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

Published in: Quaestiones Mathematicae
Volume 48 , issue 5 , 2025 , 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 R_P ∈ 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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