Co-Kasch Modules

TL;DR

This paper introduces co-Kasch modules, explores their properties, characterizes them over various rings, and establishes their relationship with classical ring structures and modules.

Contribution

It defines co-Kasch modules, characterizes them in terms of simple modules and ring properties, and provides complete descriptions for specific rings like Dedekind domains and the integers.

Findings

A module is co-Kasch iff every simple module in σ[M] is a homomorphic image of M.

For right artinian rings, co-Kasch modules relate to the ring being a right H-ring.

The structure of co-Kasch modules over ℤ is fully characterized.

Abstract

In this paper we study the modules $M$ every simple subfactors of which is a homomorphic image of $M$ and call them co-Kasch modules. These modules are dual to Kasch modules $M$ every simple subfactors of which can be embedded in $M$. We show that a module is co-Kasch if and only if every simple module in $\sigma[M]$ is a homomorphic image of $M$. In particular, a projective right module $P$ is co-Kasch if and only if $P$ is a generator for $\sigma[P]$. If $R$ is right max and right $H$-ring, then every right $R$-module is co-Kasch; and the converse is true for the rings whose simple right modules have locally artinian injective hulls. For a right artinian ring $R$, we prove that: (1) every finitely generated right $R$-module is co-Kasch if and only if every right $R$-module is a co-Kasch module if and only if $R$ is a right $H$-ring; and (2) every finitely generated projective right $R$-module is co-Kasch if and only if the Cartan matrix of $R$ is a diagonal matrix. For a Pr\"ufer domain $R$, we prove that, every nonzero ideal of $R$ is co-Kasch if and only if $R$ is Dedekind. The structure of $\mathbb{Z}$-modules that are co-Kasch is completely characterized.