On cycles of length three

Abstract

We prove that if A is a string algebra then there are not three irreducible morphisms between indecomposable A-modules such that its composition belongs to R6\R7, whenever the compositions of two of them are not in R3. Moreover, for any positive integer n≥3, we show that there are algebras where their module category have n irreducible morphisms such that their composition is in Rn+4\Rn+5.