## Commutative Ring TheoryIn 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 |

298 | |

317 | |

### Common terms and phrases

0-smooth 4-algebra adic topology Artinian associated prime assume assumption catenary chain choose closure CM ring coefficients conditions are equivalent contradiction Corollary define depth dimension element equidimensional exact sequence example exists extension field faithfully flat fc-algebra field of fractions finite number finite X-module free module functor G-ring Gorenstein graded ring hence holds homogeneous induction injective injective resolution integral domain integrally closed intersection inverse irreducible isomorphism Koszul complex Krull ring Lemma linearly independent localisation M-sequence maximal ideal minimal prime divisor Moreover multiplicative set natural map Noetherian local ring Noetherian ring non-zero p-basis P-primary PeSpec polynomial ring previous theorem principal ideal proj dim projective Proof Prove the following quotient rank regular local ring regular ring residue field ring homomorphism satisfies Spec subfield submodule subring subset Suppose surjective system of parameters theory valuation ring write