Modules over Endomorphism Rings

✍ Scribed by Theodore G. Faticoni

• Publisher: Cambridge University Press
• Year: 2009
• Tongue: English
• Leaves: 393
• Series: Encyclopedia of Mathematics and its Applications volume 130
• Edition: 1
• Category: Library

✦ Synopsis

This is an extensive synthesis of recent work in the study of endomorphism rings and their modules, bringing together direct sum decompositions of modules, the class number of an algebraic number field, point set topological spaces, and classical noncommutative localization. The main idea behind the book is to study modules G over a ring R via their endomorphism ring EndR(G). The author discusses a wealth of results that classify G and EndR(G) via numerous properties, and in particular results from point set topology are used to provide a complete characterization of the direct sum decomposition properties of G. For graduate students this is a useful introduction, while the more experienced mathematician will discover that the book contains results that are not otherwise available. Each chapter contains a list of exercises and problems for future research, which provide a springboard for students entering modern professional mathematics.

✦ Table of Contents

Cover......Page 1
MODULES OVER ENDOMORPHISM RINGS......Page 2
Title......Page 4
Copyright......Page 5
Dedication......Page 6
Contents......Page 8
Preface......Page 14
1.1 Rings, modules, and functors......Page 22
1.2 Azumaya–Krull–Schmidt theorem......Page 24
1.3 The structure of rings......Page 25
1.4 The Arnold–Lady theorem......Page 26
2.1 Preliminaries......Page 30
2.2 A functorial bijection......Page 35
2.3 Internal cancellation......Page 38
2.4 Power cancellation......Page 40
2.5 Unique decomposition......Page 42
2.6 Algebraic number fields......Page 44
2.8 Problems for future research......Page 46
3.1 The sequence of groups......Page 47
3.2 Analytic methods......Page 53
3.4 Problems for future research......Page 57
4.1 Units and sequences......Page 58
4.2 Calculations with primary ideals......Page 61
4.3 Quadratic number fields......Page 65
4.4 The Gaussian integers......Page 66
4.5 Imaginary quadratic number fields......Page 67
4.7 Problems for future research......Page 69
5.1 Introduction......Page 70
5.2 Some functors......Page 72
5.4 Local correspondence......Page 75
5.6 Problems for future research......Page 81
6.1 Rtffr groups......Page 82
6.2 Direct sum decompositions......Page 84
6.3 Locally semi-perfect rings......Page 88
6.4 Balanced semi-primary groups......Page 91
6.5 Examples......Page 93
6.6 Exercises......Page 96
6.7 Problems for future research......Page 97
7.1 Introduction......Page 98
7.2 Functors and bijections......Page 100
7.3 The square......Page 102
7.4 Isomorphism classes......Page 111
7.5 The equivalence class {I }......Page 112
7.6 Commutative domains......Page 114
7.7 Cardinality of the kernels......Page 116
7.8 Relatively prime to τ......Page 117
7.9 Power cancellation......Page 119
7.10 Algebraic number fields......Page 121
7.12 Problems for future research......Page 123
8.2 Eichler groups......Page 124
8.3 Direct sums of L-groups......Page 127
8.4 Eichler L-groups are J-groups......Page 130
8.5 Exercises......Page 132
8.6 Problems for future research......Page 133
9.1.1 The homotopy of G-plexes......Page 134
9.1.2 Homotopy and homology......Page 139
9.1.3 Endomorphism modules as G-plexes......Page 141
9.2 Two commutative triangles......Page 149
9.2.1 G-Solvable R-modules......Page 150
9.2.2 A factorization of the tensor functor......Page 152
9.3 Left endomorphism modules......Page 155
9.3.1 Duality......Page 158
9.4 Self-small self-slender modules......Page 163
9.5 (µ) Implies slender injectives......Page 164
9.6 Exercises......Page 165
