The Auslander and Ringel-Tachikawa Theorem for submodule embeddings

TL;DR

This paper extends the Auslander and Ringel-Tachikawa theorem to certain subcategories of modules over artinian rings, including subspace representations of posets, showing they decompose into finitely generated indecomposables.

Contribution

It adapts the finite representation type decomposition result to subcategories closed under subobjects and sums, broadening its applicability.

Findings

Subcategories of finite representation type decompose into indecomposables.

Results apply to subspace representations of posets.

The theorem is extended beyond the original module category context.

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 paper, 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. In particular, the results in this paper hold for subspace representations of a poset, in case this subcategory is of finite representation type.