Generalized derivations acting on Lie ideals in prime rings and Banach algebras - Hermas, A.; Oukhtite, L.; Taoufiq, L. (Matematychni Studii, 2023)
Let R be a prime ring and L a non-central Lie ideal of R. The purpose of this paper is to describe generalized derivations of R satisfying some algebraic identities locally on L. More precisely, we consider two generalized derivations F_1 and F_2 of a prime ring R satisfying one of the following identities: 1. F_1(x)° y +x ° F_2(y) =0, 2. [F_1(x),y] + F_2([x,y]) =0, for all x,y in a non-central Lie ideal L of R. Furthermore, as an application, we study continuous generalized derivations satisfying similar algebraic identities with power values on nonvoid open subsets of a prime Banach algebra A. Our topological approach is based on Baire’s category theorem and some properties from functional analysis.