Overview

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.

Full Product Details

Author:   Vesselin Drensky ,  Edward Formanek
Publisher:   Birkhauser Verlag AG
Imprint:   Birkhauser Verlag AG
Edition:   2004 ed.
Dimensions:   Width: 17.80cm , Height: 1.10cm , Length: 25.40cm
Weight:   0.830kg
ISBN:  

9783764371265

ISBN 10:   3764371269
Pages:   200
Publication Date:   25 May 2004
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

A Combinatorial Aspects in PI-Rings.- Vesselin Drensky.- 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.- B Polynomial Identity Rings.- Edward Formanek.- 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?.

Reviews

From the reviews: The book under review consists of two excellent monographs on the PI-theory by two leading researchers, V. Drensky and E. Formanek ! In summary, both expositions are very well written, and the book is recommended both for graduate students and researchers. (MATHEMATICAL REVIEWS)

"From the reviews: ""The book under review consists of two excellent monographs on the PI-theory by two leading researchers, V. Drensky and E. Formanek ! In summary, both expositions are very well written, and the book is recommended both for graduate students and researchers."" (MATHEMATICAL REVIEWS)"

From the reviews: <p>a oeThe book under review consists of two excellent monographs on the PI-theory by two leading researchers, V. Drensky and E. Formanek a ] In summary, both expositions are very well written, and the book is recommended both for graduate students and researchers.a (MATHEMATICAL REVIEWS)

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