A localization of a power series ring over a Prüfer domain
type: Item
instance of: scholarly article
title: A localization of a power series ring over a Prüfer domain
zbMATH Open document ID: 1466.13013
author: Mi Hee Park
published in: Journal of Pure and Applied Algebra
publication date: 2021-07-14T00:00:00Z
review text: An integral domain \(D\) is called Strong Finite Type domain (SFT-domain) if every ideal \(I\) of \(D\) is an SFT-ideal, that is there exist a finitely generated ideal \(J \subseteq I\) and a positive integer \(k\) such that \(a^k \in J\) for each \(a \in I\). A domain \(D\) is called piecewise Noetherian if it has Noetherian prime spectrum and satisfies the ascending chain condition on \(P\)-primary ideals for each prime ideal \(P\) of \(D\). These two properties are both Noetherian-like property. This paper is mainly interested in the study of the properties of various localizations of a power series ring over Prüfer domain. The author investigates the case where \(K[\![X]\!]\) and \(D[\![X]\!]\) have the same quotient field where \(D\) is either a valuation domain or an SFT- Prüfer domain. The author also investigates when \(D[\![X]\!]_{P[\![X]\!]}\) is Noetherian, where \(P\) is a nonzero SFT-prime ideal of \(D\). Finally, the author characterizes when \(D[\![X]\!]\) is piecewise Noetherian.
reviewed by: Ali Benhissi
Mathematics Subject Classification ID: 13E05
Mathematics Subject Classification ID: 13F05
Mathematics Subject Classification ID: 13F25
Mathematics Subject Classification ID: 13F30
zbMATH DE Number: 7371432
zbMATH Keywords: power series ring
zbMATH Keywords: anti-Archimedean domain
zbMATH Keywords: valuation domain
zbMATH Keywords: Prüfer domain
zbMATH Keywords: SFT-ring
zbMATH Keywords: piecewise Noetherian ring
MaRDI profile type: Publication
full work available at URL: j.jpaa.2021.106765
OpenAlex ID: W3165116317
Wikidata QID: Q112882086
cites work: Anti-archimedean rings and power series rings
cites work: Krull Dimension in Power Series Rings
cites work: Power series rings over Prüfer domains
cites work: Q3931523
cites work: Transcendence degree in power series rings
cites work: When $(D \lbrack\lbrack X \rbrack\rbrack)_{P \lbrack\lbrack X \rbrack\rbrack$ is a Valuation Ring
cites work: Piecewise noetherian rings
cites work: Q3933092
cites work: Conducive integral domains
cites work: Q5687544
cites work: A Note on the Quotient Field of the Domain DX
cites work: Q5661989
cites work: A localization of a power series ring over a valuation domain
cites work: Completion of a Prüfer domain
cites work: Krull dimension of mixed extensions
cites work: Krull-dimension of the power series ring over a nondiscrete valuation domain is uncountable
cites work: Q5603302
cites work: On Krull's conjecture concerning completely interally closed integrity domains, III
cites work: The piecewise Noetherian property in power series rings over a valuation domain
cites work: Noetherian-like properties in polynomial and power series rings
cites work: Krull dimension of a power series ring over a valuation domain
DOI: 10.1016/J.JPAA.2021.106765
description: scientific article; zbMATH DE number 7371432