 2 tsts = t 2 s 2 t − st 2 s 2 , using the relations s 3 = 1 and tst = sts.



We show that if we add the above three polynomials to R then the enlarged set becomes a Gröbner-Shirshov basis for C[G].  the set of defining relations R combined with the Theorem 3.2 We denote by R  forms a Gröbner-Shirshov basis for C[G]. The relations in Lemma 3.1. Then R  corresponding R-standard monomials are of the form si w where i = 0, 1, 2 and the monomial w is one of the following 8 monomials : 1, t, ts, t 2 , ts 2 , t 2 s, ts 2 t, t 2 s 2 .  is closed under composition. Since the composition with a Proof We show that R reflection polynomial is just the product with a power of a generator, we consider only the compositions between the others, except for the cases in Lemma 3.1. • (tst − sts)s 2 ts − ts(ts 2 ts − sts 2 t) = ts 2 ts 2 t − st 2 s = s 2 ts 2 t 2 − st 2 s = st 2 s − st 2 s = 0 from the relations s 3 = 1, and (a), (b) in Lemma 3.1. • (ts 2 t 2 − s 2 t 2 s)st − ts 2 t (tst − sts) = ts 2 tsts − s 2 t 2 s 2 t = t 2 s 2 − t 2 s 2 = 0 from tst = sts, s 3 = 1, and (c) in Lemma 3.1. • (ts 2 ts − sts 2 t)st 2 − ts 2 (ts 2 t 2 − s 2 t 2 s) = tst 2 s − sts 2 tst 2 = s 2 ts 2 − s 2 ts 2 = 0 from s 3 = 1, tst = sts. • (ts 2 t 2 − s 2 t 2 s)s 2 ts − ts 2 t (ts 2 ts − sts 2 t) = ts 2 tsts 2 t − 1 = 1 − 1 = 0 from tst = sts, s 3 = 1, t 3 = 1. • (ts 2 t 2 − s 2 t 2 s)s 2 t 2 − ts 2 t (ts 2 t 2 − s 2 t 2 s) = ts 2 ts 2 t 2 s − s 2 t = s 2 t − s 2 t = 0

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from s 3 = 1, t 3 = 1, (a). • (tst − sts)ts 2 t − ts(t 2 s 2 t − st 2 s 2 ) = ts 2 t 2 s 2 − ststs 2 t = s 2 t 2 − s 2 t 2 = 0 from (b), s 3 = 1, tst = sts. • (t 2 s 2 t − st 2 s 2 )st − t 2 s 2 (tst − sts) = s − s = 0 from s 3 = 1, t 3 = 1. • (t 2 s 2 t − st 2 s 2 )s 2 ts − t 2 s 2 (ts 2 ts − sts 2 t) = s 2 t − st 2 sts = s 2 t − s 2 t = 0 from s 3 = 1, t 3 = 1, tst = sts. • (ts 2 t 2 − s 2 t 2 s)s 2 t − ts 2 (t 2 s 2 t − st 2 s 2 ) = s 2 − s 2 = 0 from s 3 = 1, t 3 = 1. • (ts 2 t 2 − s 2 t 2 s)ts 2 t − ts 2 t (t 2 s 2 t − st 2 s 2 ) = ts 2 tst 2 s 2 − s 2 t 2 sts 2 t = tst − tst = 0 from tst = sts, s 3 = 1. • (t 2 s 2 t − st 2 s 2 )t − t (ts 2 t 2 − s 2 t 2 s) = ts 2 t 2 s − st 2 s 2 t = s 2 t 2 s 2 − s 2 t 2 s 2 = 0 from (b), (c). • (t 2 s 2 t − st 2 s 2 )s 2 t 2 − t 2 s 2 (ts 2 t 2 − s 2 t 2 s) = t 2 st 2 s − st 2 st 2 = sts 2 ts − s 2 ts 2 t = s 2 ts 2 t − s 2 ts 2 t = 0 from s 3 = 1, tst = sts, (a). • (t 2 s 2 t − st 2 s 2 )ts 2 t − t 2 s 2 (t 2 s 2 t − st 2 s 2 ) = ts 2 − st 2 s 2 ts 2 t = ts 2 − s 2 t 2 st = ts 2 − ts 2 = 0 from s 3 = 1, t 3 = 1, tst = sts.  reduce to 0, which means that R  is We checked that all possible compositions in R a Gröbner-Shirshov basis by Theorem 2.6.  Remark. (1) Note that the order of G is known to be 24 (See [32]). Since the number  of R-standard monomials is 3 × 8 = 24,  is Gröbnerwhich equals the dimension of C[G], Theorem 2.6 shows again that R Shirshov basis for C[G]. (2) Once a Gröbner-Shirshov basis for C[G] is obtained, we have the multiplication table between the standard monomials. For example, a monomial ts 2 t is multiplied by t from the right to be (s 2 ts 2 t)t = s 2 (ts 2 t 2 ) = s 2 s 2 t 2 s = st 2 s from the relation (b) in Lemma 3.1 and s 3 = 1.

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(3) We find that the set of the above 8 monomials in (3.1) forms the set of right coset representatives of the subgroup s in the Shephard group s, t of type L 2 . (4) The number of the coset representatives of s in L 2 is equal to the number of edges in the corresponding regular complex polygon 3{3}3, following the notation of Shephard in [32], called the Möbius-Kantor configuration, since the symmetry group is transitive on the set of edges in the regular polygon 3{3}3. (5) We note that the Shephard groups G 25 and G 26 of types L 3 and M3 , respectively, contain the group of type L 2 as a parabolic subgroup. Their sets of standard monomials were calculated in [20].

3.2 The Group G 5 Let G be the complex reflection group G 5 . Take our base field and monomial order as in the previous subsection. The group algebra C[G] is generated by s, t with the defining relations s 3 = 1 = t 3 , tsts = stst. The corresponding Dynkin-type diagram is given by 3

s

3

t

.

