E-mail: mashraf80@hotmail.com
rehman100@postmark.net
Abstract.
Let $R$ be a 2-torsion free prime ring and let $\sigma , \tau$ be automorphisms
of $R$. For any $x, y \in R$, set $[x , y]_{\sigma , \tau} =
x\sigma(y) - \tau(y)x$.
Suppose that $d$ is a $(\sigma , \tau)$-derivation defined on $R$.
In the present paper it is shown that $(i)$ if $R$ satisfies
$[d(x) , x]_{\sigma , \tau} = 0$, then either $d = 0$
or $R$ is commutative $(ii)$ if $I$ is a nonzero ideal of $R$
such that $[d(x) , d(y)] = 0$, for all $x, y \in I$, and $d$
commutes with both
$\sigma$ and $\tau$, then either $d = 0$ or $R$ is commutative.
$(iii)$ if $I$ is a nonzero ideal of $R$
such that $d(xy) = d(yx)$, for all $x, y \in I$, and $d$ commutes with
$\tau$,
then $R$ is commutative. Finally a related result has been obtain for
$(\sigma , \tau)$-derivation.
AMSclassification. 16W25, 16N60, 16U80
Keywords. Prime rings, $(\sigma , \tau)$-derivations, ideals, torsion free rings and commutativity.