The intersection theorem for orderings of higher level in rings
type: Item
instance of: scholarly article
title: The intersection theorem for orderings of higher level in rings
zbMATH Open document ID: 0758.12003
DOI: 10.1007/BF02567084
author: Ralph Berr
published in: Manuscripta Mathematica
publication date: 1993-01-16T00:00:00Z
full work available at URL: https://eudml.org/doc/155723
review text: The main result of this paper is an intersection theorem for orderings of commutative rings of arbitrary level. Let \(A\) be a commutative ring with 1 and \(T\subset A\) a preordering of level \(n\). In Theorem 6 the author proves the following equivalency for \(a\in A\): 1) \(a\in P^ +\) for all orderings \(P\supset T\), where \(P^ +=P\setminus(P\cap-P)\), 2) \(at=1+t'\) for some \(t,t'\in T\). To obtain this result he first studies \(M\)-convex ideals of \(A\), where \(M\) is a \(T\)-module. As an application he proves a Positivstellensatz of higher level for a certain class of formally real fields.
Mathematics Subject Classification ID: 12D15
Mathematics Subject Classification ID: 11E81
zbMATH DE Number: 91462
zbMATH Keywords: intersection theorem
zbMATH Keywords: orderings of commutative rings
zbMATH Keywords: preordering
zbMATH Keywords: Positivstellensatz of higher level
zbMATH Keywords: formally real fields
MaRDI profile type: Publication
cites work: Q4748231
cites work: Summen n-ter Potenzen in Körpern.
cites work: Q4289378
cites work: On rings admitting orderings and 2-primary chains of orderings of higher level
cites work: Null- and Positivstellensätze for generalized real closed fields
cites work: Positivbereiche in kommutativen Ringen
cites work: Signatures of higher level on rings with many units
description: scientific article; zbMATH DE number 91462