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An Engel condition with $b$ -generalized derivations for Lie ideals

Pao-Kuei Liau

Department of Mathematics, National Changhua, University of Education Changhua 500, Taiwan

E-mail Address: paokuei.liau@gmail.com

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and

Cheng-Kai Liu

Department of Mathematics, National Changhua, University of Education Changhua 500, Taiwan

E-mail Address: ckliu@cc.ncue.edu.tw

Corresponding author.

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https://doi.org/10.1142/S0219498818500469Cited by:6 (Source: Crossref)

Abstract

Let $R$ be a prime ring with the extended centroid $C$ , $L$ a noncommutative Lie ideal of $R$ and $g$ a nonzero $b$ -generalized derivation of $R$ . For $x, y \in R$ , let $[x, y] = x y - y x$ . We prove that if $[[\dots [[g (x^{n_{0}}), x^{n_{1}}], x^{n_{2}}], \dots], x^{n_{k}}] = 0$ for all $x \in L$ , where $n_{0}, n_{1}, \dots, n_{k}$ are fixed positive integers, then there exists $λ \in C$ such that $g (x) = λ x$ for all $x \in R$ except when $R \subseteq M_{2} (F)$ , the $2 \times 2$ matrix ring over a field $F$ . The analogous result for generalized skew derivations is also described. Our theorems naturally generalize the cases of derivations and skew derivations obtained by Lanski in [C. Lanski, An Engel condition with derivation, Proc. Amer. Math. Soc.118 (1993), 75–80, Skew derivations and Engel conditions, Comm. Algebra42 (2014), 139–152.]

Communicated by L. Bokut

Keywords:

AMSC: 16W20, 16W25, 16W55

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