Generalized derivations with power values on rings and Banach algebras - Abderrahman Hermas; Abdellah Mamouni; Lahcen Oukhtite (Mathematica Bohemica)
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)° H(y))^m=(x° y)^n$ for all $x,y \in I,$ \end{itemize} for two fixed positive integers $m≥1$, $n≥1$ 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.