Generalized derivations in prime rings and noncommutative Banach algebras

Bull. Korean Math. Soc. 2008 Vol. 45, No. 4, 621-629 Printed December 1, 2008

Vincenzo De Filippis University of Messina

Abstract : Let $R$ be a prime ring of characteristic different from $2$, $C$ the extended centroid of $R$, and $\delta$ a generalized derivations of $R$. If $[[\delta(x),x],\delta(x)]=0$ for all $x\in R$ then either $R$ is commutative or $\delta(x)=ax$ for all $x\in R$ and some $a \in C$. We also obtain some related result in case $R$ is a Banach algebra and $\delta$ is either continuous or spectrally bounded.

Keywords : prime ring, derivations, differential identities, Banach algebras