Received May 23, 2023. Published online February 21, 2024.
Abstract: 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: (1) $aF(x)^mH(y)^m=x^ny^n$ for all $x,y \in I$, (2) $ (F(x)\circ H(y))^m=(x\circ y)^n$ for all $x,y \in I,$ for two fixed positive integers $m\geq1$, $n\geq1$ 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.
Keywords: prime ring; generalized derivation; Banach algebra; Jacobson radical
Affiliations: Abderrahman Hermas, Department of Mathematics, Faculty of Science and Technology, Sidi Mohamed Ben Abdellah University, Fez, Morocco, e-mail: Abde.hermas@gmail.com; Abdellah Mamouni, Department of Mathematics, Faculty of Science, Moulay Ismail University, Meknes, Morocco, e-mail: a.mamouni.fste@gmail.com; Lahcen Oukhtite (corresponding author), Department of Mathematics, Faculty of Science and Technology, Sidi Mohamed Ben Abdellah University, Fez, Morocco, e-mail: oukhtitel@hotmail.com
