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 ${\Re }^6\backslash {\Re }^7$, whenever the compositions of two of them are not in ${\Re }^3$. Moreover, for any positive integer $n\ge 3$, we show that there are $n$ irreducible morphisms such that their composition is in ${\Re }^{n+4}\backslash {\Re }^{n+5}$.