Some results on prime rings and $(sigma,tau)$- Lie ideals

Abstract

Let R be a prime ring with characteristic not 2, $sigma,tau,alpha,beta,lambda,mu$ are auto­morphisms of R and d : $R rightarrow R $ a nonzero ($sigma,tau$)-derivation. Suppose that $aepsilon R$. In this paper, we give some results on ($sigma,tau$)-Lie ideals and prove that, (1) If $[a, d(R)]_{alpha,beta}$ = 0 and $dsigma = sigma d, dtau = tau d$, then $aepsilon C_{alpha,beta}$. (2) Let $d_1$ be a nonzero ($sigma,tau$)-derivation and $d_2$ an ($alpha,beta$)-derivation of R such that $d_2alpha = alpha d_2, d_2beta = beta d_2$. If $[d_1(R),d_2(R)]lambda,mu = 0$ then R is commutative. (3) If I is a nonzero ideal of R and d(x,y) = 0 for all x,y $epsilon$ I, then R is commutative.(4) If d(R,a) = 0 then $(d(R), a)_{sigma,tau}=0$