ADS modules

TL;DR

This paper explores the properties of ADS rings and modules, establishing connections with classical notions, characterizing their structure, and introducing the concept of completely ADS modules with specific decompositions.

Contribution

It provides new characterizations of ADS modules, links with classical module theory, and introduces the concept of completely ADS modules with structural results.

Findings

A simple ring that is ADS as a right module is either right self-injective or indecomposable.

Under certain conditions, a unique ADS hull exists up to isomorphism.

Completely ADS semiperfect modules are characterized as direct sums of semisimple and local modules.

Abstract

We study the class of ADS rings and modules introduced by Fuchs. We give some connections between this notion and classical notions such as injectivity and quasi-continuity. A simple ring R such that R is ADS as a right R-module must be either right self-injective or indecomposable as a right R-module. Under certain conditions we can construct a unique ADS hull up to isomorphism. We introduce the concept of completely ADS modules and characterize completely ADS semiperfect right modules as direct sum of semisimple and local modules.