DEFINABLE SUBCATEGORIES OVER PURE SEMISIMPLE RINGS

Abstract

Let R be a right pure semisimple ring, and [Formula: see text] be a family of sources of left almost split morphisms in mod-R. We study the definable subcategory [Formula: see text] in Mod-R determined by the family [Formula: see text], and show that [Formula: see text] has several nice properties similar to those of the category Mod-R. For example, its functor category [Formula: see text] is a module category, and preinjective objects of [Formula: see text] are sources of left almost split morphisms in [Formula: see text] and in mod-R. As an application, it is shown that if R is a right pure semisimple ring with no nonzero homomorphisms from preinjective modules to non-preinjective indecomposable modules in mod-R (in particular, if R is right pure semisimple hereditary), then any definable subcategory of Mod-R determined by a finite set of indecomposable right R-modules contains only finitely many non-isomorphic indecomposable modules.