Research ArticleNo Access

Power cocentralizing generalized derivations on left ideals of prime rings

Cheng-Kai Liu

https://orcid.org/0000-0003-3277-6690

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

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

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and

Yuan-Tsung Tsai

https://orcid.org/0009-0000-7586-8849

Department of Computer Science and Engineering, Tatung University, Taipei City 10491, Taiwan

E-mail Address: yttsai@gm.ttu.edu.tw

Corresponding author.

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https://doi.org/10.1142/S0219498825501452Cited by:0 (Source: Crossref)

Abstract

Let $R$ $R$ be a prime ring with center $Z (R)$ $Z (R)$ , let $L$ $L$ be a nonzero left ideal of $R$ $R$ and let $g, h$ $g, h$ be two generalized derivations of $R$ $R$ . In this paper, we completely characterize the structure of $L$ $L$ and all possible forms of $g, h$ $g, h$ such that $g (x^{m}) x^{n} - x^{n} h (x^{m}) \in Z (R)$ $g (x^{m}) x^{n} - x^{n} h (x^{m}) \in Z (R)$ for all $x \in L$ $x \in L$ , where $m, n$ $m, n$ are fixed positive integers. With this, several known results can be either deduced or generalized. Moreover, our result can be regarded as the one-sided ideal version of the theorem obtained by Lee and Zhou in [An identity with generalized derivations, J. Algebra Appl. 8 (2009) 307–317].

Communicated by Y. Zhou

Keywords:

AMSC: 16N60, 16W25, 16R50

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