On modules over valuation domains whose finitely generated submodules are direct sums of cyclics
type: Item
instance of: scholarly article
title: On modules over valuation domains whose finitely generated submodules are direct sums of cyclics
zbMATH Open document ID: 0644.13001
DOI: 10.1007/BF01194571
author: B. George
published in: Archiv der Mathematik
publication date: 1988-01-01T00:00:00Z
review text: Let \({\mathcal C}\) denote the class of all \(R\)-modules over a valuation domain \(R\), all finitely generated submodules of which are direct sums of cyclic \(R\)-modules. The authors prove that a torsion module \(M\) lies in \({\mathcal C}\) (in general, a module M belongs to \({\mathcal C}\) iff its torsion part belongs to \({\mathcal C})\) iff every countably generated submodule of \(M\) is isomorphic to a submodule of a direct sum of uniserial \(R\)-modules, iff \(M\) is a submodule of a pseudo-separable \(R\)-module (\(K\) is pseudo-separable if every finite subset of \(K\) is embeddable in a cyclically pure submodule that is a finite direct sum of uniserial submodules).
Mathematics Subject Classification ID: 13A18
Mathematics Subject Classification ID: 13C05
zbMATH DE Number: 4049753
zbMATH Keywords: valuation domain
zbMATH Keywords: torsion module
zbMATH Keywords: cyclically pure submodule
reviewed by: Ladislav Bican
MaRDI profile type: Publication
cites work: Arbitrarily large indecomposable divisible torsion modules over certain valuation domains
cites work: Q3027118
cites work: Separable torsion modules over valuation domains
cites work: Q3699812
cites work: Almost Maximal Valuation Rings
cites work: Modules Over Dedekind Rings and Valuation Rings
cites work: Injective Modules Over Prufer Rings
cites work: Finitely generated modules over valuation rings
full work available at URL: https://doi.org/10.1007/bf01194571
OpenAlex ID: W1993003228
Recommended article: On rings each of whose finitely generated modules is a direct sum of cyclic modules
Recommended article: Q3031083
Recommended article: On modules whose closed M-cyclic submodules are summands
Recommended article: On commutative rings whose ideals are direct sums of cyclic modules
Recommended article: On the classification of indecomposable generated modules over valuation domains
Recommended article: Indecomposable finitely generated modules over valuation domains
Recommended article: Finitely generated modules over valuation rings
Recommended article: Some properties of direct sums of uniserial modules over valuation domains
Recommended article: Cyclic modules whose quotients have all complement submodules direct summands
Recommended article: Q3807338
description: scientific article; zbMATH DE number 4049753