Finite intersection of valuation overrings of polynomial rings in at most three variables
type: Item
instance of: scholarly article
title: Finite intersection of valuation overrings of polynomial rings in at most three variables
zbMATH Open document ID: 1547.13025
author: L. Paudel
published in: Algebra and Discrete Mathematics
publication date: 2024-06-25T00:00:00Z
review text: Let \(k\) be a field, \(D\) be any of the polynomial rings \(k[x_1]\), \(k[x_1,x_2]\), \(k[x_1,x_2,x_3]\), and \(K\) be the fraction field of \(D\). We say that a lattice-ordered group (\(\ell\)-group) \(G\) is realizable over \(D\) if there is a ring \(R\), \(D\subseteq R \subseteq K\) such that \(G\) is isomorphic to the group of divisibility \(G(R)\) of \(R\) (which is defined here as the multiplivative group of nonzero principal fractional ideals of \(R\)). The group \(G(R)\) is ordered by reverse inclusion. The aim of this paper is to classify all semi-local \(\ell\)-groups which are realized over \(D\) (semi-local means that it contains a finite number of maximal filters). The proofs rely on the study of the group of divisibility of finite intersections of valuation overrings of \(D\). We start with definitions. The cardinal product of two ordered groups \(G_1\) and \(G_2\), denoted by \(G_1\times_c G_2\), is the product of the groups \(G_1\) and \(G_2\) with pointwise ordering, and \(G_1\times_l G_2\) denotes this product together with the lexicographic order. \N\NIf \(D=k[x_1]\), then a nontrivial semilocal \(\ell\)-group \(G\) is realizable over \(D\) if, and only if, it is isomorphic (as \(\ell\)-group) to \(\mathbb{Z}\times_c \mathbb{Z}\times_c \cdots \times_c\mathbb{Z}\) for some finite number of copies of \(\mathbb{Z}\). \N\NIn the case where \(D=k[x_1,x_2]\), the author assumes that \(k\) is infinite. The condition is \(G\) being isomorphic to a cardinal product \(G_1\times_c G_2\times_c G_3\) of semilocal \(\ell\)-groups, where \(G_1\) is isomorphic to a finite cardinal product of subgroups of \(\mathbb{Q}\), \(G_2\) is isomorphic to a finite cardinal product of finitely generated subgroups of \(\mathbb{R}\) having rational rank \(2\), and \(G_3\) is isomorphic to a finite cardinal product of \(\ell\)-groups of the form \(\mathbb{Z}\times_l(\mathbb{Z}\times_c\cdots\times_c\mathbb{Z})\). \N\NTurning to \(D=k[x_1,x_2,x_3]\), the author still assumes tha…
reviewed by: Gerard Leloup
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13F30
Mathematics Subject Classification ID: 20F60
Mathematics Subject Classification ID: 13B22
zbMATH DE Number: 7870927
zbMATH Keywords: group of divisibility
zbMATH Keywords: valuation domain
zbMATH Keywords: Bézout domain
zbMATH Keywords: lattice-ordered group
MaRDI profile type: Publication
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DOI: 10.12958/ADM1997
description: scientific article; zbMATH DE number 7870927