Multiplicative Theory of Ideals (eBook)

Front Cover 1
MULTIPLICATIVE THEORY OF IDEALS 4
Copyright Page 5
Contents 8
Preface 12
Prerequisites 14
Chapter I. Modules 18
1 Rings and Modules 18
2 Chain Conditions 25
3 Direct Sums 29
4 Tensor Products 32
5 Flat Modules 38
Exercises 44
Chapter II. Primary Decompositions and Noetherian Rings 53
1 Operations on Ideals and Submodules 53
2 Primary Submodules 56
3 Noetherian Rings 61
4 Uniqueness Results for Primary Decompositions 65
Exercises 69
Chapter III. Rings and Modules of Quotients 78
1 Definition 78
2 Extension and Contraction of Ideals 83
3 Properties of Rings of Quotients 88
Exercises 91
Chapter IV. Integral Dependence 99
1 Definition of Integral Dependence 99
2 Integral Dependence and Prime Ideals 101
3 Integral Dependence and Flat Modules 105
4 Almost Integral Dependence 109
Exercises 111
Chapter V. Valuation Rings 116
1 The Definition of a Valuation Ring 116
2 Ideal Theory in Valuation Rings 122
3 Valuations 124
4 Prolongation of Valuations 131
Exercises 135
Chapter Vl. Prüfer and Dedekind Domains 141
1 Fractional Ideals 141
2 Prüfer Domains 143
3 Overrings of Prüfer Domains 149
4 Dedekind Domains 151
5 Extension of Dedekind Domains 157
Exercises 161
Chapter VII. Dimension of Commutative Rings 173
1 The Krull Dimension 173
2 The Krull Dimension of a Polynomial Ring 178
3 Valuative Dimension 181
Exercises 185
Chapter VIII. Krull Domains 188
1 Krull Domains 188
2 Essential Valuations 196
3 The Divisor Class Group 202
4 Factorial Rings 207
Exercises 211
Chapter IX. Generalizations of Dedekind Domains 218
1 Almost Dedekind Domains 218
2 ZPI-Rings 222
3 Multiplication Rings 226
4 Almost Multiplication Rings 233
Exercises 237
Chapter X. Prüfer Rings 243
1 Valuation Pairs 243
2 Counterexamples 249
3 Large Quotient Rings 251
4 Prüfer Rings 253
Exercises 261
Appendix: Decomposition of ideals in Noncommutative Rings 269
Exercises 280
Bibliography 283
Subject Index 311
Pure and Applied Mathematics: A Series of Monographs and Textbooks 316