Turkish Journal of Mathematics




In this paper, the $k$-derivation is defined on a $\Gamma$-ring $M$ (that is, if $M$ is a $\Gamma$-ring, $d:M\to M$ and $k:\Gamma\to \Gamma$ are to additive maps such that $d(a\beta b )= d(a)\beta b + ak(\beta)b + a\beta d(b) $ for all $a,b\in M, \quad \beta \in \Gamma$, then $d$ is called a $k$-derivation of $M$) and the following results are proved. (1) Let $R$ be a ring of characteristic not equal to 2 such that if $xry=0$ for all $x, y\in R$ then $r=0$. If $d$ is a $k$-derivation of the $(R=)\Gamma$-ring $R$ with $k=d$, then $d$ is the ordinary derivation of $R$. (2) Let $M$ be a nonzero prime $\Gamma$-ring of characteristic not equal to 2, $\gamma$ be an element of $\Gamma$ and $a$ is an element in $M$ such that $[ [x, a]_{\gamma} , a]_{\gamma} =0$ for all $x\in M$. Then $a\gamma a = 0$ or $a\in C_{\gamma}$. (3) Let $M$ be a prime $\Gamma$-ring with Char$M \ne 2$, $d$ be a nonzero $k$-derivation of $M$, $\gamma$ be a nonzero element of $\Gamma$ and $k(\gamma) \ne 0$. If $d(M) \subseteq C_{\gamma}$, then $M$ is a commutative $\Gamma$-ring.

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