Dimensions of ring theory:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Dordrecht u.a.
Reidel
1987
|
| Series: | Mathematics and its applications
36 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XI, 360 S. |
| ISBN: | 902772461X |
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MARC
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Record in the Search Index
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|---|---|
| adam_text | Contents
SERIES EDITOR S PREFACE ix
ACKNOWLEDGMENT xi
INTRODUCTION 1
CHAPTER 1. Finiteness Conditions for Lattices 3
1.1. Lattices 3
1.2. Noetherian and Artinian Lattices 5
1.3. Lattices of Finite Length , 7
1.4. Irreducible Elements in a Lattice 10
1.5. Goldie Dimension of a Modular Lattice 11
1.6. Goldie Dimension and Chain Conditions for Modular Lattices with
Finite Group Actions 20
1.7. Complements and Pseudo Complements 24
1.8. Semiatomic Lattices and Compactly Generated Lattices 26
1.9. Semiartinian Lattices 29
1.10. Indecomposable Elements in a Lattice 31
1.11. Exercises 33
Bibliographical Comments to Chapter 1 36
CHAPTER 2. Finiteness Conditions for Modules 37
2.1. Modules 37
2.2. The Lattice of Submodules of a Module 44
2.3. Noetherian and Artinian Modules 47
Vj Contents
2.4. Modules of Finite Length 49
2.5. Semisimple Modules 51
2.6. Semisimple and Simple Artinian Rings 53
2.7. The Jacobson Radical and the Prime Radical of a Ring 56
2.8. Rings of Fractions. Goldie s Theorems 60
2.9. Artinian Modules which are Noetherian 67
2.10. Projective and Injective Modules 69
2.11. Tensor Product and Flat Modules 76
2.12. Normalizing Extensions of a Ring 83
2.13. Graded Rings and Modules 88
2.14. Graded Rings and Modules of Type TL. Internal Homogenisation 92
2.15. Noetherian Modules over Graded Rings of Type 7L. Applications 94
2.16. Strongly Graded Rings and Clifford Systems for Finite Groups 98
2.17. Invariants of a Finite Group Action 109
2.18. Exercises 112
Bibliographical Comments to Chapter 2 120
CHAPTER 3. Krull Dimension and Gabriel Dimension of an Ordered Set 121
3.1. Definitions and Basic Properties 121
3.2. The Krull Dimension of a Modular Lattice 126
3.3. Critical Composition Series of a Lattice 131
3.4. The Gabriel Dimension of a Modular Lattice 134
3.5. Comparison of Krull and Gabriel Dimension 137
3.6. Exercises 139
Bibliographical Comments to Chapter 3 141
CHAPTER 4. Krull Dimension and Gabriel Dimension of Rings and Modules.. 143
4.1. Definitions and Generalities 143
4.2. Krull and Gabriel Dimension of Some Special Classes of Rings and
Modules 146
4.2.1. The Ring of Endomorphisms of a Projective Finitely
Generated Module 146
4.2.2. Normalizing Extensions 147
4.2.3. Rings Strongly Graded by a Finite Group 151
4.2.4. The Ring of Invariants 152
4.2.5. Graded Rings of Type 2Z 152
4.2.6. Filtered Rings and Modules 156
Contents vii
4.2.7. Ore and Skew Laurent Extensions 158
4.2.8. Affine P.I. Algebras.(Addendum) 165
4.3. Exercises 168
Bibliographical Comments to Chapter 4 170
CHAPTER 5. Rings with Krull Dimension 171
5.1. Nil Ideals 171
5.2. Semiprime Rings with Krull Dimension 173
5.3. Classical Krull Dimension of a Ring 177
5.4. Associated prime Ideals 178
5.5. Fully Left Bounded Rings with Krull Dimension 182
5.6. Examples of Noetherian Rings of Arbitrary Krull Dimension 185
5.7. Exercises 188
Bibliographical Comments to Chapter 5 190
CHAPTER 6. Krull Dimension of Noetherian Rings. The Principal Ideal
Theorem 191
6.1. Fully Left Bounded Left Noetherian Rings 191
6.2. The Reduced Rank of a Module 193
6.3. Noetherian Rings Satisfying Condition H 195
6.4. Fully Bounded Noetherian Rings 199
6.5. Krull Dimension and Invertible Ideals in a Noetherian Ring 207
6.6. The Principal Ideal Theorem 210
6.7. Exercises 214
Bibliographical Comments to Chapter 6 215
CHAPTER 7. Relative Krull and Gabriel Dimensions 217
7.1. Additive Topologies and Torsion Theories 217
7.2. The Lattices C7(M) and CjJ(M) 221
7.3. Relative Krull Dimension 230
7.4. Relative Krull Dimension Applied to the Principal Ideal Theorem 235
