LOCALLY FINITELY GENERATED MODULES OVER RINGS WITH ENOUGH IDEMPOTENTS

Abstract

Let R = ⊕ Reλ = ⊕ eλ R be an associative ring with enough idempotents indexed over a possibly infinite set Λ. Assume that {eλ : λ ∈ Λ} is a set of pairwise orthogonal primitive idempotents, and that R is locally bounded, that is, the projective modules eλR and Reλ are of finite length for each λ ∈ Λ. We prove the existence of almost split sequences ending at the indecomposable finitely generated non-projective unital R-modules. Moreover, we consider the unital R-modules X that are locally finitely generated, that is, Xeλ is a finitely generated eλ Reλ-module for all λ ∈ Λ. We show that such X accept perfect decompositionsX = ⊕ Xi as direct sums of indecomposable modules.