Introduction to Rings And Modules
by: C. Musili
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language: en [ english ]
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8173190372
9788173190377
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year: 1997
pages: 215
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Introduction to Rings And Modules, Second Edition
C. Musili
en 2.69 MB 8173190372
9788173190377
F B V S O C P
year: 1997
pages: 215
edition: 2
mathematics
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This book is a self-contained elementary introduction to rings and modules, and should be useful for courses on Algebra. The emphasis is on concept development with adequate examples and counter-examples drawn from topics such as analysis, topology, etc. The entire material, including exercises, is fully class tested.
frontcover..................................... 1
Preface........................................ 8
Preface to the second edition.................. 10
Contents....................................... 12
Glossary of notation........................... 16
1 Rings........................................ 22
1.1 Terminology.......................... 22
1.2 Rings of Continuous Functions.......... 29
1.3 Matrix Rings........................... 33
1.4 Polynomial Rings....................... 35
1.5 P ower Series Rings.................... 36
1.6 Laurent Rings......................... 37
1.7 Boolean Rings.......................... 38
1.8 Some Special Rings..................... 39
1.9 Direct Products........................ 41
1.10 Several Variables..................... 42
1.11 Opposite Rings........................ 43
1.12 Characteristic of a Ring.............. 43
1.13 Exercises............................. 45
1.14 True/False Statements................. 50
2 Ideals....................................... 52
2.1 Definitions.......................... 52
2.2 Maximal Ideals......................... 53
2.3 Generators............................. 58
2.4 Basic Properties of Ideals............. 60
2.5 Algebra of Ideals...................... 67
2.6 Quotient Rings......................... 70
2.7 Ideals in Quotient Rings............... 72
2.8 Lo cal Rings........................... 76
2.9 Exercises.............................. 79
2.10 True/False Statements................. 84
3 Homomorphisms of Rings....................... 86
3.1 Definitions and Basic Properties..... 86
3.2 Fundamental Theorems................... 89
3.3 Endomorphism Rings..................... 94
3.4 Field of fractions..................... 96
3.5 Prime fields...........................101
3.6 Exercises..............................102
3.7 True/False Statements..................106
4 Factorisation in Domains.....................108
4.1 Division in Domains..................108
4.2 Euclidean Domains......................111
4.3 Principal Ideal Domains................115
4.4 Factorisation Domains..................118
4.5 Unique Factorisation Domains...........120
4.6 Eisenstein's Criterion.................125
4.7 Exercises..............................127
4.8 True/False Statements..................130
5 Modules......................................134
5.1 Definitions and Examples.............134
5.2 Direct Sums............................140
5.3 Free Modules...........................141
5.4 Vector Spaces..........................143
5.5 Some Pathologies.......................145
5.6 Quotient Modules.......................151
5.7 Homomorphisms..........................152
5.8 Simple modules.........................156
5.9 Modules over P I D's...................158
5.10 Exercises.............................163
5.11 True/False Statements.................168
Chapter 6 : Modules with chain Conditions....170
6.1 Artinian Modules.....................170
6.2 Noetherian Modules.....................173
6.3 Modules of Finite Length...............177
6.4 Artinian Rings.........................182
6.5 Noetherian Rings.......................183
6.6 Radical s.............................. 89
6.6.1 Nil Radical......................190
6.6.2 Jacobson Radical.................191
6.7 Radical of an Artinian Ring............195
6.8 Exercises..............................199
6.9 True/False Statements..................202
Answers to True/False Statements...............204
Index..........................................206
backcover......................................215
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