Left centralizers on Lie ideals in prime and semiprime gamma rings

Let $U$ be a Lie ideal of a 2-torsion free prime $\mathrm{\Gamma }$-ring $M$ such that $u\alpha u\in U$ for all $u\in U$ and $\alpha \in \mathrm{\Gamma }$. If $T:M\rightarrow M$isanadditivemappingsatifyingtherelation$T(u\alpha u)=T(u)\alpha u$forall$u\in U$and$\alpha \in\Gamma $,thenweprovethat$T(u\alpha v)=T(u)\alpha v$forall$u, v\in U$and$\alpha \in\Gamma $. Also this result is extended to semiprime $\mathrm{\Gamma }$-rings.