Bézout orders and Henselization
type: Item
instance of: scholarly article
title: Bézout orders and Henselization
zbMATH Open document ID: 0826.16010
author: Darrell Haile
author: Patrick Morandi
author: A. R. Wadsworth
published in: Journal of Algebra
publication date: 1995-07-25T00:00:00Z
review text: Let \(V\) be a valuation ring of a field \(F\), and \(B\) a Bézout maximal \(V\)-order in some central simple \(F\)-algebra \(S\). If \(F_h\), \(V_h\) denotes Henselization, the authors show that \(B_h=V_h\otimes_V B\) is a semi-hereditary maximal \(V_h\)-order in \(S_h=F_h\otimes_F S\). Moreover, they prove that \(B_h\) is Bézout if and only if \(B\) is a Dubrovin valuation ring in \(S\), i.e. if \(B/\text{Rad }B\) is simple artinian, and for any \(s\in S\setminus B\) there are \(b,c\in B\) with \(bs,sc\in B\setminus\text{Rad }B\). Let \(n\), \(t\) be the sizes of the matrix rings \(S_h\) and \(B /\text{Rad }B\), respectively. The authors use the above theorem in order to give an alternative proof of Gräter's result that \(n/t\) coincides with the extension number of \(V\) in \(S\).
reviewed by: Wolfgang Rump
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16W60
Mathematics Subject Classification ID: 16S50
zbMATH DE Number: 763951
zbMATH Keywords: Bézout maximal order
zbMATH Keywords: central simple algebras
zbMATH Keywords: Henselization
zbMATH Keywords: semi- hereditary maximal orders
zbMATH Keywords: Dubrovin valuation rings
zbMATH Keywords: simple Artinian rings
zbMATH Keywords: matrix rings
zbMATH Keywords: extension number
MaRDI profile type: Publication
full work available at URL: jabr.1995.1094
OpenAlex ID: W2002270949
DOI: 10.1006/JABR.1995.1094
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description: scientific article; zbMATH DE number 763951