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On $(\sigma, \tau)$-derivations in prime
rings

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*Mohammad Ashraf and Nadeem-ur-Rehman*

**Address.** M. Ashraf, Department of Mathematics, Faculty of
Science, King Abdul Aziz University

P.O. Box. 80203, Jeddah 21589, Saudi-Arabia

Nadeem-ur-Rehman, Department of Mathematics, University of
Kaiserslautern

P.O. Box 3049, 67653 Kaiserslautern, Germany
**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.