On Modules Whose Pure Submodules Are Essential in Direct Summands

TL;DR

This paper introduces pure extending modules, explores their properties, and demonstrates their decomposition patterns, including a generalization of a classical theorem, while also resolving an open problem in module theory.

Contribution

It defines pure extending modules, establishes their properties, and connects them to classical modules, providing new decomposition results and solving an open problem.

Findings

Pure extending and extending modules coincide over von Neumann regular rings.

Cyclic modules with pure extending proper factors decompose into pure-uniform submodules.

Constructed a centrally quasi-morphic module that is not centrally morphic.

Abstract

We introduce the notion of pure extending modules, a refinement of classical extending modules in which only pure submodules are required to be essential in direct summands. Fundamental properties and characterizations are established, showing that pure extending and extending modules coincide over von Neumann regular rings. As an application, we prove that pure extending modules admit decomposition patterns analogous to those in the classical theory, including a generalization of the Osofsky-Smith theorem: a cyclic module whose proper factor modules are pure extending decomposes into a finite direct sum of pure-uniform submodules. Additionally, we resolve an open problem of Dehghani and Sedaghatjoo by constructing a centrally quasi-morphic module that is not centrally morphic, arising from the link between pure-extending behavior and nonsingularity in finitely generated modules over Noetherian rings.