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Article

MSC: 16N60, 16W25
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Keywords:
generalized skew derivation; Lie ideal; prime ring
Summary:
Let $R$ be a prime ring of characteristic different from 2 and 3, $Q_r$ its right Martindale quotient ring, $C$ its extended centroid, $L$ a non-central Lie ideal of $R$ and $n\geq 1$ a fixed positive integer. Let $\alpha $ be an automorphism of the ring $R$. An additive map $D\colon R\to R$ is called an $\alpha $-derivation (or a skew derivation) on $R$ if $D(xy)=D(x)y+\alpha (x)D(y)$ for all $x,y\in R$. An additive mapping $F\colon R\to R$ is called a generalized $\alpha $-derivation (or a generalized skew derivation) on $R$ if there exists a skew derivation $D$ on $R$ such that $F(xy)=F(x)y+\alpha (x)D(y)$ for all $x,y\in R$. We prove that, if $F$ is a nonzero generalized skew derivation of $R$ such that $F(x)\* [F(x),x]^n = 0$ for any $x\in L$, then either there exists $\lambda \in C$ such that $F(x)=\lambda x$ for all $x\in R$, or $R\subseteq M_2(C)$ and there exist $a\in Q_r$ and $\lambda \in C$ such that $F(x)=ax+xa+\lambda x$ for any $x\in R$.
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