Indecomposable modules over pure semisimple hereditary rings

Abstract

If R is a hereditary left artinian ring, then R is left pure semisimple if and only if the family R-ind of all finitely generated indecomposable left R-modules has a (unique) Ext-injective partition R-ind=⋃α⩽δUα. This partition is used to give a complete description of the distribution of all indecomposable modules over a left pure semisimple hereditary indecomposable ring R of infinite representation type. More precisely, R-ind is the disjoint union of the countable set of all preinjective modules and the finite set of all preprojective modules, and countable sets of Auslander–Reiten components of the form ⋃k<ωUα+k, for all limit ordinals α, constructed from the Ext-injective partition of R-ind. In particular, we show that an indecomposable left R-module M is not the source of a left almost split morphism in R-mod if and only if M belongs to Uα, where α is an infinite limit ordinal; and the direct sum of modules in Uα is not endofinite for each infinite limit ordinal α. Moreover, the endomorphism ring of each indecomposable left R-module is a division ring.