Witt rings of LG-rings
type: Item
instance of: scholarly article
title: Witt rings of LG-rings
zbMATH Open document ID: 0921.11019
DOI: 10.1016/S0019-3577(98)80019-0
author: Ires Dias
author: Artibano Micali
published in: Indagationes Mathematicae. New Series
publication date: 1999-05-04T00:00:00Z
review text: An LG-ring \(A\) is a commutative ring with \(1\neq 0\) such that the following local-global principle holds: if a polynomial \(f\in A[X_1,\cdots ,X_n]\), \(n\geq 1\), represents an invertible element over the localization \(A_{\mathcal P}\) for each prime ideal \({\mathcal P}\) of \(A\), then \(f\) represents an invertible element over \(A\). Let \(W(A)\) (resp. \(W_q(A)\)) be the Witt ring of regular bilinear (resp. regular quadratic) forms over the LG-ring \(A\). It is shown that there exists a bijection between the minimal prime ideals of \(W(A)\) and the signatures of \(A\) (resp. the set of maximal orderings of \(A\) provided \(| A/{\mathcal M}| \geq 3\) for each maximal ideal of \(A\)). Furthermore, the authors prove that if \(B\) is a Galois ring extension of \(A\) whose Galois group \(G\) is of odd order, then \(W(A)=W(B)^G\) (resp. \(W_q(A)=W_q(B)^G\)) with a suitably defined action of \(G\) on \(W(B)\) (resp. \(W_q(B)\)). Some of these results have been known before in the special case of semilocal rings (see, e.g., the lecture notes of \textit{R. Baeza} [Quadratic forms over semilocal rings, Lect. Notes Math. 655 (Berlin: Springer-Verlag, 1978; Zbl 0382.10014)]), and some of the arguments used in the case of semilocal rings carry over to LG-rings without major changes.
reviewed by: Detlev Hoffmann
Mathematics Subject Classification ID: 11E08
Mathematics Subject Classification ID: 13H99
Mathematics Subject Classification ID: 13B05
Mathematics Subject Classification ID: 11E81
zbMATH DE Number: 1251263
zbMATH Keywords: quadratic forms
zbMATH Keywords: LG-rings
zbMATH Keywords: Witt ring of bilinear forms over a ring
zbMATH Keywords: Witt ring of quadratic forms over a ring
zbMATH Keywords: signatures
zbMATH Keywords: orders
zbMATH Keywords: minimal prime ideals
zbMATH Keywords: Galois ring extensions
MaRDI profile type: Publication
cites work: Quadratic forms over semilocal rings
cites work: Module equivalences: Local to global. When primitive polynomials represent units
cites work: On prime ideals of a Witt ring over a local ring
cites work: Signatures on semilocal rings
cites work: Structure of Witt Rings and Quotients of Abelian Group Rings
cites work: Real closures of commutative rings. I.
cites work: Q3712427
cites work: Q5603302
cites work: An introduction to real algebra
cites work: Projective Modules Over Rings with Many Units
full work available at URL: s0019-3577(98)80019-0
OpenAlex ID: W1967142421
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description: scientific article; zbMATH DE number 1251263