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.
Author(s): Vesselin Drensky, Edward Formanek (auth.)
Series: Advanced Courses in Mathematics CRM Barcelona
Edition: 1
Publisher: Birkhäuser Basel
Year: 2004
Language: English
Commentary: Improved version of http://library.lol/main/99f8a2e569d6cbd73673e54674679819 : added ToC, corrected page numbers, put some pages in right order
Pages: 200
Title
Contents
Foreword
Part A: Combinatorial Aspects in PI-Rings (Vesselin Drensky)
Introduction
1. Basic Properties of PI-algebras
2. Quantitative Approach to PI-algebras
3. The Amitsur-Levitzki Theorem
4. Central Polynomials for Matrices
5. Invariant Theory of Matrices
6. The Nagata-Higman Theorem
7. The Shirshov Theorem for Finitely Generated PI-algebras
8. Growth of Codimensions of PI-algebras
Bibliography
Part B: Polynomial Identity Rings (Edward Formanek)
Introduction
General References for PI-Rings
1. Polynomial Identities
2. The Amitsur-Levitzki Theorem
3. Central Polynomials
4. Kaplansky's Theorem
5. Theorems of Amitsur and Levitzki on Radicals
6. Posner's Theorem
7. Every PI-ring Satisfies a Power of the Standard Identity
8. Azumaya Algebras
9. Artin's Theorem
10. Chain Conditions
11. Hilbert and Jacobson PI-Rings
12. The Ring of Generic Matrices
13. The Generic Division Ring of Two 2 x 2 Generic Matrices
14. The Center of the Generic Division Ring
15. Is the Center of the Generic Division Ring a Rational Function Field?
Bibliography
Index
