Communications of the
Korean Mathematical Society CKMS

Let $R$ be an associative ring with involution $\ast$ and $\alpha, \beta: R\rightarrow R$ ring homomorphisms. An additive mapping $d:R\rightarrow R$ is called an $(\alpha, \beta)^\ast$-derivation of $R$ if $d(xy)=d(x)\alpha(y^\ast)+\beta(x)d(y)$ is fulfilled for any $x,y \in R$, and an additive mapping $F:R\rightarrow R$ is called a generalized $(\alpha, \beta)^\ast$-derivation of $R$ associated with an $(\alpha, \beta)^\ast$-derivation $d$ if $F(xy)=F(x)\alpha(y^\ast)+\beta(x)d(y)$ 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}, moreover, we generalize a theorem of Ali et al.~\cite{AFFK} and a theorem of Huang and Koc \cite{HK} related to generalized Jordan triple $(\alpha, \beta)^\ast$-derivations.

Keywords: semiprime rings, Lie ideals, $(\alpha,\beta)^\ast$-derivations, generalized $(\alpha,\beta)^\ast$-derivations, Jordan $(\alpha, \beta)^\ast$-derivations, generalized Jordan $(\alpha, \beta)^\ast$-derivations

MSC numbers: 16W25, 16N60, 16U80