Hereditary orders with infinitely many idempotent ideals
type: Item
instance of: scholarly article
title: Hereditary orders with infinitely many idempotent ideals
zbMATH Open document ID: 0531.16003
DOI: 10.1016/0022-4049(84)90087-2
published in: Journal of Pure and Applied Algebra
publication date: 1984-01-01T00:00:00Z
review text: In the classical theory of orders over a Dedekind domain, any hereditary order R is equivalent to a maximal order S, and R can be obtained from S as a finitely iterated idealizer. That a finite sequence of iterations suffices to get from S to R depends on R having only finitely many idempotent ideals. In the general theory of orders in simple artinian rings (in the sense of Goldie), the analogs of hereditary orders are hereditary noetherian prime (HNP) rings, and the analogs of maximal orders are Dedekind prime rings. \textit{J. C. Robson} proved that an HNP ring R is equivalent to a Dedekind prime ring if and only if R has only finitely many idempotent ideals, in which case R can be obtained from the Dedekind prime ring as a finitely iterated idealizer [J. Algebra 22, 45- 81 (1972; Zbl 0239.16003), Theorem 6.3]. Whether or not every HNP ring has this form remained an open question, which the present authors settle in the negative by constructing HNP rings with infinitely many idempotent ideals. Variations in their construction produce examples in which (a) all the idempotent maximal ideals belong to cycles of length two, or (b) all the idempotent maximal ideals belong to cycles but there is one cycle of each prime integer length, or (c) none of the idempotent maximal ideals belong to cycles. The present expository paper provides an overview of the construction process, which is a special type of infinitely iterated idealizer. Details will appear in a longer paper of the authors [''Constructions of hereditary noetherian rings and simple rings'', to appear in Proc. Lond. Math. Soc., III Ser.].
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16Dxx
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16N60
zbMATH DE Number: 3842067
zbMATH Keywords: orders over Dedekind domain
zbMATH Keywords: hereditary noetherian prime rings
zbMATH Keywords: hereditary order
zbMATH Keywords: maximal order
zbMATH Keywords: finitely iterated idealizer
zbMATH Keywords: idempotent ideals
zbMATH Keywords: orders in simple artinian rings
zbMATH Keywords: Dedekind prime rings
zbMATH Keywords: HNP rings
zbMATH Keywords: idempotent maximal ideals
author: J. T. Stafford
author: Robert B. jun. Warfield
reviewed by: Kenneth R. Goodearl
MaRDI profile type: Publication
full work available at URL: 0022-4049(84)90087-2
OpenAlex ID: W2056365280
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description: scientific article; zbMATH DE number 3842067