Commutative Ring Theory

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
5 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
13 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
Vll
ix
xiii
1
1
6
14
20
20
30
37
42
45
45
53
55
64
71
71
78
86
92
92
103
104
116
123
123
133
139
153
153
161
169
8 Flatness revisited
22 The local flatness criterion
23 Flatness and fibres
24 Generic freeness and open loci results
9 Derivations
25 Derivations and differentials
26 Separability
27 Higher derivations
10 I-smoothness
28 /-smoothness
29 The structure theorems for complete local rings
30 Connections with derivations
11 Applications of complete local rings
31 Chains of prime ideals
32 The formal fibre
33 Some other applications
Appendix A. Tensor products, direct and inverse limits
Appendix B. Some homological algebra
Appendix C. The exterior algebra
Solutions and hints for exercises
References
Index
173
173
178
185
190
190
198
207
213
213
223
230
246
246
255
261
266
274
283
287
298
315