# Coneat Injective Modules

## TL;DR

This paper introduces coneat injective modules, explores their properties and generalizations, and characterizes certain rings like von Neumann regular and right SF-rings through these modules.

## Contribution

It defines coneat injectivity, studies its generalizations, and characterizes specific rings using properties of coneat injective modules.

## Key findings

- Coneat injective modules form an enveloping class between injective and pure injective modules.
- Generalizations like relative coneat injectivity are developed and analyzed.
- Characterizations of von Neumann regular and right SF-rings are provided via coneat injective modules.

## Abstract

A module is called coneat injective if it is injective with respect to all coneat exact sequences. The class of such modues is enveloping and falls properly between injectives and pure injectives. Generalizations of coneat injectivity, like relative coneat injectivity and full invariance of a module in its coneat injective envelope, are studied. Using properties of such classes of modules, we characterize certain types of rings like von Neumann regular and right SF-rings. For instance, R is a right SF-ring if and only if every coneat injective left R-module is injective.