Jump of a domain and spectrally balanced domains
type: Item
instance of: scholarly article
title: Jump of a domain and spectrally balanced domains
zbMATH Open document ID: 1267.13006
author: Mohamed Khalifa
author: Ali Benhissi
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 2013-01-02T00:00:00Z
review text: Throughout, \(R\) is a locally finite-dimensional integral domain. The authors introduce and investigate two invariants, \(j_R(a) = \inf\{\, \mathrm{ht}P - \mathrm{ht}Q \mid P\) and \(Q\) are prime ideals of \(R\) with \(Q \subset aR \subseteq P\, \}\) for \(a \in R\) a nonzero, nonunit, and \(j(R) = \sup\{\, j_R(a) \mid a \in R\text{ is a nonzero, nonunit}\,\}\). They study these two ``jump'' invariants for several classes of integral domains such as polynomial rings, power series rings, localizations, and pullbacks. They are particularly interested in the case when \(j(R) = 1\) (such domains are called spectrally balanced (SB-) domains). For example, they show that if \(R\) is a finite-dimensional Prüfer domain, then \(R[X]\) (resp., \(RX\)) is an SB-domain if and only if \(R\) is an SB-domain (resp., an SB-domain satisfying the SFT-property), and that a finite-dimensional divided domain \(R\) is a Jaffard domain if and only if \(R[X_1, \dots, X_n]\) is an SB-domain for every \(n \geq 1\).
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13C15
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13B25
Mathematics Subject Classification ID: 13F25
Mathematics Subject Classification ID: 13G05
zbMATH DE Number: 6120476
zbMATH Keywords: Jaffard domain
zbMATH Keywords: Krull dimension
zbMATH Keywords: jump of a domain
zbMATH Keywords: Prüfer domain
zbMATH Keywords: APVD
zbMATH Keywords: polynomial ring
zbMATH Keywords: power series ring
reviewed by: David F. Anderson
MaRDI profile type: Publication
full work available at URL: s10231-011-0199-9
OpenAlex ID: W2064049092
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DOI: 10.1007/S10231-011-0199-9
description: scientific article; zbMATH DE number 6120476