Functional Identities
by
Matej Brešar
University of Maribor
Coauthors: K.I. Beidar, M.A. Chebotar, W.S. Martindale III

A functional identity on a ring R is, roughly speaking, an identity involving maps of R. A typical example is the identity

n å i=1  Ei(x1, ... , xi-1, xi+1, ... , xn)xi + xiFi(x1, ... , xi-1, xi+1, ... , xn) = 0

for all x₁, ... , x_n in R. Here, E_i, F_i:R^n-1 --> R are arbitrary maps. The usual goal when treating a functional identity on R is either to describe the form of all maps involved in the identity, or, when this is not possible, to describe the structure of R. For example, if R is a prime ring, the maps E_i, F_i satisfying the above identity can be uniquely determined unless R satisfies S_2n-2, the standard polynomial identity of degree 2n-2.

We shall briefly survey the main concepts and results of the theory of functional identities, and discuss some of its applications. In particular, we will present solutions of the long-standing Herstein's problems on Lie homomorphisms of some Lie subrings of associative rings. We shall also outline some other apllications, for instance, ring-theoretic generalizations of some results on linear preservers obtained by linear algebraists and operator theorists.

Date received: April 19, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cage-32.