# Harmanci Injectivity of Modules

## TL;DR

This paper explores Harmanci injective modules, a new class related to injective, flat, and cotorsion modules, providing characterizations, properties, and conditions under which the injective envelope of a ring is flat.

## Contribution

It introduces and studies Harmanci injective modules, establishing their properties, their role in cotorsion theory, and conditions for the flatness of injective envelopes of rings.

## Key findings

- Harmanci injective modules form an enveloping class.
- They establish a perfect cotorsion theory with Matlis injective modules.
- Conditions identified for the injective envelope of a ring to be flat.

## Abstract

In this paper, we are interested in a class of modules partaking in the hierarchy of injective and cotorsion modules, so-called Harmanci injective modules, which turn out by the motivation of relations among the concepts of injectivity, flatness and cotorsionness. We give some characterizations and properties of this class of modules. It is shown that the class of all Harmanci injective modules is enveloping, and forms a perfect cotorsion theory with the class of modules whose character modules are Matlis injective. One of the main objectives we pursue is to know when the injective envelope of a ring as a module over itself is a flat module.