Some commutativity theorems for $*$-prime rings with $(sigma,tau)$-derivation

Abstract

‎Let $R$ be a $*$-prime ring with center‎ ‎$Z(R)$‎, ‎$d$ a non-zero $(sigma,tau)$-derivation of $R$ with associated‎ ‎automorphisms $sigma$ and $tau$ of $R$‎, ‎such that $sigma$‎, ‎$tau$‎ ‎and $d$ commute with $'*'$‎. ‎Suppose that $U$ is an ideal of $R$ such that $U^*=U$‎, ‎and $C_{sigma,tau}={cin‎ ‎R~|~csigma(x)=tau(x)c~mbox{for~all}~xin R}.$ In the present paper‎, ‎it is shown that if characteristic of $R$ is different from two and‎ ‎$[d(U),d(U)]_{sigma,tau}={0},$ then $R$ is commutative‎. ‎Commutativity of $R$ has also been established in case if‎ ‎$[d(R),d(R)]_{sigma,tau}subseteq C_{sigma,tau}.$