The order structure of the real spectrum of commutative rings
type: Item
instance of: scholarly article
title: The order structure of the real spectrum of commutative rings
zbMATH Open document ID: 0962.13024
published in: Journal of Algebra
publication date: 2001-06-14T00:00:00Z
review text: The real spectrum of a commutative ring \(A\) with unit is the set of all pairs \((P,\leq)\) where \(P\) is a prime ideal of \(A\) and \(\leq\) is a total (ring)-order on \(A/P\) endowed with a basis of open sets \(H(a_1,\dots,a_n)\) consisting of all elements \((P,\leq)\) such that \(0<(a_1/P) \wedge\dots \wedge 0< (a_n/P)\), for arbitrary finite sequences of \(A\). The present paper demonstrates that any jump-dense and complete root system is order-isomorphic to the real spectrum of a commutative unitary ring as a significant contribution to the problem of providing an ``intrinsic'' characterization of those special spaces that occur as real spectra of rings. As a byproduct the authors produce from their constructions needed in the proofs of their results several classes of rings with sufficiently nice properties generally of a formal-power-series like form over the real numbers or an inverse limit of such rings over a suitably directed set which may be of interest to students who otherwise might eschew perusing this quite interesting paper.
reviewed by: J. Neggers
Mathematics Subject Classification ID: 13J25
Mathematics Subject Classification ID: 14P05
Mathematics Subject Classification ID: 06F25
zbMATH DE Number: 1496009
zbMATH Keywords: order structure
zbMATH Keywords: real spectrum
author: Max A. Dickmann
author: Daniel Gluschankof
author: F. Lucas
MaRDI profile type: Publication
full work available at URL: jabr.2000.8304
OpenAlex ID: W2020693059
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DOI: 10.1006/JABR.2000.8304
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description: scientific article; zbMATH DE number 1496009