Denote by R the set of reflection relations and the braid relation, that is, R = {s 3 − 1, t 3 − 1, tsts − stst}. Then C[G] is isomorphic to Cs, t/I , where I is the two-sided ideal generated by R in Cs, t. Identify C[G] with Cs, t/I , and we say that the algebra C[G] is defined by R. Using the same procedure as the proof in Lemma 3.1, we consider every possible composition to find more relations. Lemma 3.3 The following relations hold in C[G] : (a) (b) (c) (d) (e) (f) (g) (h) (i) (j)

ts 2 tst − stst 2 s = 0, tst 2 s 2 − s 2 t 2 st = 0, t 2 s 2 ts − sts 2 t 2 = 0, ts 2 t 2 s 2 − s 2 t 2 s 2 t = 0, tst 2 st 2 − s 2 ts 2 ts = 0, t 2 s 2 t 2 s − st 2 s 2 t 2 = 0, t 2 st 2 st − sts 2 ts 2 = 0, ts 2 ts 2 ts − sts 2 ts 2 t = 0, ts 2 ts 2 t 2 − s 2 t 2 st 2 s = 0, ts 2 t 2 st 2 s − sts 2 t 2 st 2 = 0.

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 the set of defining relations R combined with the Theorem 3.4 We denote by R  forms a Gröbner-Shirshov basis for C[G]. The relations in Lemma 3.3. Then R  corresponding R-standard monomials are of the form si w where i = 0, 1, 2 and the monomial w is one of the following 24 monomials : 1, t, ts, t 2 , ts 2 , tst, t 2 s, ts 2 t, tst 2 , t 2 s 2 , t 2 st, ts 2 ts, ts 2 t 2 , tst 2 s, t 2 s 2 t, t 2 st 2 , (ts 2 )2 , ts 2 t 2 s, tst 2 st, t 2 s 2 t 2 , (t 2 s)2 , (ts 2 )2 t, ts 2 t 2 st, ts 2 t 2 st 2 .

Proof Note that the order of G is known to be 72 = 6 × 12, the product of the  fundamental degrees from the invariant theory (See [32]). Since the number of Rstandard monomials is 3 × 24 = 72,  is Gröbnerwhich equals the dimension of C[G], Theorem 2.6 shows again that R Shirshov basis for C[G]. 

3.3 The Group G 6 Let G be the complex reflection group G 6 . Then the group algebra C[G] is generated by s, t with the defining relations s 3 = 1 = t 2 , tststs = ststst, where the corresponding Dynkin-type diagram is given by 3

s

t

.

Set our monomial order as before, and denote by R the set of reflection relations and the braid relation, that is, R = {s 3 − 1, t 2 − 1, tststs − ststst}. Then the group algebra C[G] is isomorphic to Cs, t/I , where I is the two-sided ideal generated by R in Cs, t. The algebra C[G] is identified with Cs, t/I , and we say that C[G] is defined by R. We find more relations by performing the Composition algorithm [3, 4]. Lemma 3.5 The following relations hold in C[G] : (a) ts 2 tsts − ststs 2 t = 0, (b) ts 2 ts 2 ts − sts 2 ts 2 t = 0,

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(c) tsts 2 ts 2 − s 2 ts 2 tst = 0.  the set of defining relations R combined with the Theorem 3.6 We denote by R  forms a Gröbner-Shirshov basis for C[G]. The relations in Lemma 3.5. Then R  corresponding R-standard monomials are of the form si w where i = 0, 1, 2 and the monomial w is one of the following 16 monomials : 1, t, ts, ts 2 , tst, ts 2 t, tsts, ts 2 ts, tsts 2 , tstst, (ts 2 )2 , ts 2 tst, tsts 2 t, (ts 2 )2 t, tsts 2 ts, tsts 2 tst. Proof Notice that the order of G is known to be 48 = 4 × 12, the product of the  fundamental degrees from the invariant theory (See [32]). Since the number of Rstandard monomials is 3 × 16 = 48,  is Gröbnerwhich equals the dimension of C[G], Theorem 2.6 shows again that R Shirshov basis for C[G]. 

4 The Shephard Groups of Octahedral Type In this section, a Gröbner-Shirshov basis for each of the Shephard groups of octahedral type is presented.

4.1 The Group G 8 Let G be the complex reflection group G 8 following the Shephard-Todd notations in [33]. Then the group algebra C[G] is generated by s, t with the defining relations s 4 = 1 = t 4 , tst = sts, where the corresponding Dynkin-type diagram is given by 4

s

4

t

.

Set our monomial order to be the degree-lexicographic order with s < t, and denote by R the set of reflection relations and the braid relation, that is, R = {s 4 − 1, t 4 − 1, tst − sts}.

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Then the group algebra C[G] is isomorphic to Cs, t/I , where I is the two-sided ideal generated by R in the free associative algebra Cs, t. From now on, we identify C[G] with Cs, t/I . We say that the algebra C[G] is defined by R. We apply the Composition algorithm to R, leading us to find more relations. Lemma 4.1 The following relations hold in C[G] : (a) (b) (c) (d) (e) (f) (g) (h) (i) (j)

ts 3 ts − sts 2 t 2 = 0, ts 2 t 3 − s 3 t 2 s = 0, t 3 s 2 t − st 2 s 3 = 0, ts 3 t 3 − s 3 t 3 s = 0, ts 2 t 2 s 2 − s 2 t 2 s 2 t = 0, t 2 s 2 t 2 s − st 2 s 2 t 2 = 0, t 3 s 3 t − st 3 s 3 = 0, t 2 s 3 t 2 s − st 2 s 3 t 2 = 0, ts 3 t 2 s 3 − s 3 t 2 s 3 t = 0, ts 3 t 2 s 2 t − sts 3 t 2 s 2 = 0.

 the set of defining relations R combined with the Theorem 4.2 We denote by R  forms a Gröbner-Shirshov basis relations in Lemma 3.1(a) and Lemma 4.1. Then R  for C[G]. The corresponding R-standard monomials are of the form si w where i = 0, 1, 2, 3 and the monomial w is one of the following 24 monomials : 1, t, ts, t 2 , ts 2 , t 2 s, t 3 , ts 3 , ts 2 t, t 2 s 2 , t 3 s, ts 3 t, ts 2 t 2 , t s , t s t, t 3 s 2 , ts 3 t 2 , ts 2 t 2 s, t 2 s 3 t, t 2 s 2 t 2 , t 3 s 3 , ts 3 t 2 s, t 2 s 3 t 2 , ts 3 t 2 s 2 . 2 3

2 2

Proof We note that the order of G is known to be 96 = 8 × 12, the product of the fundamental degrees from the invariant theory (See [32]). Since the number of  R-standard monomials is 4 × 24 = 96,  is Gröbnerwhich is equal to the dimension of C[G], Theorem 2.6 shows again that R Shirshov basis for C[G]. 

