Strong commutativity preserving maps on Lie ideals of semiprime rings

TL;DR

This paper characterizes when endomorphisms or antihomomorphisms of a semiprime ring preserve strong commutativity on a Lie ideal, showing such maps are centralizing if they preserve commutativity.

Contribution

It establishes a necessary and sufficient condition for endomorphisms and antihomomorphisms to be strong commutativity preserving on Lie ideals in semiprime rings.

Findings

Endomorphisms and antihomomorphisms preserving strong commutativity are exactly those that are centralizing.

The result applies to 2-torsion free semiprime rings with nonzero square closed Lie ideals.

Provides a characterization linking strong commutativity preservation to centralizing property.

Abstract

Let $R$ be a 2-torsion free semiprime ring and $U$ a nonzero square closed Lie ideal of $R$. In this paper it is shown that if $f$ is either an endomorphism or an antihomomorphism of $R$ such that $f(U)=U,$ then $f$ is strong commutativity preserving on $U$ if and only if $f$ is centralizing on $U.$