Beitr\ EMIS ELibM Electronic Journals Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 51, No. 2, pp. 345-351 (2010)

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Generalized derivations and commutativity of rings with involution

L. Oukhtite, S. Salhi and L. Taoufiq

Université Moulay Ismaïl, Faculté des Sciences et Techniques, Département de Mathématiques, Groupe d'Algèbre et Applications, B. P. 509 Boutalamine, Errachidia; Maroc, e-mail: e-mail: e-mail:

Abstract: Let $(R,\star)$ be a $2$ -torsion free ring with involution and $ F$ a generalized derivation, associated to a derivation $d$, satisfying one of the following conditions: 1) for each $x,y\in R$ either $d(x)\circ F(y)=0$ or $d(x)\circ F(y)=x\circ y$. 2) for each $x,y\in R$ either $[d(x),F(y)]=0$ or $[d(x),F(y)]= [x,y]$. In this paper it is shown that if $R$ is $\star$-prime, then $R$ is a commutative ring. Moreover, examples proving the necessity of the $\star$-primeness condition are given.

Keywords: rings with involution, generalized derivation, commutativity

Classification (MSC2000): 16W10, 16W25, 16N60

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Electronic version published on: 24 Jun 2010. This page was last modified: 8 Sep 2010.

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