Integral semihereditary orders, extremality, and Henselization
type: Item
instance of: scholarly article
title: Integral semihereditary orders, extremality, and Henselization
zbMATH Open document ID: 0871.16009
author: John S. Kauta
published in: Journal of Algebra
publication date: 1997-09-24T00:00:00Z
review text: The paper presents results of the author's Ph.D. thesis. Let \(V\) be a valuation domain with quotient field \(F\), and \(Q\) a central simple \(F\)-algebra. The author proves first that a semi-hereditary \(V\)-order in \(Q\) is extremal, i.e. coincides with the multiplier of its radical. (The converse holds for one-dimensional \(V\)!) Whereas Bézout \(V\)-orders always exist in \(Q\) (e.g. as Dubrovin valuation rings), there need not be semi-hereditary \(V\)-orders. If \(Q\) is a division algebra admitting a total valuation ring extending \(V\), there is a unique semi-hereditary \(V\)-order in \(Q\). If the skewfield part of \(Q\) admits an invariant valuation ring extending \(V\), the semi-hereditary \(V\)-orders are explicitly characterized (as in the classical situation). In this case, the maximal overorders and, in particular, the Dubrovin valuation rings are easily described. The author also shows that the semi-hereditary property is compatible with henselization and proves some pertinent facts in this context.
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16W60
zbMATH DE Number: 1013984
zbMATH Keywords: valuation domains
zbMATH Keywords: central simple algebras
zbMATH Keywords: Bézout orders
zbMATH Keywords: Dubrovin valuation rings
zbMATH Keywords: semi-hereditary \(V\)-orders
zbMATH Keywords: maximal overorders
reviewed by: Wolfgang Rump
MaRDI profile type: Publication
full work available at URL: jabr.1996.6844
OpenAlex ID: W2062440978
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DOI: 10.1006/JABR.1996.6844
description: scientific article; zbMATH DE number 1013984