# Red-injective modules

## TL;DR

This paper introduces the concept of Red-injective modules, explores their properties, and uses them to characterize important classes of rings such as quasi-Frobenius and V-rings.

## Contribution

It defines Red-injective modules inspired by Soc-injective modules and applies this concept to characterize specific classes of rings.

## Key findings

- Red-injective modules are characterized and their properties are studied.
- Red-injective modules are used to characterize quasi-Frobenius rings.
- Red-injective modules help in understanding V-rings.

## Abstract

Let $\text{Red}(M)$ be the sum of all reduced submodules of a module $M$. For modules over commutative rings, $\text{Soc}(M)\subseteq \text{Red}(M)$. By drawing motivation from how $\text{Soc}$-injective modules were defined by Amin et. al. in \cite{amin2005}, we introduce $\text{Red}$-injective modules, study their properties and use them to characterize quasi-Frobenius rings and $V$-rings.