Bulletin of the
Korean Mathematical Society BKMS

Let $m, n, r$ be nonzero fixed positive integers, $R$ a $2$-torsion free prime ring, $Q$ its right Martindale quotient ring, and $L$ a non-central Lie ideal of $R$. Let $D:R\longrightarrow R$ be a skew derivation of $R$ and $E(x)=D(x^{m+n+r})-D(x^m)x^{n+r}-x^mD(x^n)x^{r}-x^{m+n}D(x^{r})$. We prove that if $E(x)=0$ for all $x\in L$, then $D$ is a usual derivation of $R$ or $R$ satisfies $s_4(x_1,\ldots,x_4)$, the standard identity of degree $4$.

Keywords: skew derivation, automorphism, prime ring

MSC numbers: 16W25, 16W20, 16N60