ABSTRACT
The main objective of this note is to characterize the irreducible morphisms of a class C. As it is well known, these objects played a crucial role for the development of the modern theory of representations of algebras which was initially motivated by the monumental works of M. Auslander and I. Reiten. They proved the existence of almost split sequences (which now we prefer to call Auslander-Reiten sequences) for C = mod−Λ, thus solving completely, for this class, the problems of existence of almost split and irreducible morphisms arriving to or starting at an indecomposable. Several years afterwards, M. Auslander and S. Smalø solved the main
problem for general classes C (see [2]). For the representation theory of Artin algebras it was clearly enough to
study the categories of indecomposable modules, and so, to characterize irreducibles ending up or starting from an indecomposable module. On the other hand, reckoning with the fact that, in the last decades, the
study of derived categories has become an important tool in several areas and, especially, in representation theory, the notion of irreducible morphisms for a class is certainly of major importance.
