Atlantic Association for Research in
Mathematical Sciences

Memorial University of Newfoundland

Atlantic Algebra Centre

Mini Course

Associative algebras with polynomial identities (or
PI-algebras) are an important class of algebras which enjoy many of
the properties of finite dimensional algebras and commutative
algebras. The mini-course is devoted to the following topics.

Matrix algebras are among the most important and attractive to
study noncommutative objects in Ring Theory, with numerous
applications in mathematics and other branches of science. We
present two of the main theorems on polynomial identities of
matrices: the Amitsur-Levitzki theorem that the
*n*-by-*n* matrices satisfy the standard identity of
degree 2n and the existence of central polynomials.

We consider several theorems which are among the cornerstones
of the theory of PI-algebras: the Kaplansky theorem about primitive
PI-algebras, the Levitzki theorem that primitive PI-algebras have
no nil-ideals and the theorem of Posner, a localization theorem,
which generalizes the fact that a commutative domain has a quotient
field.

These two theorems (stated and proved in a purely
combinatorial way) have many applications: the positive solution
for PI-algebras of the Kurosh problem on the finite-dimensionality
of finitely generated algebraic algebras, the theorem of
Razmyslov-Kemer-Braun on the nilpotency of the radical of a
finitely generated PI-algebra, the theorem of Berele that finitely
generated PI-algebras have finite Gelfand-Kirillov dimension.