4.2 The Group G 9 Let G be the complex reflection group G 9 . Then the group algebra C[G] is generated by s, t with the defining relations s 4 = 1 = t 2 , tststs = ststst,

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where the corresponding Dynkin-type diagram is given by 4

s

t

.

Set our monomial order as before, and denote by R the set of reflection relations and the braid relation, that is, R = {s 4 − 1, t 2 − 1, tststs − ststst}. Then C[G] is isomorphic to Cs, t/I , where I is the two-sided ideal generated by R in Cs, t. Identify C[G] with Cs, t/I , and we say that the algebra C[G] is defined by R. Performing the Composition algorithm, we find more relations. Lemma 4.3 The following relations hold in C[G] : (a) (b) (c) (d) (e) (f) (g) (h) (i) (j)

ts 2 tsts − ststs 2 t = 0, ts 3 tsts − ststs 3 t = 0, ts 3 ts 2 ts − sts 2 ts 3 t = 0, tsts 2 ts 3 − s 3 ts 2 tst = 0, ts 3 ts 3 ts − sts 3 ts 3 t = 0, tsts 3 ts 3 − s 3 ts 3 tst = 0, ts 2 ts 3 ts 3 − s 3 ts 3 ts 2 t = 0, ts 2 ts 2 ts 2 ts − sts 2 ts 2 ts 2 t = 0, tsts 2 ts 2 ts 2 − s 2 ts 2 ts 2 tst = 0, ts 2 ts 2 ts 3 ts 2 − s 3 ts 2 ts 3 tst = 0.

 the set of defining relations R combined with the Theorem 4.4 We denote by R  forms a Gröbner-Shirshov basis for C[G]. The relations in Lemma 4.3. Then R  corresponding R-standard monomials are of the form si w where i = 0, 1, 2, 3 and the monomial w is one of the following 48 monomials : 1, t, ts, ts 2 , tst, ts 3 , ts 2 t, (ts)2 , ts 3 t, ts 2 ts, tsts 2 , (ts)2 t, ts 3 ts, (ts 2 )2 , ts 2 tst, tsts 3 , tsts 2 t, ts 3 ts 2 , ts 3 tst, ts 2 ts 3 , (ts 2 )2 t, 3 tsts t, tsts 2 ts, (ts 3 )2 , ts 3 ts 2 t, ts 2 ts 3 t, (ts 2 )2 ts, tsts 3 ts, ts(ts 2 )2 , tsts 2 tst, (ts 3 )2 t, ts 2 ts 3 ts, (ts 2 )3 , (ts 2 )2 tst, tsts 3 ts 2 , tsts 3 tst, ts(ts 2 )2 t, ts 2 ts 3 ts 2 , ts 2 ts 3 tst, (ts 2 )2 ts 3 , (ts 2 )3 t, tsts 3 ts 2 t, ts(ts 2 )2 ts, ts 2 ts 3 ts 2 t, (ts 2 )2 ts 3 t, ts(ts 2 )2 tst, (ts 2 )2 ts 3 ts, (ts 2 )2 ts 3 tst.

Proof Notice that the order of G is known to be 192 = 8 × 24, the product of the  fundamental degrees from the invariant theory (See [32]). Since the number of Rstandard monomials is 4 × 48 = 192,

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 is Gröbnerwhich is equal to the dimension of C[G], Theorem 2.6 shows again that R Shirshov basis for C[G]. 

4.3 The Group G 10 Let G be the complex reflection group G 10 . Then the group algebra C[G] is generated by s, t with the defining relations s 4 = 1 = t 3 , tsts = stst, where the corresponding Dynkin-type diagram is given by 4

s

3

t

.

Set our monomial order as before, and denote by R the set of reflection relations and the braid relation, that is, R = {s 4 − 1, t 3 − 1, tsts − stst}. Then C[G] is isomorphic to Cs, t/I , where I is the two-sided ideal generated by R in Cs, t. The algebra C[G] is identified with Cs, t/I , and we say that the algebra C[G] is defined by R. Applying the Composition algorithm to R leads us to find more relations. Lemma 4.5 The following relations hold in C[G] : (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) (p) (q) (r)

ts 2 tst − stst 2 s = 0, t 2 s 2 ts − sts 2 t 2 = 0, ts 3 tst − stst 2 s 2 = 0, tst 2 s 3 − s 3 t 2 st = 0, tst 2 st 2 − s 3 ts 2 ts = 0, t 2 s 3 ts − sts 3 t 2 = 0, t 2 st 2 st − sts 2 ts 3 = 0, ts 2 ts 2 ts − sts 2 ts 2 t = 0, ts 2 t 2 s 3 − s 3 t 2 s 2 t = 0, tst 2 s 2 t 2 − s 3 ts 3 ts = 0, t 2 s 3 t 2 s − st 2 s 3 t 2 = 0, t 2 s 2 t 2 st − sts 3 ts 3 = 0, ts 3 ts 2 ts − sts 2 ts 3 t = 0, ts 3 t 2 s 3 − s 3 t 2 s 3 t = 0, ts 2 ts 3 t 2 − s 3 t 2 st 2 s = 0, ts 3 ts 3 t 2 − s 3 t 2 s 2 t 2 s = 0, ts 3 t 2 st 2 s − sts 2 t 2 s 2 t 2 = 0, ts 2 ts 2 t 2 s 2 − s 3 ts 2 t 2 st = 0,

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ts 3 ts 2 t 2 st − sts 2 t 2 st 2 s 2 = 0, ts 2 ts 3 ts 3 − s 3 ts 3 ts 2 t = 0, ts 2 ts 2 t 2 st 2 − s 3 ts 2 ts 3 ts = 0, ts 2 t 2 s 2 t 2 s 2 − s 2 t 2 s 2 t 2 s 2 t = 0, ts 3 ts 3 ts 3 − s 3 t 2 st 2 s 2 t = 0, ts 3 ts 3 ts 2 t − sts 3 ts 2 t 2 s 2 = 0, ts 3 t 2 s 2 t 2 s 2 − s 2 ts 3 t 2 s 2 t 2 = 0, ts 2 ts 3 ts 2 t 2 − s 3 ts 2 t 2 st 2 s = 0, ts 3 ts 2 t 2 s 2 t 2 − s 3 ts 3 ts 3 ts 2 = 0.