7.5. Relative Gabriel Dimension 239
7.6. Relative Krull and Gabriel Dimensions of Graded Rings 244
7.7. Exercises 246
Bibliographical Comments to Chapter 7 247
viii Contents
CHAPTER 8. Homological Dimensions 249
8.1. The Projective Dimension of a Module 249
8.2. Homological Dimension of Polynomial Rings and Rings of Formal
Power Series 259
8.3. Injective Dimension of a Module 262
8.4. The Flat Dimension of a Module 274
8.5. The Artin Rees Property and Homological Dimensions 281
8.6. Regular Local Rings 285
8.7. Exercises 288
Bibliographical Comments to Chapter 8 291
CHAPTER 9. Rings of Finite Global Dimension 293
9.1. The Zariski Topology. 293
9.2. The Local Study of Homological Dimension 295
9.3. Rings Integral over their Centres 297
9.4. Commutative Rings of Finite Global Dimension 302
9.5. Exercises 307
Bibliographical Comments to Chapter 9 312
CHAPTER 10. The Gelfand Kirillov Dimension 313
10.1. Definitions and Basic Properties 314
10.2. GK dimension of Filtered and Graded Algebras 325
10.3. Applications to Special Classes of Rings 328
10.3.1. Rings of Differential Operators and Weyl Algebras 328
10.3.2. Remarks on Enveloping Algebras of Lie Algebras.(Addendum). 332
10.3.3. P.I.Algebras.(Addendum) 335
10.4. Exercises 339
Bibliographical Comments to Chapter 10 341
REFERENCES 343
INDEX 355
|
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| illustrated | Not Illustrated |
| indexdate | 2024-07-09T15:18:19Z |
| institution | BVB |
| isbn | 902772461X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-000449014 |
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| physical | XI, 360 S. |
| publishDate | 1987 |
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| publisher | Reidel |
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| series | Mathematics and its applications |
| series2 | Mathematics and its applications |
| spelling | Năstăsescu, Constantin Verfasser aut Dimensions of ring theory by Constantin Năstăsescu and Freddy van Oystaeyen Dordrecht u.a. Reidel 1987 XI, 360 S. txt rdacontent n rdamedia nc rdacarrier Mathematics and its applications 36 Ring Mathematik (DE-588)4128084-2 gnd rswk-swf Algebraischer Ring (DE-588)4141855-4 gnd rswk-swf Dimensionstheorie (DE-588)4149935-9 gnd rswk-swf Ringtheorie (DE-588)4126571-3 gnd rswk-swf Dimensionstheorie (DE-588)4149935-9 s Ring Mathematik (DE-588)4128084-2 s DE-604 Ringtheorie (DE-588)4126571-3 s Algebraischer Ring (DE-588)4141855-4 s Van Oystaeyen, Freddy 1947- Verfasser (DE-588)128792884 aut Mathematics and its applications 36 (DE-604)BV008163334 36 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000449014&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Năstăsescu, Constantin Van Oystaeyen, Freddy 1947- Dimensions of ring theory Mathematics and its applications Ring Mathematik (DE-588)4128084-2 gnd Algebraischer Ring (DE-588)4141855-4 gnd Dimensionstheorie (DE-588)4149935-9 gnd Ringtheorie (DE-588)4126571-3 gnd |
| subject_GND | (DE-588)4128084-2 (DE-588)4141855-4 (DE-588)4149935-9 (DE-588)4126571-3 |
| title | Dimensions of ring theory |
| title_auth | Dimensions of ring theory |
| title_exact_search | Dimensions of ring theory |
| title_full | Dimensions of ring theory by Constantin Năstăsescu and Freddy van Oystaeyen |
| title_fullStr | Dimensions of ring theory by Constantin Năstăsescu and Freddy van Oystaeyen |
| title_full_unstemmed | Dimensions of ring theory by Constantin Năstăsescu and Freddy van Oystaeyen |
| title_short | Dimensions of ring theory |
| title_sort | dimensions of ring theory |
| topic | Ring Mathematik (DE-588)4128084-2 gnd Algebraischer Ring (DE-588)4141855-4 gnd Dimensionstheorie (DE-588)4149935-9 gnd Ringtheorie (DE-588)4126571-3 gnd |
| topic_facet | Ring Mathematik Algebraischer Ring Dimensionstheorie Ringtheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000449014&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV008163334 |
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