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

@article{Ashraf2002,
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.},
author = {Ashraf, Mohammad, Nadeem-ur-Rehman},
journal = {Archivum Mathematicum},
keywords = {prime rings; $(\sigma , \tau )$-derivations; torsion free rings and commutativity; prime rings; -derivations; ideals; commutativity theorems},
language = {eng},
number = {4},
pages = {259-264},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On $(\sigma ,\tau )$-derivations in prime rings},
url = {http://eudml.org/doc/248943},
volume = {038},
year = {2002},
}

TY - JOUR
AU - Ashraf, Mohammad
AU - Nadeem-ur-Rehman
TI - On $(\sigma ,\tau )$-derivations in prime rings
JO - Archivum Mathematicum
PY - 2002
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 038
IS - 4
SP - 259
EP - 264
AB - 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.
LA - eng
KW - prime rings; $(\sigma , \tau )$-derivations; torsion free rings and commutativity; prime rings; -derivations; ideals; commutativity theorems
UR - http://eudml.org/doc/248943
ER -