Hereditary Noetherian prime rings. III: Infinitely generated projective modules
type: Item
instance of: scholarly article
title: Hereditary Noetherian prime rings. III: Infinitely generated projective modules
zbMATH Open document ID: 0955.16025
author: Lawrence S. Levy
author: J. C. Robson
published in: Journal of Algebra
publication date: 2000-08-13T00:00:00Z
review text: This is the last paper of a trilogy that brings the theory of projective modules over an HNP ring (= hereditary Noetherian prime ring) \(R\) roughly up to the same level as in the case where \(R\) is a commutative Dedekind domain. It describes the structure of the infinitely generated projective right \(R\)-modules. As in the finitely generated case, which is dealt with in part II [J. Algebra 218, No. 2, 338-372 (1999; Zbl 0946.16017)], the description involves the notion of the genus \(\Psi(P)\) of a projective \(R\)-module \(P\). This is a function defined on \(\{0\}\cup\mathcal W\), where \(\mathcal W\) is a set of representatives of the isomorphism classes of unfaithful simple \(R\)-modules; it is constructed as follows. Let \(Q(R)\) be the simple Artinian classical quotient ring of the HNP ring \(R\). Set \(\Psi(P)_0=Q(R)\)-rank of \(P\otimes_R Q(R)\) = the (finite or infinite) number of simple direct summands in a direct sum decomposition of the semisimple \(Q(R)\)-module \(P\otimes_R Q(R)\). If \(W\in\mathcal W\) with annihilator \(M\), then \(R/M\) is a simple Artinian ring and \(\Psi(P)_W\) is defined to be the \(R/M\)-rank of \(P/PM\). It turns out that if \(P\) and \(P'\) are infinitely generated projective \(R\)-modules, then \(P\cong P'\) if and only if \(\Psi(P)=\Psi(P')\). The main theorem of the paper establishes precisely when a family of cardinal numbers, indexed by \(\{0\}\cup\mathcal W\), is the genus \(\Psi(P)\) of some infinitely generated projective \(R\)-module \(P\); it also provides a construction of \(P\). Very explicit descriptions are obtained in two cases: (1) If \(R\) is any HNP ring and \(P_R\) is uncountably generated projective, then \(P\cong H\oplus\bigoplus_iS^{\alpha_i}_i\), where \(H_R\) is finitely generated projective, the \(\alpha_i\) are infinite cardinal numbers satisfying \(\alpha_1<\alpha_2<\cdots\) with at least one \(\alpha_i\) uncountable, and \(S_1\subset S_2\subset\cdots\) is a finite or countably infinite sequence o…
reviewed by: Günter Krause
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16D40
Mathematics Subject Classification ID: 16P50
Mathematics Subject Classification ID: 16D60
Mathematics Subject Classification ID: 16N60
Mathematics Subject Classification ID: 16E60
Mathematics Subject Classification ID: 16D70
zbMATH DE Number: 1423614
zbMATH Keywords: infinitely generated projective modules
zbMATH Keywords: hereditary Noetherian prime rings
zbMATH Keywords: genus
zbMATH Keywords: Dedekind domains
zbMATH Keywords: HNP rings
zbMATH Keywords: simple modules
zbMATH Keywords: simple Artinian classical quotient rings
zbMATH Keywords: direct summands
zbMATH Keywords: direct sums
zbMATH Keywords: hereditary orders
MaRDI profile type: Publication
full work available at URL: jabr.1999.8123
OpenAlex ID: W4210374105
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DOI: 10.1006/JABR.1999.8123
description: scientific article; zbMATH DE number 1423614