On the composition of four irreducible morphisms in the fifth power of the radical

Abstract

Given A an artin algebra we study when the non-zero composition of four irreducible morphisms between indecomposable A-modules belongs to the fifth power of the radical of its module category. In particular, when A is a finite dimensional algebra over an algebraically closed field we prove that if the composition of four irreducible morphisms between indecomposable A-modules belongs to the fifth power of the radical of their module category then any composition of four irreducible morphisms between the same indecomposable A-modules so is.