Strong commutativity preserving derivations on Lie ideals of prime $\Gamma$-rings

Okan Arslan, Berna Arslan

Let $M$ be a $\Gamma$-ring and $S\subseteq M$. A mapping $f:M\rightarrow M$ is called <em>strong commutativity preserving</em> on $S$ if $[f(x),f(y)]_{\alpha}=[x,y]_{\alpha}$, for all $x,y\in S$, $\alpha\in\Gamma$. In the present paper, we investigate the commutativity of the prime $\Gamma$-ring $M$ of characteristic not 2 with center $Z(M)\neq (0)$ admitting a derivation which is strong commutativity preserving on a nonzero square closed Lie ideal $U$ of $M$. Moreover, we also obtain a related result when a mapping $d$ is assumed to be a derivation on $U$ satisfying the condition $d(u)\circ_{\alpha}d(v)=u\circ_{\alpha}v$, for all $u,v\in U$, $\alpha\in \Gamma$.