A constructive comparison of the rings \(R(X)\) and \(R\langle X\rangle\) and application to the Lequain-Simis induction theorem
type: Item
instance of: scholarly article
title: A constructive comparison of the rings \(R(X)\) and \(R\langle X\rangle\) and application to the Lequain-Simis induction theorem
zbMATH Open document ID: 1147.13012
author: Afef Ellouz
author: Henri Lombardi
author: Ihsen Yengui
published in: Journal of Algebra
publication date: 2008-10-06T00:00:00Z
review text: Let \(R\) be a commutative unitary ring. Denote by \(S\) (respectively \(U\)) the multiplicative subset of \(R[X]\) formed by monic polynomials (respectively primitive polynomials, i.e, polynomials whose coefficients generate the whole ring). Let \(R<X>=S^{-1}R[X]\) and \(R(X)=U^{-1}R[X]\). In this paper, the authors constructively prove that for any ring \(R\) with Krull dimension \(\leq d\), the ring \(R<X>\) locally behaves like the ring \(R(X)\) or a localization of a polynomial ring of type \((S^{-1}R)[X]\) with \(S\) a multiplicative subset of \(R\) such that the Krull dimension of \(S^{-1}R\leq d-1\). As an application, they give a simple and constructive proof of the Lequain-Simis induction theorem which is an important variation of the Quillen induction theorem.
reviewed by: Ali Benhissi
Mathematics Subject Classification ID: 13F20
zbMATH DE Number: 5349020
zbMATH Keywords: constructive mathematics
zbMATH Keywords: Lequain-Simis induction theorem
zbMATH Keywords: Quillen-Suslin theorem
zbMATH Keywords: finitely generated projective modules
zbMATH Keywords: local-global principles
zbMATH Keywords: Prufer domains
zbMATH Keywords: arithmetical rings
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2007.12.004
OpenAlex ID: W4317639527
cites work: Projective modules over polynomial rings: a constructive approach
cites work: Projective modules over some non-noetherian polynomial rings
cites work: Hidden constructions in abstract algebra. Krull Dimension of distributive lattices and commutative rings
cites work: Generating non-Noetherian modules constructively
cites work: Dynamical method in algebra: Effective Nullstellensätze
cites work: Q5718574
cites work: Algorithmic theory of arithmetic rings, Prüfer rings and Dedekind rings. (Théorie algorithmique des anneaux arithmétiques, des anneaux de Prüfer et des anneaux de Dedekind.)
cites work: Finite conductor properties of \(R(X)\) and \(R\langle X\rangle\)
cites work: Q4039904
cites work: Serre's conjecture
cites work: Q4674273
cites work: Projective modules over R[\(X_ 1,\)\dots ,X//n], R a Prüfer domain
cites work: Q5454196
cites work: Constructive rereading of Artin-Schreier theory
cites work: Krull dimension, Nullstellensätze and dynamical evaluation
cites work: Hidden constructions in abstract algebra. I: Integral dependance.
cites work: Q4681541
cites work: Q3637541
cites work: Q4736172
cites work: Q3112383
cites work: Suslin's algorithms for reduction of unimodular rows
cites work: Q4138823
cites work: A course in constructive algebra
cites work: An algorithm for unimodular completion over noetherian rings
cites work: Q4413922
cites work: Projective modules over polynomial rings
cites work: Faisceaux algébriques cohérents
cites work: Q4124942
cites work: Making the use of maximal ideals constructive
cites work: Dynamical Gröbner bases
DOI: 10.1016/J.JALGEBRA.2007.12.004
description: scientific article; zbMATH DE number 5349020