Integral semihereditary orders inside a Bézout maximal order
type: Item
instance of: scholarly article
title: Integral semihereditary orders inside a Bézout maximal order
zbMATH Open document ID: 0871.16010
author: John S. Kauta
published in: Journal of Algebra
publication date: 1997-09-24T00:00:00Z
review text: This paper is a continuation of the author's article reviewed above [see Zbl 0871.16009]. For a valuation domain \(V\) with quotient field \(F\), semi-hereditary \(V\)-orders \(H\) in a central simple \(F\)-algebra \(Q\) are considered which admit a Bézout overorder. If, in addition, this overorder \(B_0\) is a Dubrovin valuation ring of \(Q\), the author shows that extremal \(V\)-orders \(R\subset B_0\) are semi-hereditary. (The converse does hold in general and has been proved in the above mentioned previous article.) If \(B_0\) is just Bézout, there are only finitely many maximal overorders of \(H\), each of them being a Bézout order. Moreover, if \(\text{Rad }V\) is principal, the intersection of all Bézout overorders coincides with \(H\). Otherwise, it is shown that \(B_0\) is the unique maximal overorder of \(H\).
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16W60
zbMATH DE Number: 1013985
zbMATH Keywords: semi-hereditary \(V\)-orders
zbMATH Keywords: central simple algebras
zbMATH Keywords: Bézout overorders
zbMATH Keywords: Dubrovin valuation rings
zbMATH Keywords: extremal \(V\)-orders
zbMATH Keywords: maximal overorders
reviewed by: Wolfgang Rump
MaRDI profile type: Publication
full work available at URL: jabr.1996.6843
OpenAlex ID: W2074709767
cites work: On Maximally Central Algebras
cites work: Extensions of Valuation Rings in Central Simple Algebras
cites work: Q3312379
cites work: Q5185040
cites work: Valuation theory
cites work: Prime PI-rings in which finitely generated right ideals are principal
cites work: The Defektsatz for Central Simple Algebras
cites work: Bézout orders and Henselization
cites work: Hereditary Orders
cites work: Integral semihereditary orders, extremality, and Henselization
cites work: Two Remarks About Hereditary Orders
cites work: An approximation theorem for Dubrovin valuation rings
cites work: Maximal orders over valuation rings
cites work: Noncommutative Prüfer rings satisfying a polynomial identity
cites work: Q4061092
cites work: Dubrovin valuation rings and Henselization
DOI: 10.1006/JABR.1996.6843
description: scientific article; zbMATH DE number 1013985