Modules in which pure submodule is essential in a direct summand

TL;DR

This paper introduces pure extending modules, a generalization of extending modules, and characterizes various ring types based on properties of these modules, providing new insights into module theory.

Contribution

It defines pure extending modules and explores their properties, offering characterizations of regular, semisimple, local, and PDS rings in terms of these modules.

Findings

Pure extending modules generalize extending modules.

Characterizations of ring types via pure extending modules.

Examples and counterexamples illustrating properties.

Abstract

In this paper, we study the class of modules have the property that every pure submodule is essential in a direct summand. These modules are termed as pure extending modules which is a proper generalisation of extending modules. Examples and counterexamples are given. We study some properties of pure extending modules and characterize regular ring, semisimple ring, local ring and PDS ring in terms of pure extending modules.