Algebraic Structure of Crossed Products (eBook)

Front Cover 1
The Algebraic Structure of Crossed Products 4
Copyright Page 5
Contents 10
Preface 8
CHAPTER 1. PRELIMINARIES 12
1. Notation and terminology 12
2. Projective, injective and flat modules 22
3. Artinian and Noetherian modules 30
4. Group actions 36
5. Cohomology groups and group extensions 39
6. Some properties of cohomology groups 52
7. Matrix rings and related results 57
CHAPTER 2. GROUP-GRADED ALGEBRAS AND CROSSED PRODUCTS: GENERAL THEORY 70
1. Definitions and elementary properties 70
2. Equivalent crossed products 82
3. Some ring–theoretic results 89
4. The centre of crossed products over simple rings 103
5. Projective crossed representations 114
6. Graded and G–invariant ideals 130
7. Induced modules 136
8. Montgomery's theorem 158
CHAPTER 3. THE CLASSICAL THEORY OF CROSSED PRODUCTS 162
1. Central simple algebras 162
2. The Brauer group 172
3. Classical crossed products and the Brauer group 180
CHAPTER 4. CLIFFORD THEORY FOR GRADED ALGEBRAS 192
1. Graded modules 192
2. Restriction to A1 197
3. Graded homomorphism modules 202
4. Extension from A1 207
5. Induction from A1 216
CHAPTER 5. PRIMITIVE AND PRIME IDEALS OF CROSSED PRODUCTS 234
1. Primitive, prime and semiprime ideals 234
2. Primitive ideals in crossed products 236
3. Prime coefficient rings 241
4. Incomparability and Going Down 257
5. A Going Up Theorem 270
6. Chains of prime and primitive ideals 278
CHAPTER 6. SEMIPRIME AND PRIME CROSSED PRODUCTS 288
1. Coset calculus 288
2. .–methods 296
3. The main theorem and its applications 305
4. Sufficient conditions for semiprimeness 316
5. Twisted group algebras 322
Bibiliography 342
Notation 350
Index 356