An introduction to rings and modules with K-theory in view:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge
Cambridge Univ. Press
2000
|
| Series: | Cambridge studies in advanced mathematics
65 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 265 S. |
| ISBN: | 0521632749 |
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Record in the Search Index
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| adam_text | Contents
PREFACE p ge xiii
1 BASICS 1
1.1 RINGS 1
1.1.1 The definition 2
1.1.2 Nonunital rings 3
1.1.3 Subrings 4
1.1.4 Ideals 4
1.1.5 Generators 5
1.1.6 Homomorphisms 5
1.1.8 Residue rings 7
1.1.10 The characteristic 8
1.1.11 Units 8
1.1.12 Constructing the field of fractions 9
1.1.13 Noncommutative polynomials 10
Exercises 11
1.2 MODULES 14
1.2.1 The definition 15
1.2.2 Some first examples 16
1.2.3 Bimodules 17
1.2.4 Homomorphisms of modules 17
1.2.5 The composition of homomorphisms 18
1.2.6 The opposite of a ring 19
1.2.7 Balanced bimodules 20
1.2.8 Submodules and generators 21
1.2.9 Kernel and image 22
1.2.10 Quotient modules 22
1.2.12 Images and inverse images 23
1.2.14 Change of rings 24
vii
viii Contents
1.2.16 Irreducible modules 24
1.2.18 Maximal elements in ordered sets 25
1.2.23 Torsion free modules and spaces over the field of fractions 27
Exercises 28
2 DIRECT SUMS AND SHORT EXACT SEQUENCES 36
2.1 DIRECT SUMS AND FREE MODULES 37
2.1.1 Internal direct sums 37
2.1.2 Examples: vector spaces 38
2.1.3 Examples: abelian groups 39
2.1.4 The uniqueness of summands 40
2.1.5 External direct sums 40
2.1.6 Standard inclusions and projections 41
2.1.8 Notation 42
2.1.9 Idempotents 42
2.1.11 Infinite direct sums 43
2.1.13 Remarks 44
2.1.14 Ordered index sets 45
2.1.15 The module LA 46
2.1.16 The module Ftr(X) 47
2.1.17 Left handed notation 47
2.1.18 Free generating sets 48
2.1.19 Free modules 48
2.1.21 Extending maps 49
Exercises 51
2.2 MATRICES, BASES, HOMOMORPHISMS OF FREE MODULES 54
2.2.1 Bases 55
2.2.2 Standard bases 56
2.2.3 Coordinates 56
2.2.5 Matrices for homomorphisms 58
2.2.7 Change of basis 59
2.2.9 Matrices of endomorphisms 60
2.2.10 Normal forms of matrices 61
2.2.12 Scalar matrices and endomorphisms 62
2.2.13 Infinite bases 64
2.2.14 Free left modules 65
Exercises 66
2.3 INVARIANT BASIS NUMBER 69
2.3.1 Some non IBN rings 70
2.3.3 Two non square invertible matrices 70
2.3.4 The type 71
Contents ix
Exercises 72
2.4 SHORT EXACT SEQUENCES 75
2.4.1 The definition 75
2.4.2 Four term sequences 76
2.4.3 Short exact sequences 76
2.4.4 Direct sums and splittings 77
2.4.6 Dual numbers 79
2.4.7 The group Ext 80
2.4.8 Pull backs and push outs 80
2.4.10 Base change for short exact sequences 82
2.4.11 The direct sum of short exact sequences 83
Exercises 84
2.5 PROJECTIVE MODULES 89
2.5.1 The definition and basic properties 89
2.5.9 Idempotents and projective modules 93
2.5.12 A cautionary example 94
2.5.13 Injective modules 94
Exercises 95
2.6 DIRECT PRODUCTS OF RINGS 97
2.6.1 The definition 98
2.6.2 Central idempotents 99
2.6.4 Remarks 100
2.6.5 An illustration 101
2.6.6 Modules 101
2.6.7 Homomorphisms 102
2.6.10 Historical note 104
Exercises 105
3 NOETHERIAN RINGS AND POLYNOMIAL RINGS 108
3.1 NOETHERIAN RINGS 108
3.1.1 The Noetherian condition 109
3.1.5 The ascending chain condition and the maximum condition 110
3.1.11 Module finite extensions 112
Exercises 114
3.2 SKEW POLYNOMIAL RINGS 115
3.2.1 The definition 116
3.2.2 Some endomorphisms 118
3.2.6 The division algorithm 120
3.2.7 Euclidean domains 120
3.2.11 Euclid s algorithm 122
3.2.12 An example 123
x Contents
3.2.13 Inner order and the centre 124
3.2.15 Ideals 125
3.2.19 Total division 126
3.2.22 Unique factorization 127
3.2.23 Further developments 128
Exercises 129
3.3 MODULES OVER SKEW POLYNOMIAL RINGS 132
3.3.1 Elementary operations 133
3.3.3 Rank and invariant factors 134
3.3.4 The structure of modules 135
3.3.7 Rank and invariant factors for modules 137
3.3.8 Non cancellation 138
3.3.12 Serre s Conjecture 140
3.3.13 Background and developments 140
Exercises 140
4 ARTINIAN RINGS AND MODULES 145
4.1 ARTINIAN MODULES 145
4.1.1 The definition 146
4.1.2 Examples 146
4.1.3 Fundamental properties 147
4.1.8 Composition series 148
4.1.11 Multiplicity 151
4.1.13 Reducibility 152
4.1.15 Complete reducibility 154
4.1.22 Fully invariant submodules 157
4.1.24 The socle series 158
