# Abelian Endoregular Modules

## TL;DR

This paper introduces abelian endoregular modules, characterizes them via their submodules, and explores their properties, including behavior under direct sums, injective hulls, and their representation as subdirect products of simple modules.

## Contribution

It defines and characterizes abelian endoregular modules, a new class of modules with abelian von Neumann regular endomorphism rings, and studies their structural properties.

## Key findings

- Abelian endoregular modules are closed under taking $M$-generated submodules.
- Direct sums and injective hulls of abelian endoregular modules are also abelian endoregular.
- Such modules can be represented as subdirect products of simple modules.

## Abstract

In this paper, we introduce the notion of abelian endoregular modules as those modules whose endomorphism rings are abelian von Neumann regular. We characterize an abelian endoregular module $M$ in terms of its $M$-generated submodules. We prove that if $M$ is an abelian endoregular module then so is every $M$-generated submodule of $M$. Also, the case when the (quasi-)injective hull of a module as well as the case when a direct sum of modules is abelian endoregular are presented. At the end, we study abelian endoregular modules as subdirect products of simple modules.