Lie Centralizers and generalized Lie derivations on prime rings by local actions

Abstract

Assume that $\mathcal{R}$ be a unital prime ring with whose characteristic is not 2 and containing a nontrivial idempotent P. In this paper, it is shown that under certain conditions, $\varphi ([A,B])=[\varphi (A),B]$ where $\varphi$ an additive map on $\mathcal{R}$ and AB = P if and only if it has the form $\varphi (A)=\lambda A+h(A)$, where h is an additive map into its center vanishing at commutators $[A,B]$ with $AB=P$ and $\lambda \in Z(\mathcal{R})$. An application of our results we characterize generalized Lie derivations on $\mathcal{R}$. Using main result we apply several classical examples of unital prime rings with nontrivial idempotents such as Banach space standard operator algebras and factor von Neumann algebras.

KEYWORDS:

• Generalized Lie derivation
• Lie centralizer
• prime ring

2020 MATHEMATICS SUBJECT CLASSIFICATION::

• 16W25
• 16W10
• 47B47

Acknowledgments

The authors wish to give their thanks to the referees.