Remarks on separativity of regular rings

TL;DR

This paper investigates conditions under which von Neumann regular rings are separative, providing new criteria and characterizations, and discusses the open problem of non-separative regular rings.

Contribution

It introduces new sufficient conditions for separativity in von Neumann regular rings and offers a characterization of regular perspective rings.

Findings

A von Neumann regular ring is separative if it is CS.

A von Neumann regular ring is separative if it is pseudo-injective.

A von Neumann regular ring is separative if it satisfies the closure extension property.

Abstract

Separative von Neumann regular rings exist in abundance. For example, all regular self-injective rings, unit regular rings, regular rings with a polynomial identity are separative. It remains open whether there exists a non-separative regular ring. In this note, we study a variety of conditions under which a von Neumann regular ring is separative. We show that a von Neumann regular ring $R$ is separative under anyone of the following cases: {\it (1)} $R$ is CS; {\it (2)} $R$ is pseudo injective (auto-injective); {\it (3)} $R$ satisfies the closure extension property: the essential closures in $R$ of two isomorphic right ideals are themselves isomorphic. We also give another characterization of a regular perspective ring (Proposition 3.3)