2005/09/22 by Divaani-Aazar, Esmkhani, Tousi
Chemistry · Mathematics · #13C05 #13E10 #Advanced Topics in Algebra #Commutative Algebra (math.AC) #FOS: Mathematics #Oxidative Organic Chemistry Reactions #Rings, Modules, and Algebras #math.AC #msc:13C05 #msc:13E10
paper · pdf · doi:10.48550/arxiv.math/0509516
8 pages, to appear in Proc. Amer. Math. Soc
arxiv created 2005/09/22 · openalex publication_date 2005/09/22 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let R be a commutative ring with identity and D an R-module. It is shown that if D is pure injective, then D is isomorphic to a direct summand of the direct product of a family of finitely embedded modules. As a result, it follows that if R is Noetherian, then D is pure injective if and only if D is isomorphic to a direct summand of the direct product of a family of Artinian modules. Moreover, it is proved that D is pure injective if and only if there is a family \Tλ\λ∈ Λ of R-algebras which are finitely presented as R-modules, such that D is isomorphic to a direct summand of a module of the form Πλ∈ ΛEλ where for each λ∈ Λ, Eλ is an injective Tλ-module.