Geodesic
On $(\sigma,\tau)$-derivations in prime rings
Archivum mathematicum, Tome 38 (2002) no. 4, pp. 259-264 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $[x , y]_{\sigma , \tau } = x\sigma (y) - \tau (y)x$. 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.
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.
Classification : 16N60, 16U70, 16U80, 16W25
Keywords: prime rings; $(\sigma, \tau )$-derivations; torsion free rings and commutativity 
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Ashraf, Mohammad; Nadeem-ur-Rehman. On $(\sigma,\tau)$-derivations in prime rings. Archivum mathematicum, Tome 38 (2002) no. 4, pp. 259-264. http://geodesic.mathdoc.fr/item/ARM_2002_38_4_a1/

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