The maximal orders which contain a given idealiser ring
type: Item
instance of: scholarly article
title: The maximal orders which contain a given idealiser ring
zbMATH Open document ID: 0964.16018
published in: Journal of Algebra
publication date: 2001-04-18T00:00:00Z
review text: Let \(C\) be a Dedekind domain, and let \(S\) be a maximal \(C\)-order in a simple Artinian ring \(Q\). Consider a proper essential right ideal \(K\) of \(S\) with \(SK=S\). The authors investigate the set \(\mathcal M\) of maximal \(C\)-orders in \(Q\) which contain the idealizer \(R\) of \(K\). They prove that the maximal \(C\)-orders in \(\mathcal M\) are in one-to-one correspondence with the right ideals \(I\) of \(S\) which satisfy \(K\subset I\) and \(RI=I\). If \(B\) denotes the bound of \(K\), that is, the largest two-sided ideal of \(S\) which is contained in \(K\), they make the further assumption that the right ideal \(K/B\) of \(S/B\) is generated by an idempotent element. (This always holds if \(R\) is hereditary.) Under this assumption they show that \(\mathcal M\) is finite, and that there is a bijection between \(\mathcal M\) and the set of ideals of \(S\) which contain \(B\). Moreover, they classify the invertible ideals of \(R\) and study their action on \(\mathcal M\) by conjugation. This generalizes classical results of Harada on hereditary orders over a Dedekind domain.
reviewed by: Wolfgang Rump
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16D25
zbMATH DE Number: 1539380
zbMATH Keywords: Dedekind domains
zbMATH Keywords: maximal orders
zbMATH Keywords: hereditary orders
zbMATH Keywords: right ideals
zbMATH Keywords: idealizers
zbMATH Keywords: invertible ideals
author: A. W. Chatters
author: Stephen Brewster
MaRDI profile type: Publication
full work available at URL: jabr.2000.8393
OpenAlex ID: W2022080488
cites work: Near-dedekind rings
cites work: A Note on Dedekind Prime Rings
cites work: Idealizers and hereditary Noetherian prime rings
DOI: 10.1006/JABR.2000.8393
description: scientific article; zbMATH DE number 1539380