On the quotients of countable direct products of modules modulo direct sums
type: Item
instance of: scholarly article
title: On the quotients of countable direct products of modules modulo direct sums
zbMATH Open document ID: 0814.13008
DOI: 10.1007/BF01875154
published in: Acta Mathematica Hungarica
publication date: 1995-06-12T00:00:00Z
review text: In this paper \(R\) is assumed to be a commutative domain with 1, and \(R \neq Q\), for the field of quotients \(Q\). Inspired by a known result of Hulanicki and Mycielski, the authors look into the module \(M^* = \prod M_ n/ \bigoplus M_ n\), for countable sets \(\{M_ 1, \dots, M_ n, \dots\}\) of \(R\)-modules. \(M^ 1 = \bigcap_{0 \neq r \in R} rM\) denotes the first Ulm submodule of \(M\). The culminating result of the paper is as follows: If p.d. \(Q = 1\), and \(M_ n\) are reduced torsion modules, then \(M^*\) satisfies: (a) \(M^{*1}\) is a divisible \(R\)-module; and (b) \(M^*/M^{*1}\) is isomorphic to a submodule of elements of countable support in a product of \(R\)-complete modules. (c) \(M^*/M^{*1}\) is \(R\)-complete, for every choice of torsion \(R\)-modules \(M_ n\), if and only if \(Q/R\) is a countably generated \(R\)-module.
Mathematics Subject Classification ID: 13C12
Mathematics Subject Classification ID: 13C05
Mathematics Subject Classification ID: 20K25
Mathematics Subject Classification ID: 13C13
Mathematics Subject Classification ID: 13D05
zbMATH DE Number: 638255
zbMATH Keywords: complete modules
zbMATH Keywords: countably generated module
zbMATH Keywords: Ulm submodule
zbMATH Keywords: reduced torsion modules
author: László Fuchs
author: L. M. Pretorius
reviewed by: Radoslav M. Dimitrić
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01875154
OpenAlex ID: W2009009790
description: scientific article; zbMATH DE number 638255