Finitely generated, non-artinian monolithic modules

TL;DR

This paper investigates the properties of noetherian rings related to the local artinian nature of injective hulls of simple modules, providing new constructions and classifying certain algebras in characteristic zero.

Contribution

It introduces a general construction for algebras lacking the local artinian property and completes the classification of down-up algebras with this property in characteristic zero.

Findings

Identified rings where injective hulls of simple modules are not locally artinian

Provided a construction method for such algebras

Classified down-up algebras in characteristic zero with this property

Abstract

We survey noetherian rings $A$ over which the injective hull of every simple module is locally artinian. Then we give a general construction for algebras $A$ that do not have this property. In characteristic 0, we also complete the classification of down-up algebras with this property.