On the non-existence of right almost split maps

Abstract

We show that, over any ring, a module $ C$ is a codomain of a~right almost split map \iff $ C$ is a finitely presented module with local endomorphism ring; thus we give an answer to a 40 years old question by M. Auslander. Using the tools developed, we also provide a useful sufficient condition for a class of modules to be non-precovering. Finally, we show a non-trivial application in the general context of morphisms determined by object.