On ( σ , τ ) -derivations in prime rings

Mohammad Ashraf; Nadeem-ur-Rehman

Archivum Mathematicum (2002)

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

Abstract

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Let R be a 2-torsion free prime ring and let σ , τ be automorphisms of R . For any x , y ∈ R , set [ x , y ] σ , τ = x σ ( y ) - τ ( y ) x . Suppose that d is a ( σ , τ ) -derivation defined on R . In the present paper it is shown that ( i ) if R satisfies [ d ( x ) , x ] σ , τ = 0 , then either d = 0 or R is commutative ( i i ) if I is a nonzero ideal of R such that [ d ( x ) , d ( y ) ] = 0 , for all x , y ∈ I , and d commutes with both σ and τ , then either d = 0 or R is commutative. ( i i i ) if I is a nonzero ideal of R such that d ( x y ) = d ( y x ) , for all x , y ∈ I , and d commutes with τ , then R is commutative. Finally a related result has been obtain for ( σ , τ ) -derivation.

How to cite

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Ashraf, Mohammad, and Nadeem-ur-Rehman. "On $(\sigma ,\tau )$-derivations in prime rings." Archivum Mathematicum 038.4 (2002): 259-264. <http://eudml.org/doc/248943>.

@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 -

References

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  9. Herstein I. N., A note on derivations, Canad. Math. Bull. 21 (1978), 369–370. (1978) Zbl0412.16018MR0506447
  10. Herstein I. N., Rings with involution, Univ. Chicago Press, Chicago 1976. (1976) Zbl0343.16011MR0442017
  11. Posner E. C., Derivations in prime rings, Proc. Amer. Math. Soc. 8 (1957), 1093–1100. (1957) MR0095863
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  13. Vukman J., Derivations on semiprime rings, Bull. Austral. Math. Soc. 53 (1995), 353–359. (1995) MR1388583

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