Let R be a prime ring of characteristic different from 2 and 3, Qr its right Martindale quotient ring, C its extended centroid, L a non-central Lie ideal of R and n ≥ 1 a fixed positive integer. Let α be an automorphism of the ring R. An additive map D: R → R is called an α-derivation (or a skew derivation) on R if D(xy) = D(x)y + α(x)D(y) for all x, y \in R. An additive mapping F: R → R is called a generalized α-derivation (or a generalized skew derivation) on R if there exists a skew derivation D on R such that F(xy) = F(x)y + α(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 λ \in C such that F(x) = λx for all x \in R, or R\subset M_{2}\left ( C \right ) and there exist a \in Qr and λ \in C such that F(x) = ax + xa + λx for any x \in R., Vincenzo De Filippis., and Obsahuje seznam literatury
Let $R$ be a prime ring with its Utumi ring of quotients $U$ and extended centroid $C$. Suppose that $F$ is a generalized derivation of $R$ and $L$ is a noncentral Lie ideal of $R$ such that $F(u)[F(u),u]^n=0$ for all $u \in L$, where $n\geq 1$ is a fixed integer. Then one of the following holds: \begin {itemize} \item [(1)] there exists $\lambda \in C$ such that $F(x)=\lambda x$ for all $x\in R$; \item [(2)] $R$ satisfies $s_4$ and $F(x)=ax+xb$ for all $x\in R$, with $a, b\in U$ and $a-b\in C$; \item [(3)] $\mathop {\rm char}(R)=2$ and $R$ satisfies $s_4$. \end {itemize} As an application we also obtain some range inclusion results of continuous generalized derivations on Banach algebras.