Absolutely Self Pure Modules

TL;DR

This paper introduces the concept of absolutely self pure modules, extending properties of quasi-injective and absolutely pure modules, and characterizes regular and left noetherian rings using this new notion.

Contribution

It defines absolutely self pure modules and explores their properties, extending existing module concepts and providing new characterizations of certain rings.

Findings

Absolutely self pure modules generalize quasi-injective and absolutely pure modules.

Regular and left noetherian rings are characterized via absolute self purity.

The paper establishes properties and implications of this new module class.

Abstract

An $R$-module $M$ is called absolutely self pure if for any finitely generated left ideal of $R$ whose kernel is in the filter generated by the set of all left ideals $L$ of $R$ with $L \supseteq$ ann $(m)$ for some $m \in M$, any map from $L$ to $M$ is a restriction of a map $R \rightarrow M$. Certain properties of quasi injective and absolutely pure modules are extended to absolute self purity. Regular and left noetherian rings are characterized using this new concept.