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

Archivum Mathematicum (2002)

• Volume: 038, Issue: 4, page 259-264
• ISSN: 0044-8753

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## Abstract

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Let $R$ be a 2-torsion free prime ring and let $\sigma ,\tau$ be automorphisms of $R$. For any $x,y\in R$, set ${\left[x,y\right]}_{\sigma ,\tau }=x\sigma \left(y\right)-\tau \left(y\right)x$. Suppose that $d$ is a $\left(\sigma ,\tau \right)$-derivation defined on $R$. In the present paper it is shown that $\left(i\right)$ if $R$ satisfies ${\left[d\left(x\right),x\right]}_{\sigma ,\tau }=0$, then either $d=0$ or $R$ is commutative $\left(ii\right)$ if $I$ is a nonzero ideal of $R$ such that $\left[d\left(x\right),d\left(y\right)\right]=0$, for all $x,y\in I$, and $d$ commutes with both $\sigma$ and $\tau$, then either $d=0$ or $R$ is commutative. $\left(iii\right)$ if $I$ is a nonzero ideal of $R$ such that $d\left(xy\right)=d\left(yx\right)$, for all $x,y\in I$, and $d$ commutes with $\tau$, then $R$ is commutative. Finally a related result has been obtain for $\left(\sigma ,\tau \right)$-derivation.

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