The embedding of the components of the Auslander-Reiten Quiver of an artin algebra containing no oriented cycle

摘要

Let ii be an artin algebra. Let Gamma be a component of the Auslander-Reiten Quiver I'(A) of A, and Gamma contain no oriented cycle. If Gamma is stable ol semi-stable, then the structure of Gamma is known, Here, we are going to consider general component which may contain both injective vertices and projective vertices at the same time. We will show that Gamma can be embedded in some Z Delta with Delta isomorphic to a section of Gamma if and only if every possible path from an injective vertex to a projective vertex is sectional. This result generalizes part of Zhang's theorem and the corresponding part of Liu's theorem. [References: 10]