 the set of defining relations R combined with the Theorem 4.6 We denote by R  forms a Gröbner-Shirshov basis for C[G]. The relations in Lemma 4.5. Then R  corresponding R-standard monomials are of the form si w where i = 0, 1, 2, 3 and the monomial w is one of the following 72 monomials : 1, t, ts, t 2 , ts 2 , tst, t 2 s, ts 3 , ts 2 t, tst 2 , t 2 s 2 , t 2 st, ts 3 t, ts 2 ts, ts 2 t 2 , tst 2 s, t 2s3,

t 2 s 2 t, t 2 st 2 , ts 3 ts, ts 3 t 2 , (ts 2 )2 , ts 2 t 2 s, tst 2 s 2 , tst 2 st, t 2 s 3 t, t 2 s 2 t 2 , (t 2 s)2 ,

ts 3 ts 2 ,

ts 3 t 2 s, ts 2 ts 3 , (ts 2 )2 t, ts 2 t 2 s 2 , ts 2 t 2 st, tst 2 s 2 t, t 2 s 3 t 2 , t 2 s 2 t 2 s, t 2 st 2 s 2 ,

(ts 3 )2 , ts 3 ts 2 t, ts 3 t 2 s 2 , ts 3 t 2 st, ts 2 ts 3 t, (ts 2 )2 t 2 , ts 2 t 2 s 2 t, ts 2 t 2 st 2 , (t 2 s 2 )2 , t 2 st 2 s 2 t, (ts 3 )2 t, ts 3 ts 2 t 2 , ts 3 t 2 s 2 t, ts 3 t 2 st 2 , ts 2 ts 3 ts, (ts 2 )2 t 2 s, ts 2 t 2 s 2 t 2 , ts 2 (t 2 s)2 , (t 2 s 2 )2 t, (ts 3 )2 ts, ts 3 ts 2 t 2 s, ts 3 t 2 s 2 t 2 , ts 2 ts 3 ts 2 , (ts 2 )2 t 2 st, ts 2 t 2 s 2 t 2 s, ts 2 t 2 st 2 s 2 , (t 2 s 2 )2 t 2 , (ts 3 )2 ts 2 , ts 3 ts 2 t 2 s 2 , ts 3 t 2 s 2 t 2 s, ts 2 ts 3 ts 2 t, ts 2 t 2 st 2 s 2 t, (t 2 s 2 )2 t 2 s, ts 3 ts 2 t 2 s 2 t.

Proof Note that the order of G is known to be 288 = 12 × 24, the product of the  fundamental degrees from the invariant theory (See [32]). Since the number of Rstandard monomials is 4 × 72 = 288,  is Gröbnerwhich is equal to the dimension of C[G], Theorem 2.6 shows again that R Shirshov basis for C[G]. 

4.4 The Group G 14 Let G be the complex reflection group G 10 . Then the group algebra C[G] is generated by s, t with the defining relations s 3 = 1 = t 2 , tstststs = stststst,

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where the corresponding Dynkin-type diagram is given by 3

8 s

t

.

Set our monomial order as before, and denote by R the set of reflection relations and the braid relation, that is, R = {s 3 − 1, t 2 − 1, tstststs − stststst}. Then C[G] is isomorphic to Cs, t/I , where I is the two-sided ideal generated by R in Cs, t. Identify C[G] with Cs, t/I , and we say that the algebra C[G] is defined by R. We use the Composition algorithm to find more relations. Lemma 4.7 The following relations hold in C[G] : (a) (b) (c) (d) (e) (f) (g) (h)

ts 2 tststs − stststs 2 t = 0, ts 2 ts 2 tsts − ststs 2 ts 2 t = 0, tststs 2 ts 2 − s 2 ts 2 tstst = 0, ts 2 ts 2 ts 2 ts − sts 2 ts 2 ts 2 t = 0, tsts 2 ts 2 ts 2 − s 2 ts 2 ts 2 tst = 0, ts 2 tsts 2 tsts − ststs 2 tsts 2 t = 0, tststs 2 tsts 2 − s 2 tsts 2 tstst = 0, tsts 2 tsts 2 ts 2 − s 2 ts 2 tsts 2 tst = 0.

 the set of defining relations R combined with the Theorem 4.8 We denote by R  forms a Gröbner-Shirshov basis for C[G]. The relations in Lemma 4.7. Then R  corresponding R-standard monomials are of the form si w where i = 0, 1, 2 and the monomial w is one of the following 48 monomials : 1, t, ts, ts 2 , tst, ts 2 t, tsts, ts 2 ts, tsts 2 , tstst, (ts 2 )2 , ts 2 tst, tsts 2 t, tststs, (ts 2 )2 t, ts 2 tsts, tsts 2 ts, tststs 2 , tststst, (ts 2 )2 ts, ts 2 tsts 2 , ts 2 tstst, ts(ts 2 )2 , tsts 2 tst, tststs 2 t, (ts 2 )3 , (ts 2 )2 tst, ts 2 tsts 2 t, ts(ts 2 )2 t, tsts 2 tsts, tststs 2 ts, (ts 2 )3 t, (ts 2 ts)2 , ts(ts 2 )2 ts, (tsts 2 )2 , tsts 2 tstst, tststs 2 tst, ts 2 tsts 2 ts 2 , (ts 2 ts)2 t, ts(ts 2 )2 tst, (tsts 2 )2 t, tststs 2 tsts, ts 2 ts(ts 2 )2 t, (tsts 2 )2 ts, tststs 2 tstst, ts 2 ts(ts 2 )2 ts, (tsts 2 )2 tst, ts 2 ts(ts 2 )2 tst.

Proof Notice that the order of G is known to be 144 = 6 × 24, the product of the  fundamental degrees from the invariant theory (See [32]). Since the number of Rstandard monomials is 3 × 48 = 144,  is Gröbnerwhich is equal to the dimension of C[G], Theorem 2.6 shows again that R Shirshov basis for C[G]. 

