Commutativity preserving maps revisited

Abstract

Dedicated to the memory of Kostia Beidar ABSTRACT K. I. Beidar and Y.-F. Lin have recently showed that under appropriate conditions a commutativity preserving map between (Jordan) algebras A and Q is of a standard form, unless it sends a certain subset of A, which one could describe (unless A is very special) as a "large" one, into the center of Q. We give a supplement to this statement by showing that this set often contains a nonzero ideal. In particular this makes it possible for us to give the definitive description of commutativity preservers in simple rings, as well as in prime rings provided that the map in question preserves commutativity in both directions.

Strong commutativity preserving generalized derivations on Lie ideals

“…Also, a simple counterexample was constructed for n ¼ 2. Since then, the study of describing maps that preserve commutativity has become an active research area in matrix theory, operator theory and ring theory (see, for instance, [1,3,8,9,[11][12][13][14]18,19,[40][41][42][43]45]). In [5], Bell and Daif investigated a special kind of commutativity preserving maps as follows: let S be a subset of R. A map f : S !…”

Functional identities and their applications

“…Theorem 4.16 unified and generalized many of the existing results, and opened the doors to the consideration of commutativity and some related linear preservers in pure algebra. Using more advanced techniques of the FI theory, its various generalizations were extensively studied (see [11,14,27,32,43,48,57,61,99] and also [53]). However, we will not discuss them here.…”

Commutativity preservers of incidence algebras

“…We draw the reader's attention to the fact that the notion of a strong commutativity preserver used here is different from that considered in [5,6]. Observe that our notion of a strong commutativity preserver is the same as a map preserving commutativity in both directions in [4,11].…”