Left centralizers on Lie ideals in prime and semiprime gamma rings

Abstract

Let $U$ be a Lie ideal of a 2-torsion free prime $\Gamma $-ring $M$ such that $u\alpha u\in U$ for all $u\in U$ and $\alpha \in\Gamma $. If $T:M\rightarrow M$ is an additive mapping satifying the relation $T(u\alpha u)=T(u)\alpha u$ for all $u\in U$ and $\alpha \in\Gamma $, then we prove that $T(u\alpha v)=T(u)\alpha v$ for all $u, v\in U$ and $\alpha \in\Gamma $. Also this result is extended to semiprime $\Gamma $-rings.