Pointwise Semicommutative Rings

TL;DR

This paper introduces the concept of pointwise semicommutative rings, explores their properties, and establishes conditions under which these rings are reduced, strongly regular, exchange, or semiperiodic.

Contribution

It defines pointwise semicommutative rings as a generalization of semicommutative rings and investigates their structural properties and conditions for reducedness.

Findings

Reduced rings are pointwise semicommutative.

Strongly regular rings are characterized by left SF property.

Exchange rings are exactly the clean rings.

Abstract

We call a ring R pointwise semicommutative if for any element a in R either l(a) or r(a) is an ideal of R. A class of pointwise semicommutative rings is a strict generalization of semicommutative rings. Since reduced rings are pointwise semicommutative, this paper studies sufficient conditions for pointwise semicommutative rings to be reduced. For a pointwise semicommutative ring R, R is strongly regular if and only if R left SF ; R is exchange if and only if R is clean; if R is semiperiodic then R/J(R) is commutative.