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# b-GENERALIZED DERIVATIONS OF SEMIPRIME RINGS HAVING NILPOTENT VALUES

Published online by Cambridge University Press:  31 March 2014

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## Abstract

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}R$ be a semiprime ring with extended centroid $C$ and with maximal right ring of quotients $Q_{mr}(R)$ . Let $d{:}\ R\to Q_{mr}(R)$ be an additive map and $b\in Q_{mr}(R)$ . An additive map $\delta {:}\ R\to Q_{mr}(R)$ is called a (left) $b$ -generalized derivation with associated map $d$ if $\delta (xy)=\delta (x)y+bxd(y)$ for all $x, y\in R$ . This gives a unified viewpoint of derivations, generalized derivations and generalized $\sigma$ -derivations with an X-inner automorphism $\sigma$ . We give a complete characterization of $b$ -generalized derivations of $R$ having nilpotent values of bounded index. This extends several known results in the literature.

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Research Article
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