A result on derivations

Abstract:

Let $R$ be a semiprime ring with a derivation $d$ and let $U$ be a Lie ideal of $R$, $a\in R$. Suppose that $ad(u)^n=0$ for all $u\in U$, where $n$ is a fixed positive integer. Then $ad(I)=0$ for $I$ the ideal of $R$ generated by $[U,U]$ and if $R$ is 2-torsion free, then $ad(U)=0$. Furthermore, $R$ is a subdirect sum of semiprime homomorphic images $R_1$ and $R_2$ with derivations $d_1$ and $d_2$, induced canonically by $d$, respectively such that $\overline ad_1(R_1)=0$ and the image of $U$ in $R_2$ is commutative (central if $R$ is 2-torsion free), where $\overline a$ denotes the image of $a$ in $R_1$. Moreover, if $U=R$, then $ad(R)=0$. This gives Bres̆ar’s theorem without the $(n-1)!$-torsion free assumption on $R$.