Endomorphism Rings with Finitely Many Maximal Right Ideals

Abstract

We show that the indecomposable R-modules whose endomorphism ring has finitely many maximal right ideals, all of them two-sided, have a surprisingly simple behavior as far as direct sums are concerned. Our main result is that these modules are completely described up to isomorphism by an easy combinatorial structure, a simple hypergraph. If 𝒞 is any full subcategory of Mod-R containing all these modules as objects, the vertices of the hypergraph are suitable ideals 𝒫 of the category 𝒞. Let SFT-R be the category of all finite direct sums of modules whose endomorphism ring has finitely many maximal right ideals. The objects of SFT-R are completely determined up to isomorphism by the dimensions of vector spaces indexed by suitable ideals 𝒫 of the category SFT-R. Several examples are given in the last section.