Composition factors for indecomposable modules

Abstract:

Let $k$ be a field and $A$ be a finite-dimensional algebra over $k$ having only a finite number of isomorphism classes of indecomposable $A$-modules. Let $M$, $N$ be two indecomposable $A$-modules. Then a homomorphism $f:M \to N$ is said to be irreducible if for every factorization $f = gh$, $g$ is split mono or $h$ is split epi [2]. The aim of this note is to give an elementary proof of the fact that the indecomposable $A$-modules are completely determined, up to isomorphism, by their composition factors if there is no chain of irreducible maps from an indecomposable module to itself. This theorem was first proved in [5] involving the theory of tilted algebras.