Gabriel-Roiter inclusions and Auslander-Reiten theory

TL;DR

This paper explores the connection between Gabriel-Roiter inclusions and Auslander-Reiten theory in artin algebras, revealing structural relationships and characterizing certain monomorphisms within homogeneous tubes.

Contribution

It establishes a link between Gabriel-Roiter submodules and indecomposable modules via irreducible monomorphisms, advancing understanding of module category structures.

Findings

Y is a factor module of an indecomposable M with an irreducible monomorphism from X

Monomorphisms in homogeneous tubes are Gabriel-Roiter inclusions under certain conditions

Provides structural insights into the relationship between Gabriel-Roiter inclusions and Auslander-Reiten theory

Abstract

Let $\Lambda$ be an artin algebra. The aim of this paper is to outline a strong relationship between the Gabriel-Roiter inclusions and the Auslander-Reiten theory. If $X$ is a Gabriel-Roiter submodule of $Y,$ then $Y$ is shown to be a factor module of an indecomposable module $M$ such that there exists an irreducible monomorphism $X \to M$. We also will prove that the monomorphisms in a homogeneous tube are Gabriel-Roiter inclusions, provided the the tube contains a module whose endomorphism ring is a division ring.