Fraction-dense algebras and spaces
type: Item
instance of: scholarly article
title: Fraction-dense algebras and spaces
zbMATH Open document ID: 0789.13005
DOI: 10.1007/BF00046636
author: Anthony W. Hager
author: Jorge Martinez
published in: Acta Applicandae Mathematicae
publication date: 1993-09-30T00:00:00Z
review text: In the following, every ring will be commutative with identity and semiprime. An \(f\)-ring is a lattice-ordered ring in which \(a \wedge b=0\) implies \(a \wedge bc=0\) for each \(c \geq 0\). For an \(f\)-ring \(A\) let \(q(A)\) be its classical ring of quotients and \(Q(A)\) be its maximal ring of quotients. Fundamental properties of these quotient rings can be found in the book by \textit{J. Lambek}: ``Lectures on rings and modules'' (2nd edition 1976; Zbl 0365.16001). The authors discuss the problem, when \(q(A)\) is rigid in \(Q(A)\). In this case, they call the \(f\)-ring \(A\) fraction-dense. Let \(C(X)\) be the ring of all real-valued continuous functions of a Tychonoff space \(X\). If \(C(X)\) is fraction-dense, then \(X\) is called fraction-dense. By theorem 1.1 an \(f\)-ring \(A\) is fraction-dense if and only if the space \(M(A)\) of minimal prime ideals is compact and extremally disconnected. This implies that a Tychonoff space is fraction-dense if and only if its Stone-Čech compactification is fraction-dense. -- In \S2 fraction- dense spaces are studied. Among other things, the following characterization is given: A compact space \(X\) is fraction-dense if and only if every regular open subset of \(X\) densely contains a cozero-set. - - The main result of \S3 is the following: Let \(G\) be an Archimedean lattice-ordered group with order unit. If \(G\) is absolute (this means that \(\text{Min} (G)\) is compact and extremally disconnected), then its \(o\)-completion and Dedekind completion coincide.
reviewed by: Dieter Remus
Mathematics Subject Classification ID: 13J25
Mathematics Subject Classification ID: 54C40
Mathematics Subject Classification ID: 13B30
Mathematics Subject Classification ID: 20F60
zbMATH DE Number: 417083
zbMATH Keywords: \(f\)-ring
zbMATH Keywords: ordered ring
zbMATH Keywords: ring of real-valued continuous functions
zbMATH Keywords: space of minimal prime ideals
zbMATH Keywords: fraction-dense Tychonoff space
zbMATH Keywords: ring of quotients
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00046636
OpenAlex ID: W3168396000
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description: scientific article; zbMATH DE number 417083