Polynomial Identity Rings

The book consists of lecture notes for the advanced course on polynomial identity (PI-) rings held in CRM Barcelona in July 2003. PI-rings form a large class, which includes all finite dimensional algebras, all commutative rings, and the Grassmann algebra. Methods used in PI-ring theory are mainly combinatorial (a study of the ideal of polynomial identities satisfied by a ring) and structural (a study of ring theoretic properties of rings satisfying a polynomial identity). The combinatorial methods are presented in part A of the book, written by V. Drensky. Here, Razmyslov's construction of central polynomials for matrices is explained as well as the Nagata-Higman theorem on nilpotency of nil algebras of bounded index, and the Shirshov and the Regev theorems. Part B, written by E. Formanek, describes fundamental theorems of Kaplansky and Posner on the structure of primitive and prime PI-rings, respectively, and the Artin theorem on Azumaya algebras. These (and many other) classical results are presented clearly and in detail. The book also contains several open problems and comments on recent results (for example, the negative solution of the Specht problem, growth of PI-algebras, and centre of the generic division ring). The book will definitely be useful for anyone interested in PI-ring theory, and more generally, in combinatorial and structural aspects of contemporary associative algebra.

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