Gabriel–Roiter inclusions and Auslander–Reiten theory

Let Λ 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→M. We also will prove that the monomorphisms in a homogeneous tube are Gabriel–Roiter inclusions, provided the tube contains a module whose endomorphism ring is a division ring.