Factor categories with applications to direct decomposition of modules:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York u.a.
Dekker
1983
|
| Schriftenreihe: | Lecture notes in pure and applied mathematics
88 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | IX, 328 S. |
| ISBN: | 0824718976 |
Internformat
MARC
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Datensatz im Suchindex
| _version_ | 1804114702518714368 |
|---|---|
| adam_text | Contents
PREFACE iii
1 PRELIMINARIES 1
1.1 Definitions Concerning Groups 1
1.2 Subgroups and Factor Groups 2
1.3 Direct Products 4
1.4 Composition Series 6
1.5 Zorn s Lemma 7
1.6 Definitions Concerning Rings 7
1.7 R Modules 9
2 CATEGORIES 13
2.1 Definitions Concerning Categories 14
2.2 Abelian Categories 25
2.3 Functors 27
2.4 Grothendieck Categories 28
2.5 The Category M 32
— K
3 GENERAL MODULES AND RINGS 34
3.1 Idempotents 54
3.2 Relatively Prime Ideals (Comaximal Ideals) 38
3.3 Jacobson Radical 39
3.4 Small Submodules and Essential Submodules 42
3.5 Limits 45
3.6 The Functors Hora and ® 48
3.7 Matrix Rings 53
4 SPECIAL MODULES AND RINGS 56
4.1 Semisimple Modules 57
4.2 Chain Conditions 58
4.3 Projective Modules and Fiat Modules 61
4.4 Injective Modules 71
4.5 Artinian Rings 75
4.6 Lifting Idempotents 83
4.7 Commutative Rings 85
viii Contents
5 MORITA CONTEXTS 100
5.1 Morita Contexts: General 100
5.2 Generators 101
5.3 Equivalences 103
5.4 Quasi injective Modules 107
5.5 Endomorphism Rings of Quasi injective Modules 111
5.6 Dualities 126
5.7 Quasi Frobenius Rings 129
5.8 QF 1, 2, 3 Rings 133
6 CATEGORIES OF INDECOMPOSABLE MODULES 137
6.1 Semiperfect Modules 138
6.2 Factor Categories 144
6.3 Categories of Indecomposable Modules 147
6.4 Categories of Injective Modules 151
6.5 Categories of Projective Modules 153
6.6 Ring Theoretical Proof 155
7 APPLICATIONS I: KRULL REMAK SCHMIDT AZUMAYA THEOREM 158
7.1 Krull Remak Schmidt Azumaya Theorem 159
7.2 Semi T nilpotency 161
7.3 Füll Submodules 166
7.4 Small Submodules in Projective Modules 173
7.5 Perfect Modules 177
7.6 Perfect Rings 180
7.7 Goldie Dimension 184
7.8 Ring Theoretical Proofs 188
8 APPLICATIONS II: EXCHANGE PROPERTY 195
8.1 Exchange Properties in Modules 195
8.2 Exchange Properties in A 198
8.3 Direct Summands with the Exchange Property 202
8.4 Direct Summands with the Xg Exchange Property 209
8.5 Modules with the Exchange Property 215
8.6 Equivalence Refinements 224
8.7 Ring Theoretical Proof 225
9 LIFTING PROPERTY AND EXTENDING PROPERTY 230
9.1 Definitions Concerning the Lifting Property
Modulo Radical 231
9.2 Lifting Property of Simple Modules Modulo Radical 233
9.3 Lifting Property of Direct Decompositions Modulo
Radical 238
9.4 Modules with the Lifting Property Modulo Radical 241
9.5 Rings with the Lifting Property 247
9.6 Extending Property of Simple Modules 249
9.7 Extending Property of Decompositions 255
9.8 Modules with the Extending Property 259
9.9 Quasi injective Modules and the Extending Property 261
9.10 Rings with the Extending Property 262
9.11 Lifting Property and Extending Property over
Commutative Rings 265
Contents ix
10 LIFTING PROPERTY FOR INJECTIVES AND EXTENDING PROPERTY
FOR PROJECTIVES 276
10.1 Functor Z* 276
10.2 Condition (*) 279
10.3 Right Artinian Rings with (*) 282
10.4 Lifting Property of Simple Modules for Injectives 285
10.5 Max. Quasi projective Modules 287
10.6 Condition (**) 292
10.7 Extending Property of Simple Modules for
Projectives 294
10.8 Mini injective Modules 297
10.9 Extending Property of Direct Decompositions of
the Socle for Projectives 298
10.10 QF Rings and Lifting (Extending) Property 303
NOTES AND REFERENCES 309
Notes 309
References 311
