CBMS Regional Conference Series in Mathematics 1991; 55 pp; softcover Number: 78 ISBN10: 0821807307 ISBN13: 9780821807309 List Price: US$29 Member Price: US$23.20 All Individuals: US$23.20 Order Code: CBMS/78
 The theory of polynomial identities, as a welldefined field of study, began with a wellknown 1948 article of Kaplansky. The field since developed along two branches: the structural, which investigates the properties of rings that satisfy a polynomial identity; and the varietal, which investigates the set of polynomials in the free ring that vanish under all specializations in a given ring. This book is based on lectures delivered during an NSFCBMS Regional Conference, held at DePaul University in July 1990, at which the author was the principal lecturer. The first part of the book is concerned with polynomial identity rings. The emphasis is on those parts of the theory related to \(n\times n\) matrices, including the major structure theorems and the construction of certain polynomial identities and central polynomials for \(n\times n\) matrices. The ring of generic matrices and its center is described. The author then moves on to the invariants of \(n\times n\) matrices, beginning with the first and second fundamental theorems, which are used to describe the polynomial identities satisfied by \(n\times n\) matrices. One of the exceptional features of this book is the way it emphasizes the connection between polynomial identities and invariants of \(n\times n\) matrices. Accessible to those with background at the level of a firstyear graduate course in algebra, this book gives readers an understanding of polynomial identity rings and invariant theory, as well as an indication of problems and research in these areas. Readership Reviews "This monograph provides an excellent overview of the subject and can serve nonexperts as an introduction to the field and serve experts as a handy reference."  Mathematical Reviews Table of Contents  Polynomial identity rings
 The standard polynomial and the AmitsurLevitzki theorem
 Central polynomials
 Posner's theorem and the ring of generic matrices
 The center of the generic division ring
 The Capelli polynomial and Artin's theorem
 Representation theory of the symmetric and general linear groups
 The first and second fundamental theorems of matrix invariants
 Applications of the first and second fundamental theorems
 The NagataHigman theorem and matrix invariants