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5 The BMR Freeness Theorem for Hecke Algebras In the remaining sections, we briefly describe a problem on the structure of Hecke algebras associated to complex reflection groups. Let G be a complex reflection group. Assume that D is a diagram for G, and let s ∈ N (D) be a node of D, representing a simple reflection, andes be the order of s. We   . define a set u = u s, j s∈N (D),0≤ j≤es −1 and also we set u−1 = u −1 s, j s∈N (D),0≤ j≤es −1

Definition 5.1 The Hecke algebra Hu associated to D is the algebra over the ring Z[u, u−1 ] generated by elements (Ts )s∈N (D) such that • the elements Ts satisfy the braid relations defined by D, • we have (Ts − u s,0 )(Ts − u s,1 ) · · · (Ts − u s,es −1 ) = 0. In [12], Broué, Malle and Rouquier conjectured that the Hecke algebra Hu associated with each irreducible complex reflection group G is a free Z[u, u−1 ]-module of rank |G|. It is known to be true for Hecke algebras of finite Coxeter groups, that is, infinite families of types A, B, D, I , and 6 exceptional ones G 23 , G 28 , G 30 , G 35 –G 37 . Also it’s proved for those of imprimitive complex reflection groups G(de, e, n) with 3 parameters [1, 2], and for the primitive group G 4 [10]. Recently, the BMR freeness has been proved completely for the other exceptional cases: G 5 –G 7 , G 9 –G 11 , G 13 –G 15 by Chavli [15]; G 8 , G 16 by Chavli [14]; G 12 , G 22 , G 24 , G 27 , G 29 , G 31 , G 33 , G 34 by Marin-Pfeiffer [30]; G 17 –G 19 by Tsuchioka [36]; G 20 , G 21 by Marin [29]; G 25 , G 32 by Marin [27]; G 26 by Marin [28].

6 The BMM Symmetrizing Trace Conjecture The Broué-Malle-Michel conjecture [11] says that, for any irreducible complex reflection group G, its Hecke algebra Hu with basis B over Z[u, u−1 ] has a linear map τ : Hu → Z[u, u−1 ] such that   • the matrix τ (bi b j ) bi ,b j ∈B is symmetric and invertible, and • the specialization of τ to the group algebra of G becomes the canonical symmetrizing trace given by τ (w) = δ1w for all w ∈ G.

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It’s well known to be valid for the finite real reflection groups, and we have the theorem for imprimitive complex reflection groups G(de, e, n) by Bremke-Malle and Malle-Mathas [9, 25]. For the exceptional complex reflection groups, a few cases have been proved: G 4 and some 2-reflection groups G 12 , G 22 , G 24 by Malle-Michel [26]; some of the rank 2 groups G 5 –G 8 by Boura-Chavli-Chlouveraki-Karvounis [8]; G 13 by Boura-Chavli-Chlouveraki [7]. Problem. The conjecture for the other exceptional groups is open: the remaining rank 2 cases of octahedral type (G 9 -G 11 , G 14 , G 15 ) and of icosahedral type (G 16 –G 21 ), the groups of rank 3 (G 25 –G 27 ) and of rank 4,5,6 (G 29 , G 31 –G 34 ).

Acknowledgements This work was supported by NRF Grant # 2018R1D1A1B07044111 and a research grant from Seoul Women’s University (2022). The author expresses his gratitude to SungSoon Kim for a helpful discussion on the complex reflection groups during her stay at KIASCMC in 2020.

References 1. S. Ariki, Representation theory of a Hecke algebra of G(r, p, n). J. Algebra 177, 164–185 (1995) 2. S. Ariki, K. Koike, A Hecke algebra of (Z/r Z)  Sn and construction of its irreducible representations. Adv. Math. 106, 216–243 (1994) 3. G.M. Bergman, The diamond lemma for ring theory. Adv. Math. 29, 178–218 (1978) 4. L.A. Bokut, Imbedding into simple associative algebras. Algebra Log. 15, 117–142 (1976) 5. L.A. Bokut, L.-S. Shiao, Gröbner-Shirshov bases for Coxeter groups. Commun. Algebra 29, 4305–4319 (2001) 6. M.A. Borges-Trenard, H. Pérez-Rosés, Complete presentations of direct products of groups. Cien. Mat. 19, 3–11 (2001) 7. C. Boura, E. Chavli, M. Chlouveraki, The BMM symmetrising trace conjecture for the exceptional 2-reflection groups of rank 2. J. Algebra 558, 176–198 (2020) 8. C. Boura, E. Chavli, M. Chlouveraki, K. Karvounis, The BMM symmetrising trace conjecture for groups G 4 , G 5 , G 6 , G 7 , G 8 . J. Symb. Comput. 96, 62–84 (2020) 9. K. Bremke, G. Malle, Reduced words and a length function for G(e, 1, n). Indag. Math. 8, 453–469 (1997) 10. M. Broué, G. Malle, Zyklotomische Heckealgebren, Représentations unipotentes génériques et blocs des groupes réductifs finis. Astérisque 212, 119–189 (1993) 11. M. Broué, G. Malle, J. Michel, Toward Spetses I. Trans. Groups 4, 157–218 (1999) 12. M. Broué, G. Malle, R. Rouquier, On complex reflection groups and their associated braid groups, in Representations of Groups (Banff, 1994). (CMS Conference Proc. 16, Am. Math. Soc. 1995), pp. 1–13 13. B. Buchberger, An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal, Ph.D. thesis, University of Innsbruck (1965, in German), translated in J. Symbolic Comput. 41, 475–511 (2006)

