Journal ArticleDOI
Bezout and prufer f-rings
Jorge Martinez,Scott Woodward +1 more
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In this paper, the authors describe Bezout and Prufer f-rings in terms of their localizations, and give a counter-example to show that the converse of the last assertion is false.Abstract:
This article describes Bezout and Prufer f-rings in terms of their localizations. All f-rings here are corrmutative, semi prime and possess an identity; they also have the bounded inversion property: a >1 implies that a is a multiplicative unit. The two main theorems are as follows: (1) A is a Bezout f-ring if and only if each localization at a maximal ideal is a (totally ordered) valuation ring; (2) Each Prufer f-ring is quasi-Bezout, and if each localization of A is a Prufer f-ring then so is A. We give a counter-example to show that the converse of the last assertion is false.read more
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Pointfree Topology and the Spectra of f-Rings
TL;DR: For various kinds of rings, either without or with additional structure, one considers several types of spectra, usually spaces of certain ideals, with some appropriate topology, such as prime ideals, maximal ideal, minimals prime ideals and closed prime ideals as discussed by the authors.
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Some algebraic characterizations of F-frames
TL;DR: In this paper, it was shown that a frame L is an F-frame precisely when the ring R of continuous real-valued functions on L is Bezout, and that a commutative ring with identity is called almost weak Baer if the annihilator of each element is generated by idempotents.
Journal ArticleDOI
Lattice-ordered algebras that are subdirect products of valuation domains
TL;DR: In this paper, it was shown that if A = C(X)) then if A is an SV-ring then it has finite rank, and if A=C(X) is the ring of all real-valued continuous functions on a Tychonoff space, the rank of X is defined to be the rank in the f-ring C(x), and X is called an SV space if C( x) is an S ring.
Book ChapterDOI
Rings of Continuous Functions as Real Closed Rings
TL;DR: Real closed rings were first introduced in this article for the purpose of establishing a new foundation for semi-algebraic geometry, and they have been used to study the topology of the real spectrum.
References
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Journal ArticleDOI
Rings of Continuous Functions.
Book
Rings of continuous functions
TL;DR: In this paper, the Stone-Czech Compactification is used to define a topological space, and a list of symbols for topological spaces is presented, including cardinal of closed sets in Beta-x, homomorphisms and continuous mapping.