Research Article

Weakly local commutativity for rings with unity

Published in: Quaestiones Mathematicae
Volume 43, issue 10, 2020 , pages: 1367–1384
DOI: 10.2989/16073606.2019.1621959
Author(s): Bakri Gadelseed, P.R. China, Junchao Wei, P.R. China, Hua Yao, P.R. China

Abstract

A ring R is called a left weakly local commutative ring (WLC, for short) if for any a ∈ N (R) and b ∈ R, (ab)2 = ba2b, which is a proper generalization of CN rings. In this paper, we show that (1) a ring R is commutative if and only if (xy)2 = yx 2 y for each x, y ∈ SN (R) = {a ∈ R|a ∉ N (R)} (2) R is a left WLC ring if and only if xyx = yx 2 for each x ∈ N (R) and y ∈ SN (R); (3) R is a reduced ring if and only if T 2 (R) is a left WLC ring; (4) R is a CN ring if and only if V 2(R) is a left WLC ring; (5) R is a commutative reduced ring if and only if V 3(R) is a left WLC ring.

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