# Article

 Title: Annihilating and power-commuting generalized skew derivations on Lie ideals in prime rings (English) Author: de Filippis, Vincenzo Language: English Journal: Czechoslovak Mathematical Journal ISSN: 0011-4642 (print) ISSN: 1572-9141 (online) Volume: 66 Issue: 2 Year: 2016 Pages: 481-492 Summary lang: English . Category: math . 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 $x\in R$. (English) Keyword: generalized skew derivation Keyword: Lie ideal Keyword: prime ring MSC: 16N60 MSC: 16W25 idZBL: Zbl 06604481 idMR: MR3519616 DOI: 10.1007/s10587-016-0270-1 . Date available: 2016-06-16T12:57:41Z Last updated: 2018-07-02 Stable URL: http://hdl.handle.net/10338.dmlcz/145738 . Reference: [1] Beidar, K. I., III., W. S. Martindale, Mikhalev, A. V.: Rings with Generalized Identities.Pure and Applied Mathematics 196 Marcel Dekker, New York (1996). Zbl 0847.16001, MR 1368853 Reference: [2] Carini, L., Filippis, V. De: Commutators with power central values on a Lie ideal.Pac. J. Math. 193 (2000), 269-278. 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