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14. E. Chavli, Universal deformations of the finite quotients of the braid group on 3 strands. J. Algbera 459, 238–271 (2016) 15. E. Chavli, The BMR freeness conjecture for the tetrahedral and octahedral families. Commun. Algebra 46, 386–464 (2018) 16. C. Chevalley, Invariants of finite groups generated by reflections. Am. J. Math. 77, 778–782 (1955) 17. H.S.M. Coxeter, The symmetry groups of the regular complex polygons. Arch. Math. 13, 86–97 (1962) 18. S.-J. Kang, I.-S. Lee, K.-H. Lee, H. Oh, Hecke algebras, Specht modules and Gröbner-Shirshov bases. J. Algebra 252, 258–292 (2002) 19. S.-J. Kang, I.-S. Lee, K.-H. Lee, H. Oh, Representations of Ariki-Koike algebras and GröbnerShirshov bases. Proc. London Math. Soc. 89, 54–70 (2004) 20. S. Kim, D.-I. Lee, Monomial bases for the primitive complex Shephard groups of rank three, Commun. Algebra 50 (2022, to appear). https://doi.org/10.1080/00927872.2021.1976200 21. D. Lee, Gröbner-Shirshov bases and normal forms for the Coxeter groups E 6 and E 7 , in Advances in Algebra and Combinatorics (Guangzhou, 2007). (Proceedings of 2nd International Congress, World Scientific Publishing, 2008), pp. 243–255 22. D.-I. Lee, Cyclotomic Hecke algebras of G(r, p, n), Algebr. Represent. Theory 13, 705–718 (2010) 23. D.-I. Lee, Standard monomials for the Weyl group F4 . ACM Commun. Comput. Algebra 49(3), 74–76 (2015) 24. J.-Y. Lee, D.-I. Lee, Gröbner-Shirshov bases for non-crystallographic Coxeter groups. Acta Cryst. A 75, 584–592 (2019) 25. G. Malle, A. Mathas, Symmetric cyclotomic Hecke algebras. J. Algebra 205, 275–293 (1998) 26. G. Malle, J. Michel, Constructing representations of Hecke algebras for complex reflection groups. LMS J. Comput. Math. 13, 426–450 (2010) 27. I. Marin, The cubic Hecke algebra on at most 5 strands. J. Pure Appl. Algebra 216, 2754–2782 (2012) 28. I. Marin, The freeness conjecture for Hecke algebras of complex reflection groups, and the case of the Hessian group G 26 . J. Pure Appl. Algebra 218, 704–720 (2014) 29. I. Marin, Proof of the BMR conjecture for G 20 and G 21 . J. Symb. Comput. 92, 1–14 (2019) 30. I. Marin, G. Pfeiffer, The BMR freeness conjecture for the 2-reflection groups. Math. Comput. 86, 2005–2023 (2016) 31. C.F. Miller, Decision problems for groups—survey and reflections, in Algorithms and Classification in Combinatorial Group Theory (Berkeley, 1989) (Math. Sci. Res. Inst. Publ. 23, Springer, 1992), pp. 1–59 32. G.C. Shephard, Regular complex polytopes. Proc. London Math. Soc. 2, 82–97 (1952) 33. G.C. Shephard, J.A. Todd, Finite unitary reflection groups. Canadian J. Math. 6, 274–304 (1954) 34. A.I. Shirshov, Some algorithmic problems for Lie algebras. Sibirk. Math. Z. 3, 292–296 (1962). (in Russian), translated in ACM SIGSAM Bull. Commun. Comput. Algebra 33 (1999) no. 2, 3–6 35. O. Svechkarenko, Gröbner-Shirshov bases for the Coxeter group E 8 , Master thesis, Novosibirsk State University (2007) 36. S. Tsuchioka, BMR freeness for icosahedral family. Exp. Math. 29, 234–245 (2020)

BZS Near-Rings and Rings Mark Farag and Ralph P. Tucci

Abstract A right near-ring (N , +, ◦) is called Boolean–zero square or BZS if, for all n ∈ N either n ◦ n = n or n ◦ n = 0. Some basic results on BZS near-rings and rings are discussed, and illustrative examples are given. Keywords Boolean · Zero square · Near-ring · Malone trivial · Nilpotent · Idempotent

1 Introduction A near-ring is an algebraic structure that generalizes the concept of an associative ring. Informally, near-rings satisfy all of the axioms for an associative ring except possibly the commutativity of the addition operation and one distributive law. Formally, a right near-ring is a triple (N , +, ◦) satisfying: (1) (N , +) is a group (written additively, though not necessarily abelian), (2) (N , ◦) is a semigroup, and (3) for all a, b, c ∈ N , (a + b) ◦ c = a ◦ c + b ◦ c. Left near-rings are defined similarly by replacing property (3) above with the leftdistributive law. Near-fields, near-rings in which the nonzero elements of the nearring form a multiplicative group, were discovered by L. E. Dickson in his seminal investigation [9] of the independence of the field axioms; Zassenhaus [28] classified all finite near-fields up to isomorphism. All near-rings considered herein are right M. Farag (B) Department of Mathematics, Fairleigh Dickinson University, Teaneck, NJ 07666, USA e-mail: [email protected] R. P. Tucci Department of Mathematics and Computer Science, Loyola University New Orleans, New Orleans, LA 70118, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 A. Badawi and J. Coykendall (eds.), Rings, Monoids and Module Theory, Springer Proceedings in Mathematics & Statistics 382, https://doi.org/10.1007/978-981-16-8422-7_17

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near-rings, and we refer the reader to [22, 24] for further general information on and examples of near-rings. Some ring theory results generalize relatively easily, mutatis mutandis, to nearrings. For example, defining a homomorphism of near-rings as is normally done for rings, the well-known Homomorphism Theorems may be proven for near-rings as well. However, the elementwise definition needed in order for a nonempty subset I of a near-ring (N , +, ◦) to be the kernel of a homomorphism must be modified to: (1) for all i ∈ I and all m ∈ N , m + i − m ∈ I (normality property), (2) for all i ∈ I and all m ∈ N , i ◦ m ∈ I (right ideal property), and (3) for all i ∈ I and all m, n ∈ N , m ◦ (n + i) − m ◦ n ∈ I (left ideal property). So a two-sided near-ring ideal is a nonempty subset I of N that satisfies all three of the preceding properties; near-ring right ideals satisfy the normality and right ideal properties; near-ring left ideals satisfy the normality and left ideal properties. Other standard ring theory concepts and constructions also require modification when generalized to near-rings. For instance, polynomials in a single commuting indeterminant with coefficients in a near-ring and matrices with entries in a nearring cannot, in general, be defined using, respectively, the usual multiplication of polynomials or of matrices (see [3, 23] for respective discussions of polynomial near-rings and matrix near-rings). The multiplicative centers of near-rings need not be additively closed (see [2] for a discussion of this phenomenon). In a manner analogous to ring theory, a Boolean near-ring (in some sources it is instead called an idempotent near-ring) (N , +, ◦) is defined as one satisfying n ◦ n = n for all n ∈ N . Boolean near-rings and their variants have been studied in some depth; see, for instance, [8, 17, 19, 25–27]. Unlike the ring theoretic case, Boolean near-rings need not have characteristic 2 and they need not have a commutative multiplication, even if the underlying additive group is abelian. Example 1.1 Let (G, +) be a nontrivial group written additively, but not necessarily abelian, and let S be a subset of G. We may define a multiplication “◦” on G via:  a◦b =

