The Auslander and Ringel–Tachikawa Theorem for Submodule Embeddings

Abstract

Auslander and Ringel–Tachikawa have shown that for an artinian ring R of finite representation type, every R-module is the direct sum of finitely generated indecomposable R-modules. In this article, we will adapt this result to finite representation type full subcategories of the module category of an artinian ring which are closed under subobjects and direct sums and contain all projective modules. If in addition ind R has left almost split morphisms, the subcategory is closed under direct products, is covariantly finite in mod R, and for modules X i in the subcategory, , we also obtain the converse. In particular, the results in this article hold for submodule representations of a poset, in case this subcategory is of finite representation type.