On Two Classes of Modules Related to CS Trivial Extensions

TL;DR

This paper introduces weakly IN and strongly CS modules, and characterizes when trivial extensions of rings by modules result in CS rings, advancing understanding of module and ring structures.

Contribution

It defines new classes of modules and provides criteria for trivial extensions to be CS rings, linking module properties to ring extensions.

Findings

Characterization of trivial extensions as CS rings

Introduction of weakly IN and strongly CS modules

Criteria connecting module properties to ring extension structure

Abstract

All rings considered are commutative. In this article we introduce and study two notions of modules which are stronger than CS modules, namely weakly IN modules and strongly CS modules. Our main aim is to characterize when a trivial extension $A=R\propto M$ (of a ring $R$ by an $R$-module $M$) is a CS ring.