Commutative Ring Theory

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Cambridge University Press, May 25, 1989 - Mathematics - 320 pages
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In addition to being an interesting and profound subject in its own right, commutative ring theory is important as a foundation for algebraic geometry and complex analytical geometry. Matsumura covers the basic material, including dimension theory, depth, Cohen-Macaulay rings, Gorenstein rings, Krull rings and valuation rings. More advanced topics such as Ratliff's theorems on chains of prime ideals are also explored. The work is essentially self-contained, the only prerequisite being a sound knowledge of modern algebra, yet the reader is taken to the frontiers of the subject. Exercises are provided at the end of each section and solutions or hints to some of them are given at the end of the book.
  

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Contents

Commutative rings and modules
1
2 Modules
6
3 Chain conditions
14
Prime ideals
20
5 The Hilbert Nullstellensatz and first steps in dimension theory
30
6 Associated primes and primary decomposition
37
Appendix to 6 Secondary representations of a module
42
Properties of extension rings
45
20 UFDs
161
21 Complete intersection rings
169
Flatness revisited
173
23 Flatness and fibres
178
24 Generic freeness and open loci results
185
Derivations
190
26 Separability
198
27 Higher derivations
207

Appendix to 7 Pure submodules
53
8 Completion and the ArtinRees lemma
55
9 Integral extensions
64
Valuation rings
71
11 DVRs and Dedekind rings
78
12 Krull rings
86
Dimension theory
92
Appendix to 13 Determinantal ideals
103
14 Systems of parameters and multiplicity
104
15 UK dimension of extension rings
116
Regular sequences
123
17 Cohen Macaulay rings
133
18 Gorenstein rings
139
Regular rings
153
Ismoothness
213
29 The structure theorems for complete local rings
223
30 Connections with derivations
230
Applications of complete local rings
246
32 The formal fibre
255
33 Some other applications
261
Tensor products direct and inverse limits
266
Some homological algebra
274
The exterior algebra
283
Solutions and hints for the exercises
287
References
298
Index
317
Copyright

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About the author (1989)

Matsumura graduated in economics from Tokyo University and from John F. Kennedy School of Government, Harvard University, where he was awarded the Lucius N. Littauer Fellowship. he was a Visiting Fellow in the Energy and Environment Programme at the Royal Institute of International Affairs in 1999-2000, seconded from the Institiute of Developing Economics, Japan External Trade Organization.

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