Strong preinjective partitions and almost split morphisms

Abstract

A family C of finitely generated indecomposable modules is said to have right almost split morphisms if there is a right almost split morphism f : N→M in add C for each indecomposable module M in C . We show that, for any ring R, if C is a family of indecomposable left R-modules of finite length such that C contains a finite cogenerating set and every subfamily of C has right almost split morphisms, then every subfamily of C has a unique strong preinjective partition of countable length. Sufficient conditions are given for a family of Noetherian modules with local endomorphism rings to have the property that each of its subfamilies has right almost split morphisms. A consequence of our results is a theorem, obtained by Zimmermann-Huisgen (Proc. Amer. Math. Soc. 109 (1990) 309–322) with different techniques, which asserts that every family of finitely generated indecomposable left modules over a left pure semisimple ring has a unique strong preinjective partition of countable length.