Some abstract maximal ideal-like spaces
type: Item
instance of: scholarly article
title: Some abstract maximal ideal-like spaces
zbMATH Open document ID: 0685.06003
DOI: 10.1007/BF01243875
author: George Georgescu
author: Ioana Voiculescu
published in: Algebra Universalis
publication date: 1989-01-01T00:00:00Z
review text: The motivation of for this paper is to find a common pattern for some facts which occur in a much similar way in both cases of distributive lattices and of commutative rings. Namely, the existence of a certain spatial frame of ideals (the o-ideals for lattices [\textit{W. H. Cornish}, Rev. Roum. Math. Pur. Appl. 22, 1059-1067 (1977; Zbl 0382.06011); the authors, J. Pure Appl. Algebra 45, 213-223 (1987; Zbl 0646.06009)], and the virginal ideals for rings [\textit{F. Borceux}, \textit{H. Simmons} and \textit{G. van den Bossche}, Proc. Lond. Math. Soc., III. Ser. 48, 230-246 (1984; Zbl 0536.18003)]) which is in particular cases isomorphic to the frame of open sets of the maximal ideal space. This, in turn, ensures interesting properties for sheaf representations. Notice that in the first case the lattice of all ideals is distributive, while in the second it is not. The abstract setting that is found to encompass both cases above is that of integral complete lattice-ordered groupoids [\textit{G. Birkhoff}, Lattice theory, 3rd ed. (1967; Zbl 0153.025)], which are algebraic lattices. For brevity, the structures studied are called multiplicative ideal structures (abbreviated mi-structures). The paper has three sections. The first one contains some preliminary material and the description of the frame of radical elements in a multiplicative ideal structure, which are an obvious generalization of the radical ideals in a ring. Notions and results related to some of those in this paragraph appear in slightly different contexts in papers of \textit{K. Keimel} [Acta Math. Acad. Sci. Hung. 23, 51-69 (1972; Zbl 0265.06016)] and \textit{J. Martinez} [Ordered algebraic structures, Lect. Notes Pure Appl. Math. 99, 125-137 (1985; Zbl 0576.06015)]. In the second section the notion of virginal element in a commutative mi-structure is introduced and the relationship between the frames of virginal elements and radical elements, respectively, is established. This permits the use of some ge…
Mathematics Subject Classification ID: 06B10
Mathematics Subject Classification ID: 06B35
Mathematics Subject Classification ID: 06D05
Mathematics Subject Classification ID: 08A30
Mathematics Subject Classification ID: 13A15
Mathematics Subject Classification ID: 06F05
zbMATH DE Number: 4122043
zbMATH Keywords: distributive lattices
zbMATH Keywords: commutative rings
zbMATH Keywords: spatial frame of ideals
zbMATH Keywords: frame of open sets
zbMATH Keywords: maximal ideal space
zbMATH Keywords: sheaf representations
zbMATH Keywords: integral complete lattice-ordered groupoids
zbMATH Keywords: multiplicative ideal structures
zbMATH Keywords: virginal elements
zbMATH Keywords: radical elements
reviewed by: Richard A. Aló
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4122043