The stable category of a left hereditary ring

TL;DR

This paper investigates the properties of the stable module category of a left hereditary ring, characterizing morphisms and providing criteria for when this category is abelian, with explicit descriptions of such rings.

Contribution

It offers a new characterization of (normal) monomorphisms and epimorphisms in stable categories and provides a criterion for the category to be abelian, including a complete classification of rings.

Findings

Characterization of (normal) monomorphisms and epimorphisms in stable categories

Criterion for the stable category of a left hereditary ring to be abelian

Explicit description of all rings with abelian stable categories

Abstract

The (co)completeness problem for the (projectively) stable module category of an associative ring is studied. (Normal) monomorphisms and (normal) epimorphisms in such a category are characterized. As an application, we give a criterion for the stable category of a left hereditary ring to be abelian. By a structure theorem of Colby-Rutter, this leads to an explicit description of all such rings.