Characterization of Multiplicative Lie Triple Derivations on Rings

Abstract

Let $\mathcal{R}$ be a ring having unit 1. Denote by $\mathcal{Z}$ $\left(\mathcal{R}\right)$ the center of $\mathcal{R}$. Assume that the characteristic of $\mathcal{R}$ is not 2 and there is an idempotent element $e\in$ $\mathcal{R}$ such that $a\mathcal{R}e$ $=\left\{0\right\}\Rightarrow a=0\text{and}a\mathcal{R}\left(1-e\right)=\left\{0\right\}\Rightarrow a=0$. It is shown that, under some mild conditions, a map $L:\mathcal{R}\to \mathcal{R}$ is a multiplicative Lie triple derivation if and only if $L\left(x\right)=\delta \left(x\right)+h\left(x\right)$ for all $x\in \mathcal{R}$, where $\delta :\mathcal{R}\to \mathcal{R}$ is an additive derivation and $h:\mathcal{R}\to \mathcal{Z}\left(\mathcal{R}\right)$ is a map satisfying $h\left(\left[\left[a,b\right],c\right]\right)=0$ for all $a,b,c\in \mathcal{R}$. As applications, all Lie (triple) derivations on prime rings and von Neumann algebras are characterized, which generalize some known results.