Introduction to Rings And Modules, Second Edition

frontcover......Page 1
Preface......Page 8
Preface to the second edition......Page 10
Contents......Page 12
Glossary of notation......Page 16
1.1 Terminology ......Page 22
1.2 Rings of Continuous Functions ......Page 29
1.3 Matrix Rings ......Page 33
1.4 Polynomial Rings ......Page 35
1.5 P ower Series Rings ......Page 36
1.6 Laurent Rings ......Page 37
1.7 Boolean Rings ......Page 38
1.8 Some Special Rings ......Page 39
1.9 Direct Products ......Page 41
1.10 Several Variables ......Page 42
1.12 Characteristic of a Ring ......Page 43
1.13 Exercises ......Page 45
1.14 True/False Statements ......Page 50
2.1 Definitions ......Page 52
2.2 Maximal Ideals ......Page 53
2.3 Generators ......Page 58
2.4 Basic Properties of Ideals ......Page 60
2.5 Algebra of Ideals ......Page 67
2.6 Quotient Rings ......Page 70
2.7 Ideals in Quotient Rings ......Page 72
2.8 Lo cal Rings ......Page 76
2.9 Exercises ......Page 79
2.10 True/False Statements ......Page 84
3.1 Definitions and Basic Properties ......Page 86
6.6 Radical s ......Page 89
3.3 Endomorphism Rings ......Page 94
3.4 Field of fractions ......Page 96
3.5 Prime fields ......Page 101
3.6 Exercises ......Page 102
3.7 True/False Statements ......Page 106
4.1 Division in Domains ......Page 108
4.2 Euclidean Domains ......Page 111
4.3 Principal Ideal Domains ......Page 115
4.4 Factorisation Domains ......Page 118
4.5 Unique Factorisation Domains ......Page 120
4.6 Eisenstein\'s Criterion ......Page 125
4.7 Exercises ......Page 127
4.8 True/False Statements ......Page 130
5.1 Definitions and Examples ......Page 134
5.2 Direct Sums ......Page 140
5.3 Free Modules ......Page 141
5.4 Vector Spaces ......Page 143
5.5 Some Pathologies ......Page 145
5.6 Quotient Modules ......Page 151
5.7 Homomorphisms ......Page 152
5.8 Simple modules ......Page 156
5.9 Modules over P I D\'s ......Page 158
5.10 Exercises ......Page 163
5.11 True/False Statements ......Page 168
6.1 Artinian Modules ......Page 170
6.2 Noetherian Modules ......Page 173
6.3 Modules of Finite Length ......Page 177
6.4 Artinian Rings ......Page 182
6.5 Noetherian Rings ......Page 183
6.6.1 Nil Radical ......Page 190
6.6.2 Jacobson Radical ......Page 191
6.7 Radical of an Artinian Ring ......Page 195
6.8 Exercises ......Page 199
6.9 True/False Statements ......Page 202
Answers to True/False Statements ......Page 204
Index ......Page 206
backcover......Page 215