Applications of reduced and coreduced modules II: Radicality of the functor $\text{Hom}_R(R/I, -)$

TL;DR

This paper investigates conditions under which the functor Hom_R(R/I, -) acts as a radical on modules over a commutative ring, leading to generalizations of Jans' correspondence and a new radical class of rings.

Contribution

It provides necessary and sufficient conditions for the functor to be a radical, extending the theory of reduced modules and introducing a new radical class of rings.

Findings

Characterization of when Hom_R(R/I, -) is a radical

Generalization of Jans' correspondence

Introduction of a new radical class of rings

Abstract

This is the second in a series of papers highlighting the applications of reduced and coreduced modules. Let $R$ be a commutative unital ring and $I$ be an ideal of $R$. We give necessary and sufficient conditions in terms of $I$-reduced and $I$-coreduced $R$-modules for the functor $\text{Hom}_R(R/I, -)$ on the abelian full subcategory of the category of $R$-modules to be a radical. These conditions further provide a setting for the generalisation of Jans' correspondence, and lead to a new radical class of rings.