Local rings [Reprint ed.] 0882752286, 9780882752280

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English Pages [240] Year 1975

Table of contents :
Preface
Contents
CHAPTER I General Commutative Rings
1. Rings, ideals, and modules
2. Prime ideals and primary ideals
3. Noetherian rings
4. Jacobson radicals
5. The definition of local rings
6. Rings of quotients
7. Prime divisors
8. Primary decomposition of ideals
9. The notions of height and altitude
10. Integral dependence
11. Valuation rings
12. Noetherian normal rings
13. Unique factorization rings
14. A normalization theorem
CHAPTER II Completions
15. Formal power series ring
16. An ideal-adic topology
17. Completions
18. Exact tensor products
19. The theorem of transition
CHAPTER III Multiplicities
20. Homogeneous rings
21. Lambda-Polynomials
22. Superficial elements
23. Multiplicities
24. System of parameters
25. Macaulay rings
CHAPTER IV The Theory of Syzygies
26. Definition of syzygies
27. Change of Rings
28. Regular local rings
29. Syzygies of graded modules
CHAPTER V Theory of Complete Local Rings and Its Application
30. Some properties of complete local rings
31. The structure theorem of complete local rings
32. Finiteness of derived normal rings
33. Derived normal rings of Noetherian integral domains
34. Chains of prime ideals
CHAPTER VI Geometric Local Rings
35. Localities
36. Pseudo-geometric rings
37. Analytical normality
38. Some types of ring extensions
39. Separably generated extensions
40. Multiplicity of a local ring
41. Purity of branch loci
42. Tensor products
CHAPTER VII Henselian Rings and Weierstrass Rings
43. Henselization
44. Hensel lemma
45. Convergent power series rings
46. Jacobian criterion of simple points
47. Analytic tensor product
Appendix
A1. Examples of bad Noetherian rings
A2. Historical Note
References
TABLE OF NOTATION
INDEX

Local rings [Reprint ed.]
0882752286, 9780882752280