On classes of C3 and D3 modules

TL;DR

This paper investigates the properties of $ ext{C3}$ and $ ext{D3}$ modules relative to a class $ ext{A}$, providing characterizations and applications to describe certain classes of rings and modules, especially in the context of artinian rings.

Contribution

It introduces new notions of $ ext{A}$-C3 and $ ext{A}$-D3 modules, offers characterizations, and applies these to classify rings such as serial artinian rings with specific properties.

Findings

A regular right $R$-module $F$ is a $V$-module iff every $F$-cyclic module is $ ext{A}$-C3.

If all right $R$-modules are $ ext{A}$-injective, then certain $ ext{A}$-C3 modules are quasi-injective or C3.

Characterizations of rings where $ ext{A}$-C3 modules are quasi-injective or C3, especially for artinian rings.

Abstract

The aim of this paper is to study the notions of $\mathcal{A}$-C3 and $\mathcal{A}$-D3 modules for some class $\mathcal{A}$ of right modules. Several characterizations of these modules are provided and used to describe some well-known classes of rings and modules. For example, a regular right $R$-module $F$ is a $V$-module if and only if every $F$-cyclic module $M$ is an $\mathcal{A}$-C3 module where $\mathcal{A}$ is the class of all simple submodules of $M$. Moreover, let $R$ be a right artinian ring and $\mathcal{A}$, a class of right $R$-modules with local endomorphisms, containing all simple right $R$-modules and closed under isomorphisms. If all right $R$-modules are $\mathcal{A}$-injective, then $R$ is a serial artinian ring with $J^{2}(R)=0$ if and only if every $\mathcal{A}$-C3 right $R$-module is quasi-injective, if and only if every $\mathcal{A}$-C3 right $R$-module is C3.