Exercises 159
4.2 ARTINIAN SEMISIMPLE RINGS 161
4.2.1 Definitions and the statement of the Wedderburn Artin
Theorem 162
4.2.4 Division rings 163
4.2.5 Matrix rings 163
4.2.6 Products of matrix rings 164
4.2.11 Finishing the proof of the Wedderburn Artin Theorem 166
4.2.16 Recapitulation of the argument 169
Exercises 170
4.3 ARTINIAN RINGS 173
4.3.1 The Jacobson radical 173
4.3.2 Examples 174
4.3.3 Basic properties 174
Contents xi
4.3.11 Alternative descriptions of the Jacobson radical 176
4.3.19 Nilpotent ideals and a characterization of Artinian rings 179
4.3.22 Semilocal rings 180
4.3.24 Local rings 181
4.3.27 Further reading 182
Exercises 182
5 DEDEKIND DOMAINS 186
5.1 DEDEKIND DOMAINS AND INVERTIBLE IDEALS 187
5.1.1 Prime ideals 187
5.1.4 Coprime ideals 188
5.1.7 Fractional ideals 189
5.1.10 Dedekind domains the definition 190
5.1.11 The class group 191
5.1.13 An exact sequence 192
5.1.15 Ideal theory in a Dedekind domain 193
5.1.25 Principal ideal domains 196
5.1.28 Primes versus irreducibles 197
Exercises 197
5.2 ALGEBRAIC INTEGERS 200
5.2.1 Integers 200
5.2.6 Quadratic fields 203
5.2.9 Separability and integral closure 205
Exercises 208
5.3 QUADRATIC FIELDS 210
5.3.1 Factorization in general 210
5.3.2 Factorization in the quadratic case 211
5.3.3 Explicit factorizations 213
5.3.4 A summary 214
5.3.5 The norm 215
5.3.9 The Euclidean property 216
5.3.10 The class group again 217
5.3.13 The class number 218
5.3.15 Computations of class groups 218
5.3.21 Function fields 221
Exercises 222
6 MODULES OVER DEDEKIND DOMAINS 224
6.1 PROJECTIVE MODULES OVER DEDEKIND DOMAINS 225
6.1.7 The standard form 228
6.1.11 The noncommutative case 229
Exercises 229
xii Contents
6.2 VALUATION RINGS 230
6.2.1 Valuations 231
6.2.4 Localization 232
6.2.9 Uniformizing parameters 234
6.2.10 The localization as a Euclidean domain 234
6.2.13 Rank and invariant factors 235
Exercises 235
6.3 TORSION MODULES OVER DEDEKIND DOMAINS 237
6.3.1 Torsion modules 238
6.3.6 The rank 239
6.3.7 Primary modules 239
6.3.10 Elementary divisors 241
6.3.13 Primary decomposition 243
6.3.17 Elementary divisors again 244
6.3.18 Homomorphisms 245
6.3.21 Alternative decompositions 246
6.3.22 Homomorphisms again 247
Exercises 249
References 252
Index 257
|
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| spelling | Berrick, Alan Jon 1948- Verfasser (DE-588)115281460 aut An introduction to rings and modules with K-theory in view A. J. Berrick & M. E. Keating Cambridge Cambridge Univ. Press 2000 XV, 265 S. txt rdacontent n rdamedia nc rdacarrier Cambridge studies in advanced mathematics 65 Anneaux (Algèbre) ram Modulen (wiskunde) gtt Modules (Algèbre) ram Ringen (wiskunde) gtt Modules (Algebra) Rings (Algebra) Keating, M. E. Verfasser aut Cambridge studies in advanced mathematics 65 (DE-604)BV000003678 65 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009618670&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Berrick, Alan Jon 1948- Keating, M. E. An introduction to rings and modules with K-theory in view Cambridge studies in advanced mathematics Anneaux (Algèbre) ram Modulen (wiskunde) gtt Modules (Algèbre) ram Ringen (wiskunde) gtt Modules (Algebra) Rings (Algebra) |
| title | An introduction to rings and modules with K-theory in view |
| title_auth | An introduction to rings and modules with K-theory in view |
| title_exact_search | An introduction to rings and modules with K-theory in view |
| title_full | An introduction to rings and modules with K-theory in view A. J. Berrick & M. E. Keating |
| title_fullStr | An introduction to rings and modules with K-theory in view A. J. Berrick & M. E. Keating |
| title_full_unstemmed | An introduction to rings and modules with K-theory in view A. J. Berrick & M. E. Keating |
| title_short | An introduction to rings and modules with K-theory in view |
| title_sort | an introduction to rings and modules with k theory in view |
| topic | Anneaux (Algèbre) ram Modulen (wiskunde) gtt Modules (Algèbre) ram Ringen (wiskunde) gtt Modules (Algebra) Rings (Algebra) |
| topic_facet | Anneaux (Algèbre) Modulen (wiskunde) Modules (Algèbre) Ringen (wiskunde) Modules (Algebra) Rings (Algebra) |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009618670&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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