Centralizers on Prime and Semiprime Gamma Rings

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

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