Contents
0 Abstract iii
1 Notation and Preliminary Results 1
1.1 Abelian Groups . . . . . . . . . . . . . . . . . . . . . 1
1.2 Associative Rings . . . . . . . . . . . . . . . . . . . . 5
1.3 Local-global and Chinese Remainder . . . . . . . . . 9
1.4 Finite Dimensional Q-Algebras . . . . . . . . . . . . 11
1.5 Localization in Commutative Rings . . . . . . . . . . 13
1.6 Integrally Closed Rings . . . . . . . . . . . . . . . . . 15
1.7 Semi-perfect Rings . . . . . . . . . . . . . . . . . . . 17
1.8 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . 20
2 Motivation by Example 21
2.1 Some Well Behaved Direct Sums . . . . . . . . . . . 22
2.2 Some Badly Behaved Direct Sums . . . . . . . . . . . 27
2.3 Corner¿s Theorem . . . . . . . . . . . . . . . . . . . . 29
2.4 The Arnold-Lady-Murley Theorem . . . . . . . . . . 32
2.4.1 A Category Equivalence . . . . . . . . . . . . 32
2.4.2 The Functor A(·) . . . . . . . . . . . . . . . . 34
2.4.3 Elementary Uses of the Functors . . . . . . . 35
2.5 Local Isomorphism . . . . . . . . . . . . . . . . . . . 39
2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.7 Questions for Future Research . . . . . . . . . . . . . 50
3 Local Isomorphism is Isomorphism 53
3.1 Integrally Closed Rings . . . . . . . . . . . . . . . . . 53
3.2 The Conductor of an Rtffr Ring . . . . . . . . . . . . 58
3.3 A Local Correspondence . . . . . . . . . . . . . . . . 62
3.4 A Canonical Decomposition . . . . . . . . . . . . . . 67
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3.5 Arnold¿s Theorem . . . . . . . . . . . . . . . . . . . . 69
3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.7 Questions for Future Research . . . . . . . . . . . . . 74
4 Commuting Endomorphisms 77
4.1 Nilpotent sets . . . . . . . . . . . . . . . . . . . . . . 77
4.2 Commutative Rtffr Rings . . . . . . . . . . . . . . . . 88
4.2.1 Modules over Commutative Rings . . . . . . . 88
4.2.2 Projectives over Commutative Rings . . . . . 92
4.2.3 Direct Sums Over Commutative Rings . . . . 94
4.2.4 Commutative Endomorphism Rings . . . . . . 95
4.3 E-Properties . . . . . . . . . . . . . . . . . . . . . . . 99
4.4 Square-free Ranks . . . . . . . . . . . . . . . . . . . . 103
4.4.1 Rank Two Groups . . . . . . . . . . . . . . . 104
4.4.2 Groups of Rank 3 . . . . . . . . . . . . . . 106
4.5 Refinement and Square-free Rank . . . . . . . . . . . 110
4.6 Hereditary Endomorphism Rings . . . . . . . . . . . 113
4.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.8 Questions for Future Research . . . . . . . . . . . . . 115
5 Refinement Revisited 119
5.1 Counting Isomorphism Classes . . . . . . . . . . . . . 120
5.1.1 Class Groups and Class Numbers . . . . . . . 120
5.1.2 The Class Group of the Integral Closure . . . 122
5.1.3 Class Number and Refinement . . . . . . . . . 125
5.1.4 Quadratic Number Fields . . . . . . . . . . . 129
5.1.5 Counting Theorems . . . . . . . . . . . . . . . 130
5.2 Integrally Closed Groups . . . . . . . . . . . . . . . . 136
5.2.1 A Review Of Notation and Terminology . . . 137
5.2.2 Locally Semi-perfect Rings . . . . . . . . . . . 138
5.2.3 Semi-primary Rtffr Groups . . . . . . . . . . . 141
5.2.4 Applications to Refinement . . . . . . . . . . 145
5.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 148
5.4 Questions for Future Research . . . . . . . . . . . . . 150
6 The Baer Splitting Property 153
6.1 Baer¿s Lemma . . . . . . . . . . . . . . . . . . . . . . 154
6.2 The Splitting of Exact Sequences . . . . . . . . . . . 158
6.3 G-Compressed Projectives . . . . . . . . . . . . . . . 161CONTENTS xiii
6.4 Some Examples . . . . . . . . . . . . . . . . . . . . . 166
6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.6 Questions for Future Research . . . . . . . . . . . . . 170
7 J -groups, L-groups, and S-groups 171
7.1 Background on Ext . . . . . . . . . . . . . . . . . . . 171
7.2 Finite Projective Properties . . . . . . . . . . . . . . 173
