Extensions of Simple Modules and the Converse of Schur's Lemma

TL;DR

This paper explores the properties of module categories satisfying the Converse of Schur's Lemma (CSL) for modules of finite length, extending previous research on CSL conditions.

Contribution

It develops fundamental properties of module categories under the CSL condition specifically for finite-length modules, advancing theoretical understanding.

Findings

Characterization of module categories with CSL for finite-length modules

Extension of previous CSL results to broader module classes

Foundational properties established for CSL-based module categories

Abstract

The converse of Schur's lemma (or CSL) condition on a module category has been the subject of considerable study in recent years. In this note we extend that work by developing basic properties of module categories in which the CSL condition governs modules of finite length.