Lectures on modules and rings:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
1999
|
| Series: | Graduate texts in mathematics
189 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 543 - 548 |
| Physical Description: | XXI, 557 S. graph. Darst. |
| ISBN: | 0387984283 9780387984285 |
Staff View
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Record in the Search Index
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| adam_text | T. Y. LAM LECTURES ON MODULES AND RINGS WITH 43 FIGURES CONTENTS PREFACE
VII NOTES TO THE READER XVII PARTIAL LIST OF NOTATIONS XIX PARTIAL LIST
OF ABBREVIATIONS XXIII 1 FREE MODULES, PROJECTIVE, AND INJECTIVE MODULES
1 1. FREE MODULES 2 1A. INVARIANT BASIS NUMBER (IBN) 2 IB. STABLE
FINITENESS 5 1C. THE RANK CONDITION 9 ID. THE STRONG RANK CONDITION 12
IE. SYNOPSIS 16 EXERCISES FOR §1 17 2. PROJECTIVE MODULES 21 2A. VBASIC
DEFINITIONS AND EXAMPLES 21 2B. DUAL BASIS LEMMA AND INVERTIBLE MODULES
23 2C. INVERTIBLE FRACTIONAL IDEALS 30 2D. THE PICARD GROUP OF A
COMMUTATIVE RING 34 2E. HEREDITARY AND SEMIHEREDITARY RINGS 42 2F. CHASE
SMALL EXAMPLES 45 2G. HEREDITARY ARTINIAN RINGS 48 2H. TRACE IDEALS 51
EXERCISES FOR §2 54 3. INJECTIVE MODULES 60 3A. BAER S TEST FOR
INJECTIVITY 60 3B. SELF-INJECTIVE RINGS 64 3C. INJECTIVITY VERSUS
DIVISIBILITY 69 XII CONTENTS 3D. ESSENTIAL EXTENSIONS AND INJECTIVE
HULLS 74 3E. INJECTIVES OVER RIGHT NOETHERIAN RINGS 80 3F.
INDECOMPOSABLE INJECTIVES AND UNIFORM MODULES 83 3G. INJECTIVES OVER
SOME ARTINIAN RINGS 90 3H. SIMPLE INJECTIVES 96 31. MATLIS THEORY 99
3J. SOME COMPUTATIONS OF INJECTIVE HULLS 105 3K. APPLICATIONS TO CHAIN
CONDITIONS 110 EXERCISES FOR §3 113 2 FLAT MODULES AND HOMOLOGICAL
DIMENSIONS 121 4. FLAT AND FAITHFULLY FLAT MODULES 122 4A. BASIC
PROPERTIES AND FLATNESS TESTS 122 4B. FLATNESS, TORSION-FREENESS, AND
VON NEUMANN REGULARITY 127 4C. MORE FLATNESS TESTS 129 4D. FINITELY
PRESENTED (F.P.) MODULES 131 4E. FINITELY GENERATED FLAT MODULES 135 4F.
DIRECT PRODUCTS OF HAT MODULES 136 4G. COHERENT MODULES AND COHERENT
RINGS 140 4H. SEMIHEREDITARY RINGS REVISITED 144 41. FAITHFULLY FLAT
MODULES 147 4J. PURE EXACT SEQUENCES 153 EXERCISES FOR §4 159 5.
HOMOLOGICAL DIMENSIONS 165 5A. SCHANUEL S LEMMA AND PROJECTIVE
DIMENSIONS 165 5B. CHANGE OF RINGS 173 5C. INJECTIVE DIMENSIONS 177 5D.
WEAK DIMENSIONS OF RINGS 182 5E. GLOBAL DIMENSIONS OF SEMIPRIMARY RINGS
187 5F. GLOBAL DIMENSIONS OF LOCAL RINGS 192 5G. GLOBAL DIMENSIONS OF
COMMUTATIVE NOETHERIAN RINGS 198 EXERCISES FOR §5 201 3 MORE THEORY OF
MODULES 207 6. UNIFORM DIMENSIONS, COMPLEMENTS, AND CS MODULES 208 6A.
