On Semi(prime) Rings and Algebras with Automorphisms and Generalized Derivations

Let R be a ring. An additive mapping \(F : R\rightarrow R\) is called a generalized derivation if there exists a derivation \(d : R\rightarrow R \) such that \( F(x y) = F(x)y + xd(y)\) for all \( x, y \in R\). In this paper, first we describe the structure of prime rings involving automorphisms and then characterized generalized derivations on semiprime rings which satisfy certain differential identities. As applications, and apart from proving the other results, many known theorems can be either generalized or deduced. Moreover, we apply our results to functional analysis, and to study the analogous conditions for continuous linear generalized derivations on Banach algebras.