Indecomposable modules over pure semisimple hereditary rings

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 = boolean OR U-alpha <=delta(alpha) . 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 boolean OR U-k<w(alpha+k). for all limit ordinals alpha, 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 U-alpha, where alpha is an infinite limit ordinal; and the direct sum of modules in U-alpha is not endofinite for each infinite limit ordinal alpha. Moreover, the endomorphism ring of each indecomposable left R-module is a division ring. (C) 2012 Elsevier Inc. All rights reserved.