BASIC DEFINITIONS AND PROPERTIES 208 6B. COMPLEMENTS AND CLOSED
SUBMODULES 214 6C. EXACT SEQUENCES AND ESSENTIAL CLOSURES 219 6D. CS
MODULES: TWO APPLICATIONS 221 6E. FINITENESS CONDITIONS ON RINGS 228 6F.
CHANGE OF RINGS 232 6G. QUASI-INJECTIVE MODULES 236 EXERCISES FOR §6 241
CONTENTS XIII 7. SINGULAR SUBMODULES AND NONSINGULAR RINGS 246 7A. BASIC
DEFINITIONS AND EXAMPLES 246 7B. NILPOTENCY OF THE RIGHT SINGULAR IDEAL
252 7C. GOLDIE CLOSURES AND THE REDUCED RANK 253 7D. BAER RINGS AND
RICKART RINGS 260 7E. APPLICATIONS TO HEREDITARY AND SEMIHEREDITARY
RINGS 265 EXERCISES FOR §7 268 8. DENSE SUBMODULES AND RATIONAL HULLS
272 8A. BASIC DEFINITIONS AND EXAMPLES 272 8B. RATIONAL HULL OF A MODULE
275 8C. RIGHT KASCH RINGS 280 EXERCISES FOR §8 284 4 RINGS OF QUOTIENTS
287 9. NONCOMMUTATIVE LOCALIZATION 288 9A. THE GOOD 288 9B. THE BAD
290 9C. THE UGLY 294 9D. AN EMBEDDING THEOREM OF A. ROBINSON 297
EXERCISES FOR §9 298 10. CLASSICAL RINGS OF QUOTIENTS 299 10A. ORE
LOCALIZATIONS 299 10B. RIGHT ORE RINGS AND DOMAINS 303 IOC. POLYNOMIAL
RINGS AND POWER SERIES RINGS 308 10D. EXTENSIONS AND CONTRACTIONS 314
EXERCISES FOR § 10 317 11. RIGHT GOLDIE RINGS AND GOLDIE S THEOREMS 320
11 A. EXAMPLES OF RIGHT ORDERS 320 1 IB. RIGHT ORDERS IN SEMISIMPLE
RINGS 323 11C. SOME APPLICATIONS OF GOLDIE S THEOREMS 331 11D. SEMIPRIME
RINGS 334 HE. NIL MULTIPLICATIVELY CLOSED SETS 339 EXERCISES FOR §11 342
12. ARTINIAN RINGS OF QUOTIENTS 345 12A. GOLDIE S P-RANK L 345 12B.
RIGHT ORDERS IN RIGHT ARTINIAN RINGS 347 12C. THE COMMUTATIVE CASE 351
12D. NOETHERIAN RINGS NEED NOT BE ORE 354 EXERCISES FOR §12 355 5 MORE
RINGS OF QUOTIENTS 357 13. MAXIMAL RINGS OF QUOTIENTS 358 XIV CONTENTS
13 A. ENDOMORPHISM RING OF A QUASI-INJECTIVE MODULE 358 13B.
CONSTRUCTION OF Q RMAX (R) 365 13C. ANOTHER DESCRIPTION OF Q RMM (R) 369
13D. THEOREMS OF JOHNSON AND GABRIEL 374 EXERCISES FOR §13 380 14.
MARTINDALE RINGS OF QUOTIENTS 383 14A. SEMIPRIME RINGS REVISITED 383
14B. THE RINGS Q R (R) AND Q (R) 384 14C. THE EXTENDED CENTROID 389 14D.
CHARACTERIZATIONS OF Q R (R) AND Q (R) 392 14E. X-INNER AUTOMORPHISMS
394 14F. A MATRIX RING EXAMPLE 401 EXERCISES FOR § 14 403 6 FROBENIUS
AND QUASI-FROBENIUS RINGS 407 15. QUASI-FROBENIUS RINGS 408 15 A. BASIC
DEFINITIONS OF QF RINGS 408 15B. PROJECTIVES AND INJECTIVES 412 15C.
