Endomorphism rings of simple modules and block decomposition

TL;DR

This paper investigates the structure of simple modules over semiperfect rings, revealing conditions under which these modules have uniform properties and exploring implications for ring decomposability and quasi-Frobenius rings.

Contribution

It provides a new characterization of indecomposable semiperfect rings via their Jacobson radical and analyzes simple modules' endomorphism rings and cardinalities.

Findings

Indecomposability of rings relates to their radical quotient.

Simple modules over indecomposable rings are either finite or share the same infinite cardinality.

Endomorphism rings of simple modules have consistent characteristics in indecomposable rings.

Abstract

A left and right noetherian semiperfect ring R is known to be indecomposable if and only if its factor by the second power of Jacobson radical is. This characterisation is used to study simple R-modules in terms of their Ext groups. It is shown that if R is indecomposable, all its simple modules are either finite or have the same infinite cardinality and their endomorphism rings have the same characteristics. The results are further strengthened in the case when R is quasi-Frobenius.