### Generalized derivations with power values in rings and Banach algebras

#### Abstract

Let $R$ be a prime ring, $U$ the Utumi quotient ring of $R$, $C=Z(U)$ the extended centroid of $R$ and $F$ a generalized derivation with associated derivation $d$ of $R$. Suppose that $(F([x,y]))^m=[x,y]^n$ for all $x, y\in I$, a nonzero ideal of $R$, where $m\geq 1$ and $n\geq 1$ are fixed integers, then one of the following holds:

(1) $R$ is commutative;

(2) there exists $a\in C$ such that $F(x)=ax$ for all $x\in R$ with $a^m=1$. Moreover, in this case if $m\neq n$, then either char $(R)=2$ or char $(R)=2^{|m-n|}-1$.

We also extend the result to the one sided case for $m=n$. Finally as an application we obtain a range inclusion result of continuous generalized derivations on Banach algebras.

**How to Cite this Article:**Asma Ali, Basudeb Dhara, Shahoor Khan, Generalized derivations with power values in rings and Banach algebras, J. Math. Comput. Sci., 7 (2017), 912-926 Copyright © 2017 Asma Ali, Basudeb Dhara, Shahoor Khan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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