Characterizations of regular modules

TL;DR

This paper explores various notions of regularity for modules, providing new characterizations in terms of morphic and reduced modules, and identifies conditions under which these notions coincide.

Contribution

It introduces novel characterizations of different regularity notions for modules and examines conditions where these notions become equivalent.

Findings

Different regularity notions coincide over commutative rings.

New characterizations of regular modules via morphic and reduced modules.

Identifies module-theoretic settings where regularity notions are indistinguishable.

Abstract

Different and distinct notions of regularity for modules exist in the literature. When these notions are restricted to commutative rings, they all coincide with the well-known von-Neumann regularity for rings. We give new characterizations of these distinct notions for modules in terms of both (weakly-)morphic modules and reduced modules. Furthermore, module theoretic settings are established where these in general distinct notions turn out to be indistinguishable.