7.3 Finitely Projective Groups . . . . . . . . . . . . . . . 175
7.4 Finitely Faithful S-groups . . . . . . . . . . . . . . . 179
7.5 Isomorphism versus Local Isomorphism . . . . . . . . 182
7.6 Some Analytic Number Theory . . . . . . . . . . . . 184
7.7 Rtffr L-Groups are J -Groups . . . . . . . . . . . . . 190
7.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 192
7.9 Questions for Future Research . . . . . . . . . . . . . 193
8 Gabriel Filters 195
8.1 Filters of Divisibility . . . . . . . . . . . . . . . . . . 195
8.1.1 Hereditary Torsion Classes . . . . . . . . . . . 197
8.1.2 Gabriel Filters of Right Ideals . . . . . . . . . 200
8.2 Idempotent Ideals . . . . . . . . . . . . . . . . . . . . 204
8.2.1 Traces of Covers . . . . . . . . . . . . . . . . 205
8.2.2 Bounded Gabriel Filters . . . . . . . . . . . . 210
8.2.3 Finite Filters of Divisibility . . . . . . . . . . 212
8.3 Gabriel Filters on Rtffr Rings . . . . . . . . . . . . . 217
8.3.1 Applications to Endomorphism Rings . . . . . 217
8.3.2 Constructing Examples . . . . . . . . . . . . . 220
8.3.3 Faithful Eings . . . . . . . . . . . . . . . . . . 222
8.4 Gabriel Filters on QEnd(G) . . . . . . . . . . . . . . 225
8.4.1 Central Quasi-summands . . . . . . . . . . . . 225
8.4.2 Q-faithful Groups . . . . . . . . . . . . . . . . 231
8.4.3 Q-faithful E-flat Groups . . . . . . . . . . . . 234
8.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 235
8.6 Questions for Future Research . . . . . . . . . . . . . 239
9 Endomorphism Modules 241
9.1 Additive Structure of Rings . . . . . . . . . . . . . . 241
9.2 E-Properties . . . . . . . . . . . . . . . . . . . . . . . 244
9.2.1 E-Rings . . . . . . . . . . . . . . . . . . . . . 244
9.2.2 E-Finitely Generated Groups . . . . . . . . . 247
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9.2.3 E-Projective Groups . . . . . . . . . . . . . . 253
9.2.4 E-Generator Groups . . . . . . . . . . . . . . 257
9.2.5 E-Projective Rtffr Groups Characterized . . . 259
9.2.6 Noetherian Endomorphism Modules . . . . . . 262
9.3 Homological Dimensions . . . . . . . . . . . . . . . . 266
9.3.1 E-Projective Dimensions . . . . . . . . . . . . 267
9.3.2 E-Flat Dimensions . . . . . . . . . . . . . . . 270
9.3.3 E-Injective Dimensions . . . . . . . . . . . . . 273
9.4 Self-injective Rings . . . . . . . . . . . . . . . . . . . 276
9.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 278
9.6 Questions for Future Research . . . . . . . . . . . . . 280
A Pathological Direct Sums 283
A.1 Nonunique Direct Sums . . . . . . . . . . . . . . . . 283
B An Acd Group Example 287
B.1 An Example by Corner . . . . . . . . . . . . . . . . . 287
C Power Cancellation 291
C.1 Failure of Power Cancellation . . . . . . . . . . . . . 291
D Cancellation 293
D.1 Failure of Cancellation . . . . . . . . . . . . . . . . . 293
E Corner Rings and Modules 297
E.1 Topological Preliminaries . . . . . . . . . . . . . . . . 297
E.2 The Construction of G . . . . . . . . . . . . . . . . . 299
E.3 Endomorphisms of G . . . . . . . . . . . . . . . . . . 301
F Corner¿s Theorem 305
F.1 Countable Rings as Endomorphism Rings . . . . . . . 305
G Torsion Torsion-free Groups 307
G.1 E-Torsion Groups . . . . . . . . . . . . . . . . . . . . 307
G.2 Self-small Corner Modules . . . . . . . . . . . . . . . 308
H E-flat Groups 311
H.1 Ubiquity . . . . . . . . . . . . . . . . . . . . . . . . . 311
H.2 Unfaithful Groups . . . . . . . . . . . . . . . . . . . . 312CONTENTS xv
I Zassenhaus and Butler 317
I.1 The Statement . . . . . . . . . . . . . . . . . . . . . 317
I.2 The Proof . . . . . . . . . . . . . . . . . . . . . . . . 318
J Countable E-Rings 323
J.1 Countable Torsion-free E-rings . . . . . . . . . . . . 323
K Dedekind E-Rings 329
K.1 Number Theoretic Preliminaries . . . . . . . . . . . . 329
K.2 Integrally Closed Rings . . . . . . . . . . . . . . . . . 330