Centralizers on Prime and Semiprime Gamma Rings

Let $M$ be a noncommutative 2-torsion free semiprime $\mathrm{\Gamma }$-ring satisfying a certain assumption and let $S$ and $T$ be left centralizers on $M$. We prove the following results: \(i) If $[S(x),T(x)]{\alpha }\beta S(x)+S(x)\beta [S(x),T(x)]{\alpha }$=$0$holdsforall$x\in M$and$\alpha ,\beta \in \Gamma $,then$[S(x),T(x)]_{\alpha }$=$0$\mathrm{.}(ii)If$S\neq 0 (T\neq 0)$, then there exists $\lambda \in C$,(the extended centroid of $M$) such that $T$=$\lambda \alpha S(S=\lambda \alpha T)$forall$\alpha \in \Gamma $. \(iii) Suppose that $[[S(x),T(x){]}_{\alpha },S(x){]}_{\beta }$=$0$ holds for all $x\in M$ and $\alpha ,\beta \in \mathrm{\Gamma }$. Then $[S(x),T(x){]}_{\alpha }$=$0$ for all $x\in M$ and $\alpha \in \mathrm{\Gamma }$. \(iv) If $M$ is a prime $\mathrm{\Gamma }$-ring satisfying a certain assumption and $S\mathrm{\ne }0(T\mathrm{\ne }0)$, then there exists $\lambda \in C$, the extended centroid, such that $T$=$\lambda \alpha S(S=\lambda \alpha T)$.