Does Int \((\mathbb Z)\) have the stacked bases property?
type: Item
instance of: scholarly article
title: Does Int \((\mathbb Z)\) have the stacked bases property?
zbMATH Open document ID: 1263.13018
author: Jean-Luc Chabert
published in: Arabian Journal of Mathematics
publication date: 2012-12-18T00:00:00Z
review text: The author attempts to answer the question from the title posed by \textit{J. Brewer} and \textit{L. Klinger} [New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 231, 65-75 (2003; Zbl 1080.13508)], in some cases. One of the results is as follows: Theorem 4.5. The two-dimensional Prüfer domain \(R=\mathrm{Int}(\mathbb Z)=\{ f\in\mathbb Q[X] | f(\mathbb Z)\subseteq\mathbb Z\}\) has the stacked bases property iff, for every matrix \(B\in\mathcal M_{m\times 2}(R)\) (\(m\) any natural number) with the unit content, there exists a matrix \(C\in\mathcal M_{2\times 2}(R)\) such that \(BC\) has unit content and all \(2\times 2\) minors of \(BC\) are zero. It is known that a Prüfer domain either with dimension 1 or with finite character has the stacked bases property. Thus, the author has given another way (via the so-called UCS-property) one could look into the original question, namely by way of the existence of certain \(2\times 2\) matrices.
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13C10
Mathematics Subject Classification ID: 13F05
zbMATH DE Number: 6116563
zbMATH Keywords: stacked bases property
zbMATH Keywords: content
zbMATH Keywords: unit content
zbMATH Keywords: unit content summand property
zbMATH Keywords: UCS-property
zbMATH Keywords: BCS-property
zbMATH Keywords: local-global ring
zbMATH Keywords: almost local-global ring
zbMATH Keywords: the ring of integer-valued polynomials
reviewed by: Radoslav M. Dimitrić
MaRDI profile type: Publication
full work available at URL: s40065-012-0021-6
OpenAlex ID: W2133364392
cites work: Pole assignability in polynomial rings, power series rings, and Prüfer domains
cites work: Pole assignability and the invariant factor theorem in Prüfer domains and Dedekind domains
cites work: Q4736147
cites work: High dimension Prüfer domains of integer-valued polynomials
cites work: Q4513600
cites work: On \(\text{Pic}(R[X])\) for \(R\) seminormal
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Wikidata QID: Q59270819
DOI: 10.1007/S40065-012-0021-6
description: scientific article; zbMATH DE number 6116563