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.

Key Words:

• Auslander–Reiten sequences
• Chains and lattices of subgroups
• Invariant subspaces
• Representation theory
• Representation type

2000 Mathematics Subject Classification:

• Primary 16G60
• Secondary 16G70, 20E15, 47A15

ACKNOWLEDGMENTS

This article is part of the author's Ph.D. thesis advised by Markus Schmidmeier at Florida Atlantic University. The author would like to thank the representation theory group in Bielefeld, where this article was completed, for their hospitality, the NSF for supporting this international exchange through DDEP Grant No. 0831369, and the referee for helpful comments and for suggesting a proof for the converse statement.

This research was partially supported by NSF DDEP Grant No. 0831369.

Notes

Communicated by D. Zacharia.