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Rings whose indecomposable modules are pure-projective or pure-injective

Published in: Quaestiones Mathematicae
Volume 48 , issue 5 : Entrepreneurship and Africa’s Cultural Context , 2024 , pages: 795–809
DOI: 10.2989/16073606.2025.2457684
Author(s): François Couchot Normandie Univ, France,
Keywords: 13C05, 13C13, 13F05, 13F30, 16L30, 16S50, Pure-projective module, pure-injective module, maximal valuation ring, discrete valuation domain of rank one, Gelfand ring, clean ring, arithmetical ring,
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Abstract

Let be the class of rings for which every indecomposable right module is pure-projective or pure-injective. When R is a Noetherian local commutative ring of maximal ideal P, it is proven that R ∈ if and only if R is either an artinian valuation ring or a discrete valuation domain of rank one with rank where is the completion of R in its P-adic topology. Let R be a commutative ring. Then R ∈ if and only if R is a clean arithmetical ring with RP ∈ for each maximal ideal P of R. Moreover, R is a semi-perfect ring when it is Noetherian. Some examples of commutative rings of the class are given.
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