Polynomial Identities in Ring Theory (eBook)

Front Cover 1
Polynomial Identities in Ring Theory 4
Copyright Page 5
Contents 8
Preface 14
Prerequisites 20
Chapter 1. The Structure of PI-Rings 22
1.1. Basic Concepts and Examples 23
1.2. Facts about Normal Polynomials 32
1.3. Matrix Algebras 35
1.4. Identities and Central Polynomials for Matrix Algebras, and Their Applications to Arbitrary Pl-Algebras 41
1.5. Primitive Rings, Kaplansky’s Theorem, and Semiprimitive Rings 52
1.6. Injections of Algebras, Featuring Various Nil Radicals 61
1.7. Central Localization of PI-Algebras 72
1.8. Tensor Products and the Artin–Procesi Theorem 80
1.9. The Prime Spectrum 93
1.10. Valuation Rings, Idempotent Lifting, and Their Applications 101
1.11. Identities of Rings without 1 121
Exercises 124
Chapter 2. The General Theory of Identities, and Related Theories 130
2.1. Basic Concepts 130
2.2. PI-Rings Which Have an Involution 140
2.3. Sets of Identities of Related Rings (with Involution) 144
2.4. Relatively Free PI-Rings and T-Ideals 154
2.5. Identities of Matrix Rings with Involution 161
2.6. Elementary Sentences of Algebraic Systems 166
Exercises 169
Chapter 3. Central Simple Algebras 172
3.1. Fundamental Results 172
3.2. Positive General Results about Maximal Subfields of Division Rings 195
3.3. The Generic Division Rings 208
Exercises 220
Chapter 4. Extensions of PI-Rings 223
4.1. Integral and Algebraic Extensions of PI-Rings 223
4.2. Formal Words and Shirshov’s Solution to the Kurosch Problem 225
4.3. The Characteristic Closure of a Prime PI-Ring 229
4.4. Finitely Generated PI-Extensions 231
4.5. Generalizing the Razmyslov–Schelter Construction 239
Exercises 241
Chapter 5. Noetherian PI-Rings 245
5.1. Sufficient Conditions for a PI-Ring to Be Noetherian 245
5.2. The Theory of Noetherian PI-Rings 250
Exercises 258
Chapter 6. The Theory of the Free Ring, Applied to Polynomial Identities 260
6.1. The Solution of the Tensor Product Question 260
6.2. Representations of Sym(n) 264
6.3. Finite Generation of Certain T-Ideals 269
Exercises 273
Chapter 7. The Theory of Generalized Identities 275
7.1. Semiprime Rings with Socle 275
7.2. The Basic Theorem of Generalized Polynomials and Its Consequences 278
7.3. Primitive Rings with Involution 286
7.4. Identities and Generalized Identities of Rings with Involution 292
7.5. Ultraproducts and Their Application to GI-Theory 297
7.6. Martindale’s Central Closure 303
Exercises 307
Chapter 8. Rational Identities, Generalized Rational Identities, and Their Applications 310
8.1. Definitions and Examples 310
8.2. Generalized Rational Identities of Division Rings 312
8.3. Rational Identities of Division Rings of Finite Degree 320
8.4. Applications of the Theory of Rational Identities 323
Appendix A: Central Polynomials of Formanek 336
Exercises 340
Appendix B: The Theory of AE Elementary Conditions on Rings 341
Exercises 347
Appendix C: Nonassociative PI-Theory 348
Exercises 359
Postscript: Some Aspects of the History 360
Bibliography 362
Major Theorems Concerning Identities 376
Major Counterexamples 379
List of Principal notation 380
Index 382
Pure and Applied Mathematics 387