DUALITY PROPERTIES 414 15D. COMMUTATIVE QF RINGS, AND EXAMPLES 417
EXERCISES FOR §15 420 16. FROBENIUS RINGS AND SYMMETRIC ALGEBRAS 422
16A. THE NAKAYAMA PERMUTATION 422 16B. DEFINITION OF A FROBENIUS RING
427 16C. FROBENIUS ALGEBRAS AND QF ALGEBRAS 431 16D. DIMENSION
CHARACTERIZATIONS OF FROBENIUS ALGEBRAS 434 16E. THE NAKAYAMA
AUTOMORPHISM 438 16F. SYMMETRIC ALGEBRAS 441 16G. WHY FROBENIUS? 450
EXERCISES FOR §16 453 7 MATRIX RINGS, CATEGORIES OF MODULES, AND MORITA
THEORY 459 17. MATRIX RINGS 461 17A. CHARACTERIZATIONS AND EXAMPLES 461
17B. FIRST INSTANCE OF MODULE CATEGORY EQUIVALENCES 470 17C. UNIQUENESS
OF THE COEFFICIENT RING 473 EXERCISES FOR §17 478 18. MORITA THEORY OF
CATEGORY EQUIVALENCES 480 18A. CATEGORICAL PROPERTIES 480 18B.
GENERATORS AND PROGENERATORS 483 18C. THE MORITA CONTEXT 485 18D. MORITA
I, II, III 488 CONTENTS XV 18E. CONSEQUENCES OF THE MORITA THEOREMS 490
18F. THE CATEGORY A[M] 496 EXERCISES FOR § 18 501 19. MORITA DUALITY
THEORY 505 19A. FINITE COGENERATION AND COGENERATORS 505 19B.
COGENERATOR RINGS 510 19C. CLASSICAL EXAMPLES OF DUALITIES 515 19D.
MORITA DUALITIES: MORITA I 518 19E. CONSEQUENCES OF MORITA I 522 19F.
LINEAR COMPACTNESS AND REFLEXIVITY 527 19G. MORITA DUALITIES: MORITA II
534 EXERCISES FOR §19 537 REFERENCES 543 NAME INDEX 549 SUBJECT INDEX
553
|
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| isbn | 0387984283 9780387984285 |
| language | English |
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| spelling | Lam, Tsit-Yuen Verfasser aut Lectures on modules and rings T. Y. Lam New York [u.a.] Springer 1999 XXI, 557 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 189 Literaturverz. S. 543 - 548 Modul Ring <Mathematik> Modules (Algebra) Rings (Algebra) Modul (DE-588)4129770-2 gnd rswk-swf Ring Mathematik (DE-588)4128084-2 gnd rswk-swf Modul (DE-588)4129770-2 s DE-604 Ring Mathematik (DE-588)4128084-2 s Graduate texts in mathematics 189 (DE-604)BV000000067 189 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008408954&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lam, Tsit-Yuen Lectures on modules and rings Graduate texts in mathematics Modul Ring <Mathematik> Modules (Algebra) Rings (Algebra) Modul (DE-588)4129770-2 gnd Ring Mathematik (DE-588)4128084-2 gnd |
| subject_GND | (DE-588)4129770-2 (DE-588)4128084-2 |
| title | Lectures on modules and rings |
| title_auth | Lectures on modules and rings |
| title_exact_search | Lectures on modules and rings |
| title_full | Lectures on modules and rings T. Y. Lam |
| title_fullStr | Lectures on modules and rings T. Y. Lam |
| title_full_unstemmed | Lectures on modules and rings T. Y. Lam |
| title_short | Lectures on modules and rings |
| title_sort | lectures on modules and rings |
| topic | Modul Ring <Mathematik> Modules (Algebra) Rings (Algebra) Modul (DE-588)4129770-2 gnd Ring Mathematik (DE-588)4128084-2 gnd |
| topic_facet | Modul Ring <Mathematik> Modules (Algebra) Rings (Algebra) Ring Mathematik |
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