Orderings of higher level and semilocal rings
type: Item
instance of: scholarly article
title: Orderings of higher level and semilocal rings
zbMATH Open document ID: 0438.13022
DOI: 10.1007/BF01214766
author: Thomas Craven
published in: Mathematische Zeitschrift
publication date: 1981-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/183665
review text: The concept of ordering of a field has been extended to that of a ``signature'' on a commutative semilocal ring \(A\), this being a homomorphism to the integers from the Witt ring of equivalence classes of nondegenerate symmetric bilinear forms over \(A\). This work has been carried out very successfully by M. Knebusch, A. Rosenberg and others. Recently E. Becker has developed a theory of ``higher level orderings'' for fields based on \(2^n\)-th powers rather than squares and Kleinstein and Rosenberg have defined corresponding Witt rings. In this paper these concepts are extended to semilocal rings. The main theorem states that if \(A\) is a local ring with residue field having more than two elements, then every ``signature of higher level'' has a corresponding prime ideal of \(A\) such that the signature extends uniquely to an ordering (of higher level) on the field of fractions of the quotient ring. Partial results are given for semilocal rings. From this theorem many of the usual results of Artin-Schreier theory can be extended to theorems concerning existence of these signatures and sums of \(2^n\)-th powers. For example, a local ring \(A\) has a signature of level \(n\) if and only if \(-1\) is not a sum of \(2^n\)-th powers. A corollary of this states the somewhat surprising result that if \(-1\) is a sum of squares, then it is a sum of \(2^n\)-th powers for every \(n\). Autorreferat.
Mathematics Subject Classification ID: 11E08
Mathematics Subject Classification ID: 11E81
Mathematics Subject Classification ID: 12D15
Mathematics Subject Classification ID: 12J10
zbMATH DE Number: 3683744
zbMATH Keywords: higher level orderings
zbMATH Keywords: semilocal rings
zbMATH Keywords: signature of higher level
zbMATH Keywords: Artin-Schreier theory
zbMATH Keywords: sums of 2n-th powers
zbMATH Keywords: sum of squares
MaRDI profile type: Publication
cites work: Q4748231
cites work: On prime ideals of a Witt ring over a local ring
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cites work: Signatures on semilocal rings
cites work: Structure of Witt Rings and Quotients of Abelian Group Rings
cites work: Q5790931
OpenAlex ID: W2085680857
description: scientific article; zbMATH DE number 3683744