On Weakly von Neumann regular rings

TL;DR

This paper introduces and investigates weak von Neumann regular rings, characterizing when polynomial rings are such and exploring conditions for ring constructions and quotients to be weak von Neumann regular.

Contribution

It defines weak von Neumann regular rings and provides necessary and sufficient conditions for polynomial rings and certain ring constructions to have this property.

Findings

Polynomial ring $R[x]$ is weak von Neumann regular iff $R$ has exactly two idempotents.

Conditions for $R=A\propto E$ to be weak von Neumann regular are established.

If $I$ is a primary ideal, then $R/I$ is weak von Neumann regular.

Abstract

In this paper, we define and study a particular case of von Neumann regular notion called a weak von Neumann regular ring. It shown that the polynomial ring $R[x]$ is weak von Neumann regular if and only if $R$ has exactly two idempotent elements. We provide necessary and sufficient conditions for $ R=A\propto E $ to be a weak von Neumann ring. It is also shown that $I$ is a primary ideal imply $R/I$ is a weak von Neumann regular ring.