A question about saturated chains of primes in Serre conjecture rings
type: Item
instance of: scholarly article
title: A question about saturated chains of primes in Serre conjecture rings
zbMATH Open document ID: 1406.13003
author: Basma Gasmi
author: Noômen Jarboui
published in: Ricerche di Matematica
publication date: 2018-11-22T00:00:00Z
review text: Let \(R, S\) be commutative rings with identity of finite dimensional. A domain \(R\) is said to be an \(S\)-domain if for each height one prime ideal \(p\) of \(R\) the extended prime \(p[X]\) is also height one. Let \(R\) be a ring, we say that \(R\) is a strong \(S\)-ring if, \(R/p\) is an \(S\)-domain for each prime ideal \(p\) of \(R\). The Serre conjecture ring \(R\langle X \rangle\) is defined as follows: \[ R\langle X \rangle:= \{ f/g \mid f, g \in R[X] \text{ and } g \text{ is monic} \}. \] By induction, one can define the Serre coots, conjecture ring in \(n\) indeterminates as \(R\langle n \rangle=R\langle X_1, \ldots, X_n \rangle:= R\langle X_1, \ldots, X_{n-1} \rangle \langle X_n \rangle\). The authors of the paper, give a positive answer to the following question with some extra assumptions. {Question}. Is \(R \langle 2 \rangle\) strong \(S\) when \(R \langle 1 \rangle\) is strong \(S\)?
reviewed by: Majid Eghbali
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 13B25
zbMATH DE Number: 6983435
zbMATH Keywords: integral domain
zbMATH Keywords: prime ideal
zbMATH Keywords: algebraic extension
zbMATH Keywords: residually algebraic extension
zbMATH Keywords: Nagata ring
zbMATH Keywords: Serre conjecture ring
zbMATH Keywords: strong S-domain
zbMATH Keywords: Prüfer domain
zbMATH Keywords: quasi-Prüferian domain
MaRDI profile type: Publication
full work available at URL: s11587-017-0324-1
OpenAlex ID: W2604726159
Wikidata QID: Q122964606
cites work: About the Spectrum of the RingsR(n) andR⟨n⟩
cites work: The rings R(X) and \(R<X>\)
cites work: On Jaffard domains
cites work: Universally Catenarian Domains of D + M Type
cites work: About a conjecture on Nagata rings
cites work: Valuative Heights and Infinite Nagata Rings
cites work: Q4887548
cites work: Residually algebraic pairs of rings
cites work: On questions related to stably strong S-domains
cites work: A counterexample for a conjecture about the catenarity of polynomial rings.
cites work: About the spectrum of Nagata rings
cites work: Universally catenarian integral domains
cites work: Couples d'anneaux partageant un idéal. (Couples of rings sharing an ideal)
cites work: Construction B, I, D et anneaux localement ou residuellement de Jaffard. (B, I, D construction and locally or residually Jaffard rings)
cites work: Krull and valuative dimension of the Serre conjecture ring \(R\langle n \rangle\)
cites work: Quelques proprietes des chaines dideaux dans les anneaux A + XB[X]
cites work: Q5661989
cites work: Pullbacks and universal catenarity
cites work: Q3795764
cites work: Sur les S-domaines forts de Kaplansky. (On strong Kaplansky S-domains)
cites work: Q4049183
cites work: Strong S-domains
cites work: On the Chain Problem of Prime Ideals
cites work: Projective modules over polynomial rings
cites work: The ring \(R<X>\)
cites work: One counterexample for two open questions about the rings \(R(X)\) and \(R\langle X\rangle\)
cites work: Two counterexamples about the Nagata and Serre conjecture rings
cites work: On questions related to saturated chains of primes in polynomial rings
DOI: 10.1007/S11587-017-0324-1
description: scientific article; zbMATH DE number 6983435