Hypersimple Rings and Modules

TL;DR

This paper introduces the concepts of hypersimple modules and rings, exploring their properties and relation to hypercyclic rings, and revisits related classical results in ring theory.

Contribution

It defines hypersimple modules and rings, establishing foundational properties and connecting to existing concepts like hypercyclic rings, thus opening new avenues in module and ring theory.

Findings

Hypersimple modules have cyclic injective hulls.

Right hypersimple rings ensure all simple modules are hypersimple.

Connections to hypercyclic rings are established.

Abstract

In this paper a simple right R-module S over a ring R is called hypersimple if its injective hull E(S) is cyclic, and a ring R is called right hypersimple if every simple right R-module is hypersimple. We initiate a study of these new notions, and revisit Osofsky's work on hypercyclic rings, i.e. rings whose cyclic right modules have cyclic injective hulls.