Sublattices of the lattice of pre-natural classes of modules
type: Item
instance of: scholarly article
title: Sublattices of the lattice of pre-natural classes of modules
zbMATH Open document ID: 0964.16003
author: John Dauns
author: Yiqiang Zhou
published in: Journal of Algebra
publication date: 2001-07-02T00:00:00Z
review text: Let \(R\) be a ring with identity. A class of right \(R\)-modules is a pre-natural class if it is closed under isomorphic copies, submodules, arbitrary direct sums, and certain essential extensions. The collection \({\mathcal N}^p_r(R)\) of all pre-natural classes was invented by \textit{Y. Zhou} [Commun. Algebra 25, No. 2, 543-557 (1997; Zbl 0871.16016)] and studied in detail [in J. Pure Appl. Algebra 140, No. 2, 191-207 (1999; Zbl 0947.16003)]. This is a lattice, which contains many previously studied lattices of \(R\)-module classes such as \(M\)-natural classes for a fixed module \(M\), natural classes, and hereditary pre-torsion classes. In this paper the sublattice structure of \({\mathcal N}^p_r(R)\) is studied and is related to ring and module properties of \(R\). Let \({\mathcal T}^p_r(R)\) be the set of all hereditary pre-torsion classes. For an arbitrary pre-natural class \(\mathcal K\), let \({\mathcal T}^p_r({\mathcal K},R)=\{{\mathcal K}\cap{\mathcal T}:{\mathcal T}\in{\mathcal T}^p_r(R)\}\). Then it is shown that some of the theory of \({\mathcal T}^p_r(R)\) is extended to \({\mathcal T}^p_r({\mathcal K},R)\). Indeed this is a complete sublattice of \({\mathcal N}^p_r(R)\) and is algebraic and upper continuous. For the set \({\mathcal N}_r(R)\) of all natural classes, let \({\mathcal N}(R,{\mathcal K})=\{{\mathcal L}\cap{\mathcal K}:{\mathcal L}\in{\mathcal N}_r(R)\}\). Then this is also sublattice of \({\mathcal N}^p_r(R)\) and is a complete Boolean lattice. Let \({\mathcal T}_r(R)\) be the set of all hereditary torsion classes. Handelman and Viola-Prioli have conjectured that \(R\) is right Noetherian if \({\mathcal T}^p_r(R)={\mathcal T}_r(R)\). Fenrick proved that if \(R\) is right Noetherian such that \({\mathcal T}^p_r(R)={\mathcal T}_r(R)\), then \(R\) is a right V-ring. In spite of Fenrick's result, even in the Noetherian case the characterization of the rings \(R\) for which \({\mathcal T}^p_r(R)={\mathcal T}_r(R)\) remains open. The authors …
Mathematics Subject Classification ID: 16D80
Mathematics Subject Classification ID: 16S90
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 06B23
Mathematics Subject Classification ID: 06E20
zbMATH DE Number: 1525269
zbMATH Keywords: pre-natural classes
zbMATH Keywords: direct sums
zbMATH Keywords: essential extensions
zbMATH Keywords: lattices of module classes
zbMATH Keywords: natural classes
zbMATH Keywords: hereditary pre-torsion classes
zbMATH Keywords: complete sublattices
zbMATH Keywords: complete Boolean lattices
zbMATH Keywords: hereditary torsion classes
zbMATH Keywords: right V-rings
zbMATH Keywords: right QI-rings
zbMATH Keywords: contravariant functors
reviewed by: Yoshiki Kurata
MaRDI profile type: Publication
full work available at URL: jabr.2000.8360
OpenAlex ID: W2041982527
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DOI: 10.1006/JABR.2000.8360
description: scientific article; zbMATH DE number 1525269