Strong commutativity preserving maps on rings

Abstract

Suppose $\R$ is a unital ring having an idempotent element $e$ which satisfies $a{\R}e=0$ implies $a=0$ and $a{\R}(1-e)=0$ implies $a=0$. In this paper, we aim to characterize the map $f:{\R}\rightarrow {\R}$, $f$ is surjective and $[f(x),f(y)]=[x,y]$ for all $x,y\in {\R}$. It is shown that $f(x)=\alpha x +\xi (x)$ for all $x\in {\R}$, where $\alpha \in {\Z}({\R})$, $\alpha^2=1$, and $\xi$ is a map from ${\R}$ into ${\Z}({\R})$. As an application, a characterization of nonlinear surjective maps preserving strong commutativity on von Neumann algebras with no central summands of type $I_1$ is obtained.