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 $n \geq 3$, we show that there are $n$ irreducible morphisms such that their composition is in $\Re^{n+4} \backslash \Re^{n+5}$.