Ideal theory

D. G. Northcott9780521604833, 9780521058407, 0521604834, 0521058406

Ideal theory is important not only for the intrinsic interest and purity of its logical structure but because it is a necessary tool in many branches of mathematics. In this introduction to the modern theory of ideals, Professor Northcott assumes a sound background of mathematical theory but no previous knowledge of modern algebra. After a discussion of elementary ring theory, he deals with the properties of Noetherian rings and the algebraic and analytical theories of local rings. In order to give some idea of deeper applications of this theory the author has woven into the connected algebraic theory those results which play outstanding roles in the geometric applications.

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Table of contents :
Front cover……Page 1
CONTENTS……Page 5
Author’s Preface……Page 7
Preliminaries……Page 9
1.2 Ideals and their calculus……Page 12
1.3 The ideal generated by a set……Page 16
1.4 Prime ideals……Page 17
1.5 Primary ideals……Page 18
1.6 Ideals with a primary decomposition……Page 21
1.7 The isolated components of an ideal……Page 25
1.8 Noetherian rings……Page 27
1.9 Some additional properties of Noetherian rings……Page 30
1.10 Some different kinds of rings……Page 32
Examples……Page 36
2.1 Homomorphisms and isomorphisms……Page 39
2.2 Residue rings……Page 41
2.3 Further properties of residue rings……Page 44
2.4 Extension rings……Page 46
2.5 The full ring of quotients……Page 47
2.6 Rings of quotients……Page 49
2.7 Generalized rings of quotients……Page 52
2.8 Local rings……Page 55
3.1 The intersection theorem……Page 57
3.2 Symbolic prime powers……Page 58
3.3 The length of a primary ideal……Page 59
3.4 Primary rings……Page 62
3.5 Rank and dimension……Page 64
4.2 Systems of parameters……Page 71
4.3 Indeterminates……Page 73
4.4 Analytic independence……Page 75
4.5 Minimal ideal bases……Page 77
4.6 Regular local rings……Page 78
4.7 Integral dependence and integral closure……Page 81
4.8 One-dimensional regular local rings……Page 83
4.9 Divisors……Page 84
4.10 Another theorem on divisors……Page 88
4.11 Continuation of §4.3……Page 89
5.1 Convergence……Page 92
5.2 Complete rings……Page 93
5.3 Concordant rings……Page 95
5.4 Formal power series……Page 97
5.5 The completion of a local ring……Page 100
5.6 Some transition theorems……Page 105
Notes……Page 109
References……Page 118
Index of Definitions……Page 119
Back cover……Page 120