Valuation rings in finite-dimensional division algebras
type: Item
instance of: scholarly article
title: Valuation rings in finite-dimensional division algebras
zbMATH Open document ID: 0667.16017
DOI: 10.1016/0021-8693(89)90190-7
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: By a valuation ring in a division algebra \(D\) the authors understand a subring containing for each \(x\) in \(D\) either \(x\) or \(x^{-1}\). Let \(D\) be a division algebra of dimension \(n^ 2\) over its centre \(K\) and let \(B\) be a valuation ring in \(K\). Then the authors show that in \(D\) there are at most \(n\) valuation rings \(B'\) such that \(B'\cap K=B\), and any two such extensions are conjugate by an element of \(D\). Further the integral closure \(T\) of \(B\) in \(D\) is a subring if and only if \(B\) has extensions to \(D\) and when this is so then \(T\) is the intersection of all these extensions. In the special case when \(K\) is a global field it is shown that only a finite number of valuation rings of \(K\) are extendable to \(D\). These results are illustrated by some examples.
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 16W60
zbMATH DE Number: 4091698
zbMATH Keywords: valuation ring
zbMATH Keywords: division algebra
zbMATH Keywords: integral closure
zbMATH Keywords: extensions
author: H. H. Brungs
author: Joachim Graeter
reviewed by: Paul M. Cohn
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90190-7
OpenAlex ID: W2025622708
cites work: Q3216754
cites work: Q4745965
cites work: Valuation theory
cites work: Über die Bewertungen endlich dimensionaler Divisionsalgebren
cites work: Zur Theorie nicht kommutativer Prüferringe
cites work: Bewertungen nicht kommutativer Körper
cites work: Q5796078
cites work: Extending Valuations to Finite Dimensional Division Algebras
description: scientific article; zbMATH DE number 4091698