Remarks on generalized (α, β)-derivations in semiprime rings

Abstract

Let $R$ be an associative ring and $\alpha, \beta: R\rightarrow R$ ring homomorphisms. An additive mapping $d:R\rightarrow R$ is called an $(\alpha, \beta)$-derivation of $R$ if $d(xy)=d(x)\alpha(y)+\beta(x)d(y)$ is fulfilled for any $x,y \in R$, and an additive mapping $D:R\rightarrow R$ is called a generalized $(\alpha, \beta)$-derivation of $R$ associated with an $(\alpha, \beta)$-derivation $d$ if $D(xy)=D(x)\alpha(y)+\beta(x)d(y)$ is fulfilled for all $x,y \in R$. In this note, we intend to generalize a theorem of Vukman \cite{V}, and a theorem of Daif and El-Sayiad \cite{DS}.