Weakly local commutativity for rings with unity

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.