A Characterization of Derivations in Prime Rings with Involution

Abstract

The purpose of this paper is to investigate $\ast$-differential identities satisfied by pair of derivations on prime rings with involution. In particular, we prove that if a 2-torsion free noncommutative ring $R$ admit nonzero derivations $d_1,d_2$ such that $[d_1(x),d_2(x^{\ast })]=0$ for all $x\in R$, then $d_1=\lambda d_2$, where $\lambda \in C$. Finally, we provide an example to show that the condition imposed in the hypothesis of our results are necessary.