Polynomial identities in ring theory, Volume 84 (Pure and Applied Mathematics)

✍ Scribed by Louis Halle Rowen

• Publisher: Academic Press
• Year: 1980
• Tongue: English
• Leaves: 387
• Category: Library

✦ Table of Contents

Polynomial Identities in Ring Theory
Copyright Page
Contents
Preface
Prerequisites
Chapter 1. The Structure of PI-Rings
1.1. Basic Concepts and Examples
1.2. Facts about Normal Polynomials
1.3. Matrix Algebras
1.4. Identities and Central Polynomials for Matrix Algebras, and Their Applications to Arbitrary Pl-Algebras
1.5. Primitive Rings, Kaplansky’s Theorem, and Semiprimitive Rings
1.6. Injections of Algebras, Featuring Various Nil Radicals
1.7. Central Localization of PI-Algebras
1.8. Tensor Products and the Artin–Procesi Theorem
1.9. The Prime Spectrum
1.10. Valuation Rings, Idempotent Lifting, and Their Applications
1.11. Identities of Rings without 1
Exercises
Chapter 2. The General Theory of Identities, and Related Theories
2.1. Basic Concepts
2.2. PI-Rings Which Have an Involution
2.3. Sets of Identities of Related Rings (with Involution)
2.4. Relatively Free PI-Rings and T-Ideals
2.5. Identities of Matrix Rings with Involution
2.6. Elementary Sentences of Algebraic Systems
Exercises
Chapter 3. Central Simple Algebras
3.1. Fundamental Results
3.2. Positive General Results about Maximal Subfields of Division Rings
3.3. The Generic Division Rings
Exercises
Chapter 4. Extensions of PI-Rings
4.1. Integral and Algebraic Extensions of PI-Rings
4.2. Formal Words and Shirshov’s Solution to the Kurosch Problem
4.3. The Characteristic Closure of a Prime PI-Ring
4.4. Finitely Generated PI-Extensions
4.5. Generalizing the Razmyslov–Schelter Construction
Exercises
Chapter 5. Noetherian PI-Rings
5.1. Sufficient Conditions for a PI-Ring to Be Noetherian
5.2. The Theory of Noetherian PI-Rings
Exercises
Chapter 6. The Theory of the Free Ring, Applied to Polynomial Identities
6.1. The Solution of the Tensor Product Question
6.2. Representations of Sym(n)
6.3. Finite Generation of Certain T-Ideals
Exercises
Chapter 7. The Theory of Generalized Identities
7.1. Semiprime Rings with Socle
7.2. The Basic Theorem of Generalized Polynomials and Its Consequences
7.3. Primitive Rings with Involution
7.4. Identities and Generalized Identities of Rings with Involution
7.5. Ultraproducts and Their Application to GI-Theory
7.6. Martindale’s Central Closure
Exercises
Chapter 8. Rational Identities, Generalized Rational Identities, and Their Applications
8.1. Definitions and Examples
8.2. Generalized Rational Identities of Division Rings
8.3. Rational Identities of Division Rings of Finite Degree
8.4. Applications of the Theory of Rational Identities
Appendix A: Central Polynomials of Formanek
Exercises
Appendix B: The Theory of AE Elementary Conditions on Rings
Exercises
Appendix C: Nonassociative PI-Theory
Exercises
Postscript: Some Aspects of the History
Bibliography
Major Theorems Concerning Identities
Major Counterexamples
List of Principal notation
Index
Pure and Applied Mathematics