Rings That are Nearly Associative (eBook)

Front Cover 1
Rings That Are Nearly Associative 4
Copyright Page 5
Contents 6
PREFACE 10
Chapter 1. Varieties of Algebras 14
1.1 Free Algebras 14
1.2 Varieties. Algebras Free in a Variety 16
1.3 Homogeneous Identities and Homogeneous Varieties 19
1.4 Partial Linearizations of Identities 22
1.5 Multilinear Identities. Complete Linearization of Identities 27
1.6 Adjoining an Identity Element 30
Chapter 2. Composition Algebras 35
2.1 Definition and Simple Properties of Composition Algebras 36
2.2 The Cayley-Dickson Process. Generalized Theorem of Hurwitz 41
2.3 Simple Quadratic Alternative Algebras 47
2.4 Further Properties of Composition Algebras 53
Chapter 3. Special and Exceptional Jordan Algebras 64
3.1 Definition and Examples of Jordan Algebras 64
3.2 Free Special Jordan Algebras 71
3.3 Shirshov's Theorem 79
Chapter 4. Solvability and Nilpotency of Jordan Algebras 94
4.1 Definitions and Examples 94
4.2 The Normal Form for Elements of the Multiplication Algebra 97
4.3 Zhevlakov's Theorem 101
4.4 Jordan Nil-Algebras 104
4.5 The Locally Nilpotent Radical 106
Chapter 5. Algebras Satisfying Polynomial Identities 111
5.1 Shinhov's Lemma 111
5.2 Associative PI-Algebras 115
5.3 Algebraic Elements and Local Finiteness 119
5.4 Special Jordan PI-Algebras 122
5.5 Alternative PI-Algebras 127
Chapter 6. Solvability and Nilpotency of Alternative Algebras 136
6.1 The Nagata–Higman Theorem 136
6.2 Dorofeev's Example 140
6.3 Zhevlakov's Theorem 143
Chapter 7. Simple Alternative Algebras 148
7.1 Preliminary Results 148
7.2 Elements of the Associative Center 157
7.3 Kleinfeld's Theorem 162
Chapter 8. Radicals of Alternative Algebras 166
8.1 Elements of the General Theory of Radicals 167
8.2 Nil-Radicals 174
8.3 The Andrunakievich Radical 181
Chapter 9. Semiprime Alternative Algebras 189
9.1 Ideals of Semiprime Algebras 189
9.2 Nondegenerate Alternative Algebras 195
9.3 Prime Alternative Algebras 205
Chapter 10. The Zhevlakov Radical 211
10.1 Primitive Alternative Algebras 212
10.2 The Kleinfeld Radical 214
10.3 The Smiley Radical 217
I 0.4 Zhevlakov's Theorem on the Coincidence of the Kleinfeld and Smiley Radicals 220
I 0.5 Radicals of Cayley–Dickson Rings 223
Chapter 11. Representations of Alternative Algebras 230
11.1 Definitions and Preliminary Results 231
11.2 Representations of Composition Algebras 240
11.3 Irreducible Alternative Modules 246
Chapter 12. Alternative Algebras with Finiteness condition 255
12.1 Alternative Algebras with Minimal Condition on Two-sided Ideals 256
12.2 Alternative Artinian Algebras 261
12.3 Alternative Algebras with Maximal Condition 267
Chapter 13. Free Alternative Algebras 271
13.1 Identities in Finitely Generated Alternative Algebras 271
13.2 Radicals and Nilpotent Elements of Free Alternative Algebras 281
13.3 Centers of Alternative Algebras 292
13.4 Alternative Algebras from Three Generators 301
Chapter 14. Radicals of Jordan Algebras 310
14.1 The McCrimmon Radical 310
14.2 Invertible Elements, Isotopy, and Homotopy 315
14.3 The Quasi-Regular Radical 320
14.4 Heredity of Radicals 331
Chapter 15. Structure Theory of Jordan Algebras with Minimal Condition 336
15.1 Quadratic Ideals 337
15.2 Radical Ideals of Jordan Algebras with Minimal Condition 340
15.3 Semisimple Jordan Algebras with Minimal Condition 346
Chapter 16. Right Alternative Algebras 355
16.1 Algebras without Nilpotent Elements 355
16.2 Nil-Algebras 360
Bibliography 365
LIST OF NOTATIONS 376
SUBJECT INDEX 380
Pure and Applied Mathematics 385