NOTATION 323
INDEX 325
|
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| id | DE-604.BV000239690 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T15:10:55Z |
| institution | BVB |
| isbn | 0824718976 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-000143961 |
| oclc_num | 9761753 |
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| physical | IX, 328 S. |
| publishDate | 1983 |
| publishDateSearch | 1983 |
| publishDateSort | 1983 |
| publisher | Dekker |
| record_format | marc |
| series | Lecture notes in pure and applied mathematics |
| series2 | Lecture notes in pure and applied mathematics |
| spelling | Harada, Manabu Verfasser aut Factor categories with applications to direct decomposition of modules New York u.a. Dekker 1983 IX, 328 S. txt rdacontent n rdamedia nc rdacarrier Lecture notes in pure and applied mathematics 88 Categorieën (wiskunde) gtt Catégories (Mathématiques) Catégories (mathématiques) ram Module Modulen (wiskunde) gtt Modules (Algèbre) Modules (algèbre) ram Théorie catégorie Théorème Krull-Remak-Schmidt-Azumaya Categories (Mathematics) Modules (Algebra) Dekomposition (DE-588)4149030-7 gnd rswk-swf Modul (DE-588)4129770-2 gnd rswk-swf Kategorie Mathematik (DE-588)4129930-9 gnd rswk-swf Kategorie Mathematik (DE-588)4129930-9 s Modul (DE-588)4129770-2 s Dekomposition (DE-588)4149030-7 s DE-604 Lecture notes in pure and applied mathematics 88 (DE-604)BV005871364 88 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000143961&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Harada, Manabu Factor categories with applications to direct decomposition of modules Lecture notes in pure and applied mathematics Categorieën (wiskunde) gtt Catégories (Mathématiques) Catégories (mathématiques) ram Module Modulen (wiskunde) gtt Modules (Algèbre) Modules (algèbre) ram Théorie catégorie Théorème Krull-Remak-Schmidt-Azumaya Categories (Mathematics) Modules (Algebra) Dekomposition (DE-588)4149030-7 gnd Modul (DE-588)4129770-2 gnd Kategorie Mathematik (DE-588)4129930-9 gnd |
| subject_GND | (DE-588)4149030-7 (DE-588)4129770-2 (DE-588)4129930-9 |
| title | Factor categories with applications to direct decomposition of modules |
| title_auth | Factor categories with applications to direct decomposition of modules |
| title_exact_search | Factor categories with applications to direct decomposition of modules |
| title_full | Factor categories with applications to direct decomposition of modules |
| title_fullStr | Factor categories with applications to direct decomposition of modules |
| title_full_unstemmed | Factor categories with applications to direct decomposition of modules |
| title_short | Factor categories with applications to direct decomposition of modules |
| title_sort | factor categories with applications to direct decomposition of modules |
| topic | Categorieën (wiskunde) gtt Catégories (Mathématiques) Catégories (mathématiques) ram Module Modulen (wiskunde) gtt Modules (Algèbre) Modules (algèbre) ram Théorie catégorie Théorème Krull-Remak-Schmidt-Azumaya Categories (Mathematics) Modules (Algebra) Dekomposition (DE-588)4149030-7 gnd Modul (DE-588)4129770-2 gnd Kategorie Mathematik (DE-588)4129930-9 gnd |
| topic_facet | Categorieën (wiskunde) Catégories (Mathématiques) Catégories (mathématiques) Module Modulen (wiskunde) Modules (Algèbre) Modules (algèbre) Théorie catégorie Théorème Krull-Remak-Schmidt-Azumaya Categories (Mathematics) Modules (Algebra) Dekomposition Modul Kategorie Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000143961&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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