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.

https://doi.org/10.28919/jmcs/3305