Communications of the
Korean Mathematical Society CKMS

Let $R$ be a $2$-torsion free prime ring, $U$ a nonzero Lie ideal of $R$ such that $u^{2}\in U$ for all $u\in U$. In the present paper, it is proved that if $d$ is a nonzero derivation and $[[d(u),u],u]=0$ for all $u\in U$, then $U\subseteq Z(R)$. Moreover, suppose that $d_{1}, d_{2}, d_{3}$ are nonzero derivations of $R$ such that $d_{3}(y)d_{1}(x)=d_{2}(x)d_{3}(y)$ for all $x,y\in U$, then $U \subseteq Z(R)$. Finally, some examples are given to demonstrate that the restrictions imposed on the hypothesis of the above results are not superfluous.

Keywords: prime ring, derivation, Lie ideal

MSC numbers: 16W10, 16W25, 16U80