9.7 Problems for future research......Page 167
10.1 Small projective generators......Page 169
10.2 Quasi-projective modules......Page 174
10.3.1 A category equivalence for submodules of free modules......Page 178
10.3.2 Right ideals in endomorphism rings......Page 182
10.3.3 A criterion for E-flatness......Page 183
10.4 Orsatti and Menini’s *-modules......Page 184
10.5 Dualities from injective properties......Page 187
10.5.1 G-Cosolvable R-modules......Page 188
10.5.2 A factorization of HomR(·,G)......Page 189
10.5.3 Dualities for the dual functor......Page 190
10.6 Exercises......Page 192
10.7 Problems for future research......Page 194
11.1 Flat endomorphism modules......Page 196
11.2.1 Definitions and examples......Page 198
11.2.2 The exact dimension of a G-plex......Page 200
11.2.3 The projective dimension of a G-plex......Page 201
11.3 The flat dimension......Page 205
11.4 Global dimensions......Page 209
11.5.1 Baer’s lemma......Page 212
11.5.2 Semi-simple rings......Page 215
11.5.3 Right hereditary rings......Page 216
11.5.4 Global dimension at most 3......Page 222
11.6.1 A review of G-coplexes......Page 223
11.6.2 Injective endomorphism rings......Page 227
11.6.3 Left homological dimensions......Page 230
11.7 A glossary of terms......Page 233
11.8 Exercises......Page 235
11.9 Problems for future research......Page 239
12.1 Projectives......Page 240
12.2 Finitely generated modules......Page 244
12.4 Problems for future research......Page 249
13.1 Beaumont–Pierce......Page 250
13.2 Noetherian modules......Page 256
13.3 Generators......Page 258
13.5 Problems for future research......Page 262
14.1 Introduction......Page 263
14.2 The UConn ’81 Theorem......Page 266
14.4 Problems for future research......Page 269
15.1 G-Monomorphisms......Page 270
15.2 Injective properties......Page 272
15.3 G-Cogenerators......Page 279
15.4 Projective modules revisited......Page 282
15.5 Examples......Page 283
15.6 Exercises......Page 284
15.7 Problems for future research......Page 285
16.1 The diagram......Page 286
16.2 Smallness and slenderness......Page 290
16.4 The construction function......Page 293
16.5 The Greek maps......Page 295
16.6.1 Complete sets of invariants......Page 296
16.6.2 Unique topological decompositions......Page 297
16.6.3 Homological dimensions......Page 300
16.8 Problems for future research......Page 303
17 Diagrams of abelian groups......Page 305
17.1 The ring EndC(X)......Page 306
17.2 Topological complexes......Page 307
17.3 Categories of complexes......Page 309
17.4 Commutative triangles......Page 312
17.5.1 A diagram for an abelian groups......Page 315
17.5.2 Self-small and self-slender......Page 317
17.5.3 Coherent complexes......Page 319
17.6 Prism diagrams......Page 321
17.7 Direct sums......Page 322
17.8 Algebraic number fields......Page 325
17.9 Exercises......Page 330
17.10 Problems for future research......Page 332
18.1 Ore localization......Page 333
18.1.1 Preliminary concepts and examples......Page 334
18.1.2 Noncommutative localization......Page 336
18.2 Marginal isomorphisms......Page 343
18.2.1 Margimorphism and localizations......Page 344
18.2.2 Marginal summands......Page 348
18.2.3 Marginal summands as projectives......Page 350
18.2.4 Projective QG-modules......Page 352
18.3.1 Totally indecomposable modules......Page 354
18.3.2 Morphisms of totally indecomposables......Page 355
18.3.3 Semi-simple marginal summands......Page 358
18.3.4 Jónsson’s theorem and margimorphisms......Page 360
18.4 Nilpotent sets and margimorphism......Page 363
18.5 Isomorphism from margimorphism......Page 367
18.6 Semi-simple endomorphism rings......Page 374
18.7 Exercises......Page 379
18.8 Problems for future research......Page 381
Bibliography......Page 383
Index......Page 389