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Keywords:
prime ring; generalized derivation; Banach algebra; Jacobson radical
prime ring; generalized derivation; Banach algebra; Jacobson radical
Summary:
Let $R$ be a prime ring and $I$ a nonzero ideal of $R.$ The purpose of this paper is to classify generalized derivations of $R$ satisfying some algebraic identities with power values on $I.$ More precisely, we consider two generalized derivations $F$ and $H$ of $R$ satisfying one of the following identities: \begin {itemize} \item [(1)] $aF(x)^mH(y)^m=x^ny^n$ for all $x,y \in I,$ \item [(2)] $ (F(x)\circ H(y))^m=(x\circ y)^n$ for all $x,y \in I,$ \end {itemize} for two fixed positive integers $m\geq 1$, $n\geq 1$ and $a$ an element of the extended centroid of $R$. Finally, as an application, the same identities are studied locally on nonvoid open subsets of a prime Banach algebra.
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