Polynomial Identity Rings

✍ Scribed by Vesselin Drensky, Edward Formanek (auth.)

• Publisher: Birkhäuser Basel
• Year: 2004
• Tongue: English
• Leaves: 196
• Series: Advanced Courses in Mathematics CRM Barcelona
• Edition: 1
• Category: Library

✦ Synopsis

A ring R satisfies a polynomial identity if there is a polynomial f in noncommuting variables which vanishes under substitutions from R. For example, commutative rings satisfy the polynomial f(x,y) = xy - yx and exterior algebras satisfy the polynomial f(x,y,z) = (xy - yx)z - z(xy - yx). "Satisfying a polynomial identity" is often regarded as a generalization of commutativity.

These lecture notes treat polynomial identity rings from both the combinatorial and structural points of view. The former studies the ideal of polynomial identities satisfied by a ring R. The latter studies the properties of rings which satisfy a polynomial identity.

The greater part of recent research in polynomial identity rings is about combinatorial questions, and the combinatorial part of the lecture notes gives an up-to-date account of recent research. On the other hand, the main structural results have been known for some time, and the emphasis there is on a presentation accessible to newcomers to the subject.

The intended audience is graduate students in algebra, and researchers in algebra, combinatorics and invariant theory.

✦ Table of Contents

Front Matter....Pages i-vii
Front Matter....Pages 1-1
Introduction....Pages 3-4
Basic Properties of PI-algebras....Pages 5-18
Quantitative Approach to PI-algebras....Pages 19-35
The Amitsur—Levitzki Theorem....Pages 37-47
Central Polynomials for Matrices....Pages 49-58
Invariant Theory of Matrices....Pages 59-74
The Nagata—Higman Theorem....Pages 75-86
The Shirshov Theorem for Finitely Generated PI-algebras....Pages 87-101
Growth of Codimensions of PI-algebras....Pages 103-117
Bibliography....Pages 119-130
Front Matter....Pages 131-131
Introduction....Pages 133-136
Polynomial Identities....Pages 137-142
The Amitsur—Levitzki Theorem....Pages 143-146
Central Polynomials....Pages 147-150
Kaplansky’s Theorem....Pages 151-154
Theorems of Amitsur and Levitzki on Radicals....Pages 155-158
Posner’s Theorem....Pages 159-160
Every PI-ring Satisfies a Power of the Standard Identity....Pages 161-162
Azumaya Algebras....Pages 163-167
Artin’s Theorem....Pages 169-171
Front Matter....Pages 131-131
Chain Conditions....Pages 173-176
Hilbert and Jacobson PI-Rings....Pages 177-178
The Ring of Generic Matrices....Pages 179-181
The Generic Division Ring of Two 2 x 2 Generic Matrices....Pages 183-184
The Center of the Generic Division Ring....Pages 185-187
Is the Center of the Generic Division Ring a Rational Function Field?....Pages 189-191
Bibliography....Pages 193-196
Back Matter....Pages 197-200

✦ Subjects

Associative Rings and Algebras; Combinatorics