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Commutative ring theory

Commutative ring theory

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7.10лв


Contents
Preface
 
Introduction
 
Conventions and terminology
 
1 Commutative rings and modules
 
1 Ideals
 
2 Modules
 
3 Chain conditions
 
2 Prime ideals
 
4 Localisation and Spec of a ring
 
^ The Hilbert Nullstellensatz and first steps in dimension theory
 
6 Associated primes and primary decomposition Appendix to §6. Secondary representations of a module
 
3 Properties of extension rings
 
7 Flatness
 
Appendix to §7. Pure submodules
 
8 Completion and the Artin—Rees lemma
 
9 Integral extensions
 
4 Valuation rings
 
10 General valuations
 
11 DVRs and Dedekind rings
 
12 Krull rings
 
5 Dimension theory
 
^ Graded rings, the Hilbert function and the Samuel function Appendix to §13. Determinantal ideals
 
14 Systems of parameters and multiplicity
 
15 The dimension of extension rings
 
6 Regular sequences
 
16 Regular sequences and the Koszul complex
 
17 Cohen-Macaulay rings
 
18 Gorenstein rings
 
7 Regular rings
 
19 Regular rings
 
20 UFDs
 
21 Complete intersection rings
vii
 
22 The local flatness criterion 17g
 
23 Flatness and fibres 185
 
24 Generic freeness and open loci results ^
 
9 Derivations 190
 
25 Derivations and differentials
 
26 Separability 207
 
27 Higher derivations 213
 
10 I-smoothness 213
 
28 1-smoothness 223
 
29 The structure theorems for complete oca rings 230
 
30 Connections with derivations 246
 
11 Applications of complete local rings 246
 
31 Chains of prime ideals 255
 
32 The formal fibre 261
 
33 Some other applications
 
Appendix Л. Tensor products, direct and inverse limits ^66
 
Appendix B. Some homological algebra 283
 
Appendix C. The exterior algebra 2g7
 
Solutions and hints for exercises 298
 
References 315
 
Index
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