Communications of the
Korean Mathematical Society CKMS

Let $R$ be a prime ring with center $Z$ and characteristic different from two, $U$ a nonzero Lie ideal of $R$ such that $u^{2}\in U$ for all $u\in U$ and $d$ be a nonzero $\left( \sigma,\tau\right)$-derivation of $R.$ We prove the following results: (i) If $[d(u),u]_{\sigma,\tau}=0$ or $[d(u),u]_{\sigma,\tau}\in C_{\sigma,\tau}$ for all $u\in U,$ then $U\subseteq Z.$ (ii) If $a\in R$ and $[d(u),a]_{\sigma,\tau}=0$ for all $u\in U,$ then $U\subseteq Z$ or $a\in Z.$ (iii) If $d([u,v])=\pm\lbrack u,v]_{\sigma,\tau}$ for all $u\in U,$ then $U\subseteq Z$.

Keywords: derivations, Lie ideals, $\left( \sigma,\tau\right)$-derivations, centralizing mappings, prime rings

MSC numbers: 16W25, 16W10, 16U80