In the past 15 years, the theory of crossed products has enjoyed a period of vigorous development. The foundations have been strengthened and reorganized from new points of view, especially from the viewpoint of graded rings. The purpose of this monograph is to give, in a self-contained manner, an up-to-date account of various aspects of this development, in an effort to convey a comprehensive picture of the current state of the subject. It is assumed that the reader has had the equivalent of a standard first-year graduate course, thus familiarity with basic ring-theoretic and group-theoretic concepts and an understanding of elementary properties of modules, tensor products and fields. A chapter on algebraic preliminaries is included, which briefly surveys topics required later in the book.
Author(s): Gregory Karpilovsky, Leopoldo Nachbin
Series: Notas De Matematica, 118
Publisher: Elsevier Science Ltd
Year: 1987
Language: English
Pages: 359
The Algebraic Structure of Crossed Products......Page 4
Copyright Page......Page 5
Preface......Page 8
Contents......Page 10
1. Notation and terminology......Page 12
2. Projective, injective and flat modules......Page 22
3. Artinian and Noetherian modules......Page 30
4. Group actions......Page 36
5. Cohomology groups and group extensions......Page 39
6. Some properties of cohomology groups......Page 52
7. Matrix rings and related results......Page 57
1. Definitions and elementary properties......Page 70
2. Equivalent crossed products......Page 82
3. Some ring–theoretic results......Page 89
4. The centre of crossed products over simple rings......Page 103
5. Projective crossed representations......Page 114
6. Graded and G–invariant ideals......Page 130
7. Induced modules......Page 136
8. Montgomery's theorem......Page 158
1. Central simple algebras......Page 162
2. The Brauer group......Page 172
3. Classical crossed products and the Brauer group......Page 180
1. Graded modules......Page 192
2. Restriction to A1......Page 197
3. Graded homomorphism modules......Page 202
4. Extension from A1......Page 207
5. Induction from A1......Page 216
1. Primitive, prime and semiprime ideals......Page 234
2. Primitive ideals in crossed products......Page 236
3. Prime coefficient rings......Page 241
4. Incomparability and Going Down......Page 257
5. A Going Up Theorem......Page 270
6. Chains of prime and primitive ideals......Page 278
1. Coset calculus......Page 288
2. Δ–methods......Page 296
3. The main theorem and its applications......Page 305
4. Sufficient conditions for semiprimeness......Page 316
5. Twisted group algebras......Page 322
Bibiliography......Page 342
Notation......Page 350
Index......Page 356
