On $(\sigma,\tau)$-derivations in prime rings
Ashraf, Mohammad ; Nadeem-ur-Rehman
Archivum Mathematicum, Tome 038 (2002), p. 259-264 / Harvested from 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.

Publié le : 2002-01-01
Classification:  16N60,  16U70,  16U80,  16W25 
@article{107839,
     author = {Mohammad Ashraf and Nadeem-ur-Rehman},
     title = {On $(\sigma,\tau)$-derivations in prime rings},
     journal = {Archivum Mathematicum},
     volume = {038},
     year = {2002},
     pages = {259-264},
     zbl = {1068.16047},
     mrnumber = {1942655},
     language = {en},
     url = {http://dml.mathdoc.fr/item/107839}
}

Ashraf, Mohammad; Nadeem-ur-Rehman. On $(\sigma,\tau)$-derivations in prime rings. Archivum Mathematicum, Tome 038 (2002) pp. 259-264. http://gdmltest.u-ga.fr/item/107839/

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