Derivations in semiprime rings and Banach algebras

Subject Areas : Rings theory

Sh. Sahebi ¹ , V. Rahmani ²

1 - Department of Mathematics, Islamic Azad University, Central Tehran Branch, P. O. Box 14168-94351, Tehran, Iran
2 - Department of Mathematics, Islamic Azad University, Central Tehran Branch, P. O. Box 14168-94351, Tehran, Iran

Received: 2013-01-01 Accepted : 2013-01-01 Published : 2013-09-01

Keywords: prime ring, semiprime ring, derivation, Utumi quotient ring, Banach algebra,

Abstract :

Let $R$ be a 2-torsion free semiprime ring with extended centroid $C$, $U$ the Utumi quotient ring of $R$ and $m,n>0$ are fixed integers. We show that if $R$ admits derivation $d$ such that $b[[d(x), x]_n,[y,d(y)]_m]=0$ for all $x,y\in R$ where $0\neq b\in R$, then there exists a central idempotent element $e$ of $U$ such that $eU$ is commutative ring and $d$ induce a zero derivation on $(1-e)U$. We also obtain some related result in case $R$ is a non-commutative Banach algebra and d continuous or spectrally bounded.

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