Abstract

Let $R$ be a non-necessarily commutative ring and $M$ an $R$-module. We use the category $\sigma[M]$ to introduce the notion of $I$-module who is a generalization of $I$-ring. It is well known that every artinian object of $\sigma [M]$ is co-hopfian but the converse is not true in general.

The aim of this paper is to characterize for a fixed ring, the left (right) $R$-modules $M$ for which every co-hopfian object of $\sigma[M]$ is artinian.

We obtain some characterization of finitely generated $I$-modules  over a commutative ring, faithfully balanced finitely generated $I$-modules, and left serial finitely generated $I$-modules over a duo-ring.

This work is licensed under a Creative Commons Attribution 4.0 License.