Research ArticleNo Access

An Engel condition with two generalized derivations on Lie ideals of prime rings

Cheng-Kai Liu

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

https://doi.org/10.1142/S0219498823501153Cited by:2 (Source: Crossref)

Abstract

Let $R$ be a prime ring, let $L$ be a noncentral Lie ideal of $R$ and let $g, h$ be two generalized derivations of $R$ . In this paper, we characterize the structure of $R$ and all possible forms of $g$ and $h$ such that ${[g (x^{m}) x^{n} - x^{s} h (x^{t}), x^{r}]}_{k} = 0$ for all $x \in L$ , where $m, n, s, t, r, k$ are fixed positive integers. With this, several known results can be either deduced or generalized. In particular, we give a Lie ideal version of the theorem obtained by Lee and Zhou in [An identity with generalized derivations, J. Algebra Appl. 8 (2009) 307–317] and describe a more complete version of the theorem recently obtained by Dhara and De Filippis in [Engel conditions of generalized derivations on left ideals and Lie ideals in prime rings, Comm. Algebra 48 (2020) 154–167].

Communicated by D. Herbera

Keywords:

AMSC: 16N60, 16W25, 16R50

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