a if b ∈ S . 0 if b ∈ /S

Then (G, +, ◦) is a near-ring, which we now call the Malone trivial near-ring on G determined by the set S (see, for example, [21] and [6]). If we take the particular case of S = G or S = G ∗ = G \ {0}, then the Malone trivial near-ring on G determined by S is, in fact, a Boolean near-ring. In general, however, Malone trivial near-rings need not be Boolean, and Boolean near-rings need not be Malone trivial.  While the preceding example shows that the multiplicative semigroup of a Boolean near-ring need not be commutative, a Boolean near-ring (N , +, ◦) does satisfy the “weak commutative” property (see [17]): for all a, b, c ∈ N , a ◦ b ◦ c = a ◦ c ◦ b.

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Many of the results and examples that follow derive from our consideration of low order BZS near-rings and BZS rings in SONATA [1], a package for the well-known GAP software [16].

2 The BZS Property and Basic Results We call a near-ring (N , +, ◦) Boolean - zero square or BZS if, for all n ∈ N either n ◦ n = n or n ◦ n = 0. BZS near-rings generalize both Boolean and Malone trivial near-rings, which were introduced in the preceding section, and zero square nearrings, which are those near-rings having the property that every nonzero element is nilpotent of index 2. Zero square near-rings have been investigated in [4, 15, 18]. The study of BZS near-rings was initiated in [11]; the first studies of the special case of BZS rings can be found in [13] and [14]. Many of the results on BZS nearrings and rings focus on the case of proper BZS near-rings, i.e., those that are neither Boolean nor zero square. Unsurprisingly, BZS rings exhibit significantly more structure than BZS near-rings do. For example, the additive groups of proper BZS rings turn out to be the same as those for Boolean rings. Proposition 2.1 The additive group of a proper BZS ring is an elementary abelian 2-group. The Malone trivial near-ring construction given in the last section illustrates that there is no such restriction on the additive groups of proper BZS near-rings. We can say more when the additive group of a proper BZS near-ring is cyclic of prime order. Theorem 2.2 A proper BZS near-ring (N , +, ◦) with additive group of prime order must satisfy n ◦ 0 = 0 for any n ∈ N , and is a Malone trivial near-ring with S = {n ∈ N ∗ | n ◦ n = n}. A near-ring (N , +, ◦) satisfying: for all n ∈ N , n ◦ 0 = 0 is called zero-symmetric. The following class of examples show that the conclusions of the preceding theorem can fail to hold for a BZS near-ring with additive group (Zm , +) in which m is composite. Example 2.3 Let p be any odd prime and let (N , +) be (Z2 p , +). Define the sets O = {1, 3, . . . 2 p − 1} and E = {0, 2, . . . , 2 p − 2}, and a multiplication on N via: ⎧ ⎪ ⎨a if b ∈ O \ { p} a ◦ b = p if b ∈ E ∪ { p} and a ∈ O ⎪ ⎩ 0 otherwise

.

It can be shown that this multiplication makes (N , +, ◦) into a near-ring. Since 1 ◦ 0 = p and a ◦ a = a if a ∈ O ∪ { p} while a ◦ a = 0 if a ∈ E, (N , +, ◦) is a non-zero-symmetric proper BZS near-ring that is not Malone trivial. 

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It is possible for a proper BZS near-ring with additive group (Zm , +) to be zerosymmetric and also not Malone trivial. Example 2.4 Let (N , +, ◦) be the near-ring with additive group (Z4 , +) and the multiplication given by ⎧ ⎪ ⎨a if b ∈ {1, 3} a ◦ b = 2 if a ∈ {1, 3}, b = 2 ⎪ ⎩ 0 otherwise

.

It can be verified that this multiplication makes (N , +, ◦) into a zero-symmetric near-ring, and since 0 ◦ 0 = 2 ◦ 2 = 0, 1 ◦ 1 = 1, and 3 ◦ 3 = 3, N is a proper BZS near-ring. However, N is not a Malone trivial near-ring since 1 ◦ 2 = 2 is neither 1 nor 0.  We note that the construction of Example 2.3 is similar to the TSI near-ring construction introduced in [6]. Furthermore, in the notation of [6], Example 2.4 is precisely a TSI near-ring with T = I = {1, 3}, and S = S1 = {2} (so that q1 = 2).

3 More Results on BZS Near-Rings and Rings Theorem 2.2 may be used to determine the number of distinct non-isomorphic proper BZS near-rings on Z p , for p a prime. This number is established in the following result. Theorem 3.1 For a given prime p, the number of isomorphism classes of nonBoolean BZS near-rings with additive group (Z p , +) is: p−2  k=1

 p−1  1 φ(d) dk , p − 1 d | gcd( p−1,k) d

where φ is the Euler totient function. We note that the number  p−1  1 φ(d) dk p − 1 d | gcd( p−1,k) d appearing within the summation expression of Theorem 3.1 appears in classical combinatorial contexts [20] as, among others, the number of necklaces with k black beads and p − k − 1 white beads or, equivalently, the number of ways to express 0 as a sum of k elements in Z/(( p − 1)Z).

