Left centralizers on Lie ideals in prime and semiprime gamma rings

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

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