On locally divided domains of the form Int(\(D\))
type: Item
instance of: scholarly article
title: On locally divided domains of the form Int(\(D\))
zbMATH Open document ID: 0995.13004
author: Paul-Jean Cahen
author: Jean-Luc Chabert
author: David E. Dobbs
author: Francesca Tartarone
published in: Archiv der Mathematik
publication date: 2002-10-17T00:00:00Z
review text: Here is the authors' abstract: ``Let \(D\) be an integral domain which is not a field. If either \(D\) is Noetherian or \(D\) is a Prüfer domain, then Int\((D)\), the ring of integer-valued polynomials of \(D\), is a treed domain if and only if it is a going-down domain. Suppose henceforth that \((D,m)\) is Noetherian local and one-dimensional, with \(D/ m\) finite. Then Int\((D)\) is a going-down domain if and only if \(D\) is unibranched (inside its integral closure); and Int\((D)\) is locally divided if and only if \(D\) is analytically irreducible. Thus, if \(D\) is unibranched but not analytically irreducible, then Int\((D)\) provides an example of a two-dimensional going-down domain which is not locally divided. Also, Int\((D)\) is a locally pseudo-valuation domain if and only if \(D\) is itself a pseudo-valuation domain. Thus, Int\((D)\) also provides an example of a two-dimensional locally divided domain which is not an LPVD.'' LPVDs (locally pseudovaluation domains), locally divided domains, etc., are generalizations of Prüfer domains.
reviewed by: Moshe Roitman
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13G05
Mathematics Subject Classification ID: 13B24
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13B22
Mathematics Subject Classification ID: 13F30
zbMATH DE Number: 1449198
zbMATH Keywords: going-down domain
zbMATH Keywords: interpolation domain
zbMATH Keywords: integer-valued polynomials
zbMATH Keywords: locally divided domain
zbMATH Keywords: LPVD
zbMATH Keywords: Prüfer domain
zbMATH Keywords: pseudovaluation domain
zbMATH Keywords: treed ring
zbMATH Keywords: unibranched domain
MaRDI profile type: Publication
full work available at URL: s000130050429
OpenAlex ID: W2063370450
DOI: 10.1007/S000130050429
description: scientific article; zbMATH DE number 1449198