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n

311

By Proposition 2.1, finite proper BZS rings all have additive groups of the form Z2 for some integer n ≥ 2. The number of isomorphism classes of such rings has

k=1

recently been established in [14], by proving and using the fact that the set of all idempotent elements of a finite proper BZS ring is a completely simple semigroup. Theorem 3.2 For any integer n ≥ 2, there are n isomorphism classes of proper BZS n

rings on the additive group Z2 . k=1

Example 3.3 Up to isomorphism, the two proper BZS rings with additive group Z2 ⊕ Z2 are      00 01 00 0 1

0, 1 ∈ Z T = , , , 2 , 00 00 01 01 taken with standard matrix addition and multiplication, and T op , its opposite ring.  We now turn to a consideration of the set of nilpotent elements of a BZS near-ring, which is “nice” in the following sense. Proposition 3.4 If (N , +, ◦) is a BZS near-ring then the set of all nilpotent elements of N is a subsemigroup of (N , ◦). In the case when the BZS near-ring is a BZS ring, the set of all nilpotent elements is even nicer. Proposition 3.5 If (N , +, ◦) is a BZS ring then the set of all nilpotent elements of N is the unique maximal ideal of N , and it has index 2 as an ideal. In general, however, the set of all nilpotent elements of a BZS near-ring may not even be an additive subgroup of that near-ring, as the following example shows. Example 3.6 Let p be an odd prime and let n be any positive integer. Define (N , +, ◦) as the Malone trivial near-ring on the group ((Z p )n , +), where “+” is the usual addition of n−tuples over Z p , and with S = {s} for some 0 = s ∈ (Z p )n . Then the set of all nilpotent elements of N is N \ S, which has p n − 1 elements, and hence cannot be a subgroup of N by Lagrange’s Theorem.  Although the set of all nilpotent elements of a BZS near-ring need not be an ideal, we do have the following necessary and sufficient condition for this to occur. Theorem 3.7 If (N , +, ◦) is a BZS near-ring then the set of all nilpotent elements of N is a near-ring ideal of N if and only if the set of all nilpotent elements of N is a left ideal of N . The next example shows that the set of all nilpotent elements of a BZS near-ring may be a right ideal that is not a two-sided ideal.

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Example 3.8 Let G = ((Z2 )3 , +), where “+” is the usual addition on the group (Z2 )3 . Let A = {(1, α, β) | α, β ∈ Z2 } and B = {(0, 0, 0), (1, 0, 0), (0, 1, 0)}. Define N = (G, +, ◦), where “◦” is the multiplication on N given by: ⎧ ⎪ ⎨(1, 0, 0) if a ∈ A and b ∈ B a · b = (0, 0, 0) if a ∈ / A and b ∈ B ⎪ ⎩ a otherwise

.

It can be shown that, for all x, y, z ∈ N , (x + y) ◦ z = x ◦ z + y ◦ z so that N is a right BZS near-ring under the given operations. Furthermore, the set of all nilpotent elements of N is {(0, 0, 0), (0, 1, 0)}, which is a right ideal of N that is not a left ideal of N .  A standard argument shows that, in a Boolean ring, every prime ideal must be maximal. Proper BZS rings also have this property. Theorem 3.9 In a proper BZS ring, every prime ideal is a maximal ideal. Thus, the set of nilpotent elements of a proper BZS ring N is the unique prime ideal as well as the unique maximal ideal of N . Since they are always commutative, Boolean rings all have the property that every one-sided ideal is also a two-sided ideal. BZS rings, which are far from commutative, do not share this property with Boolean rings, as the following example illustrates. Example 3.10 Let (N , +, ◦) be the Malone trivial near-ring on Z2 ⊕ Z2 with S = {(0, 1), (1, 0)}, so that multiplication given by:  a◦b =

a if b ∈ S . (0, 0) if b ∈ /S

We may verify that, in fact this multiplication satisfies the left as well as the right distributive law, so that N is a BZS ring. That N is a proper BZS ring follows since (1, 1) ◦ (1, 1) = (0, 0) and (0, 1) ◦ (0, 1) = (0, 1). Now let I = {(0, 0), (0, 1)}. Then (I, +) is clearly a subgroup of Z2 ⊕ Z2 , and since (0, 0) ◦ r and (0, 1) ◦ r are both in I for every r ∈ N , I is a right ideal of N . However, I is not a left ideal of N since (1, 0) ◦ (0, 1) = (1, 0) ∈ / I.  We note that the (near-)ring N in the preceding example is actually the ring T from Example 3.3 mildly “in disguise”; i.e., N is ring isomorphic to T in a rather obvious fashion. Turning to multiplicative centers of proper BZS near-rings, we find that such near-rings are as non-commutative as possible when the additive group is cyclic of prime order. Proposition 3.11 A proper BZS near-ring of prime order has a trivial multiplicative center.

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We have a similar result for proper BZS rings. Proposition 3.12 A proper BZS ring has a trivial center. So while BZS rings and Boolean rings exhibit similar additive structures, their multiplicative structures are significantly different. Interestingly, there exist BZS near-rings of composite order that have non-trivial multiplicative centers. Example 3.13 Let (N , +, ◦) be the TSI near-ring defined in Example 2.4. Then the element 2 is in the center of (N , ◦) since 2 ◦ 0 = 0 ◦ 2 = 0, 2 ◦ 1 = 1 ◦ 2 = 2, and 2 ◦ 3 = 3 ◦ 2 = 2.  Given a near-ring (N , +, ◦), we call Nd = {a ∈ N | for all x, y ∈ N , a ◦ (x + y) = a ◦ x + a ◦ y} the set of left distributive elements of the near-ring N , and GC(N ) = {c ∈ N | for all n d ∈ Nd , cn d = n d c} the generalized center of N . The generalized center coincides with the center in the case that N is a ring, and, unlike the center of a near-ring, the generalized center of a near-ring must be a subnear-ring. The generalized center has been the focus of several recent papers; see, for instance, [5–7, 10, 12]. Our final stated result characterizes the set of distributive elements and the generalized center for proper BZS near-rings with additive group of prime order. Proposition 3.14 Let (N , +, ◦) be a proper BZS near-ring with additive group of prime order p. Then Nd = {0} and GC(N ) = N .

4 Avenues for Future Work We end by listing briefly several possible avenues for future work related to the results presented in this paper: (1) Count the number of isomorphism classes of BZS near-rings on non-cyclic groups, perhaps beginning with BZS near-rings on groups of the form Z p ⊕ Z p for a given prime p. (2) Study the generalized version of the BZS property in a near-ring (N , +, ◦): for all n ∈ N , either n idempotent or n is nilpotent. (3) Study the BZS property in other algebraic structures, such as semigroups with 0.

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