Minimal right determiners of irreducible morphisms in tree string algebras

Let [Formula: see text] be a finite dimensional string algebra over a field with the quiver [Formula: see text] such that the underlying graph of [Formula: see text] is a tree, and let [Formula: see text] be the number of the minimal right determiners of all irreducible morphisms between indecomposable left [Formula: see text]-modules. Then we have [Formula: see text] where [Formula: see text] is the number of vertices in [Formula: see text], [Formula: see text] is a source in [Formula: see text] with two neighbors[Formula: see text] and [Formula: see text] is the number of vertices such that the associated vertex ideals are not zero.

Composition of irreducible morphisms in quasi-tubes

We study the composition of irreducible morphisms between indecomposable modules lying in quasi-tubes of the Auslander–Reiten quivers of artin algebras in relation with the powers of the radical of their module category.