Sums of squares in nonreal commutative rings
type: Item
instance of: scholarly article
title: Sums of squares in nonreal commutative rings
zbMATH Open document ID: 1401.11080
author: David B. Leep
author: Detlev Hoffmann
published in: Israel Journal of Mathematics
publication date: 2017-11-17T00:00:00Z
review text: Since 50 years there appeared a lot of papers on sums of squares in a (commutative) ring \(R\) and on the invariants \(s(R)\) (level) and \(p(R)\) (Pythagoras number). We should perhaps mention the important papers by \textit{M. Peters} [Math. Ann. 195, 309--314 (1972; Zbl 0217.04201); J. Reine Angew. Math. 268/269, 318--323 (1974; Zbl 0287.10034)], \textit{R. Baeza} [Math. Ann. 207, 121--131 (1974; Zbl 0248.13031); Math. Ann. 215, 13--21 (1975; Zbl 0287.13006); Quadratic forms over semilocal rings. Springer, Cham (1978; Zbl 0382.10014); Arch. Math. 33, 226--231 (1979; Zbl 0421.10015)], \textit{Z. D. Dai} et al. [Bull. Am. Math. Soc., New Ser. 3, 845--848 (1980; Zbl 0435.10016)] and \textit{D. W. Hoffmann} [J. Am. Math. Soc. 12, No. 3, 839--848 (1999; Zbl 0921.11018)] in the list of references. In the present paper the authors introduce two new invariants of \(R\) which they call ``metalevel'' and ``hyperlevel''. They allow a more systematic treatment and slightly sharper inequalities between the (four) invariants of \(R\). Several of the earlier proofs can now be simplified. Nevertheless the paper also contains a lot of new examples where the ``exact values'' of the invariants could not be determined.
reviewed by: Albrecht Pfister
Mathematics Subject Classification ID: 11E25
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13F25
zbMATH DE Number: 6808420
MaRDI profile type: Publication
full work available at URL: s11856-017-1575-y
OpenAlex ID: W2757997822
cites work: On the level of principal ideal domains
cites work: Über die Torsion der Witt-Gruppe \(W_q(A)\) eines semi-lokalen Ringes
cites work: Über die Stufe eines semi-lokalen Ringes
cites work: Quadratic forms over semilocal rings
cites work: Über die Stufe von Dedekind Ringen
cites work: The Pythagoras number of some affine algebras and local algebras.
cites work: Levels in algebra and topology
cites work: Sums of Three Integer Squares in Complex Quadratic Fields
cites work: Pythagoras numbers of fields
cites work: Q5670744
cites work: Round forms over semilocal rings
cites work: Q2778950
cites work: Q3160258
cites work: The level of orders of integers in algebraic number fields
cites work: Summen von Quadranten in Zahlringen.
cites work: Zur Darstellung von -1 Als Summe von Quadraten in Einem Körper
cites work: Q4003413
DOI: 10.1007/S11856-017-1575-Y
description: scientific article; zbMATH DE number 6808420