Herstein's theorem for pair of generalized derivations on prime rings with involution

[摘要] The aim of this paper is to study the $*$-identities with a pair of generalized derivations on $*$-ideals of prime rings with involution. In particular, we prove that if a noncommutative prime $*$-ring admit two generalized derivations $\mathcal{F}$ and $G$ such that $[\mathcal{F}(x), \mathcal{G}(x^*)]=0$ for all $x\in \mathcal{I}$, where $\mathcal{I}$ is a nonzero $*$-ideal of $\mathcal{R}$, then there exists $\lambda\in C$ such that $\mathcal{F}=\lambda \mathcal{G}.$ Finally, we provide an example which shows that the primeness of $\mathcal{R}$ is crucial in our results.

[学科分类] 数学（综合）

[关键词] prime ring;∗-ideal;involution;generalized derivation