Hereditary Noetherian prime rings. I: Integrality and simple modules
type: Item
instance of: scholarly article
title: Hereditary Noetherian prime rings. I: Integrality and simple modules
zbMATH Open document ID: 0946.16016
author: Lawrence S. Levy
author: J. C. Robson
published in: Journal of Algebra
publication date: 1999-12-20T00:00:00Z
review text: This paper is the first of a trilogy with the aim of raising the structure theory of projective modules over a Noetherian hereditary prime (HNP) ring \(R\) to a level roughly equivalent to what has been achieved for the classical case of a (commutative) Dedekind domain. Part~II [J. Algebra~218, No. 2, 338-372 (1999)] is reviewed below (see Zbl 0946.16017); Part~III has appeared [J. Algebra~225, No. 1, 275-298 (2000; Zbl 0955.16025)]. The first part of this series lays the foundation of the theory by exploring the relationship between \(R\) and its Dedekind closures which play a role similar to that of maximal orders in the theory of classical orders over a Dedekind domain. The notion of a Dedekind closure of \(R\) is due to \textit{T. J. Hodges} [Trans. Am. Math. Soc.~ 302, No. 2, 751-767 (1987; Zbl 0634.16011)] who characterized these objects as the minimal Dedekind prime overrings of \(R\) (where ``overring'' means a ring between \(R\) and its Goldie quotient ring). He also showed that each one of them is a directed union of overrings that are finitely generated as right \(R\)-modules. In the paper under review, Dedekind closures are characterized as Dedekind prime right integral overrings, where an overring \(S\) of \(R\) is called right integral over \(R\) if any subring generated by \(R\) together with finitely many elements of \(S\) is finitely generated as a right \(R\)-module. Furthermore, Dedekind closures are shown to coincide with the maximal right integral overrings of \(R\). The authors provide a precise description the simple modules over a Dedekind closure in terms of certain sets of simple \(R\)-modules, called towers, and thus get a grip on factors of finite length of projective \(R\)-modules, which is vital for the later parts of this project.
reviewed by: Günter Krause
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 16S20
Mathematics Subject Classification ID: 16P50
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16D60
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16E60
zbMATH DE Number: 1348162
zbMATH Keywords: projective modules
zbMATH Keywords: hereditary rings
zbMATH Keywords: HNP rings
zbMATH Keywords: integral extensions
zbMATH Keywords: Dedekind domains
zbMATH Keywords: uniserial modules
zbMATH Keywords: hereditary Noetherian prime rings
zbMATH Keywords: Dedekind closures
zbMATH Keywords: Goldie quotient rings
zbMATH Keywords: integral overrings
zbMATH Keywords: simple modules
MaRDI profile type: Publication
full work available at URL: jabr.1999.7884
OpenAlex ID: W1984999890
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cites work: Modules over Dedekind prime rings
cites work: Hereditary Noetherian prime rings
cites work: Serial rings
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cites work: Simple modules over hereditary Noetherian prime rings
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cites work: Q3786586
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DOI: 10.1006/JABR.1999.7884
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description: scientific article; zbMATH DE number 1348162