Co-Kasch Modules
Rafail Alizade, Engin Büyükaşık, Yılmaz Durgun
TL;DR
The paper introduces co-Kasch modules as the dual notion to Kasch modules, defining $M$ to be a right co-Kasch module when every simple subfactor of $M$ is a homomorphic image of $M$, and proves that this is equivalent to every simple module in $σ[M]$ appearing as a homomorphic image of $M$. It then develops the structure of co-Kasch modules, showing that a projective module $P$ is co-Kasch iff it generates $σ[P]$, and establishing closure properties under direct sums but not under summands or quotients in general. The work then characterizes rings whose modules are all co-Kasch, including right $V$-rings, right $H$-rings, and right max-rings, with converses under additional hypotheses; it also derives detailed equivalences for artinian rings, linking co-Kaschness to Ext vanishing and to a diagonal Cartan matrix. In the commutative realm, it connects co-Kasch modules to Dedekind and DVR structures for Prüfer domains and provides a full description of co-Kasch $\mathbb{Z}$-modules, tying the theory to primary components and localization behavior.
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 $σ[M]$ 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 $σ[P]$. 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 Prüfer 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.