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.