Two Functors Induced by Certain Ideals of Function Rings

Dube, Themba; Ighedo, Oghenetega

doi:10.1007/s10485-013-9320-0

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Some Properties Of Z(rl)1

Introduction1

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2017

2020

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Cited by 4 publications

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“…It is shown in [8], Proposition 3.10 that Z(RL) is regular if and only if L is a P -frame. On the other hand, D(RL) is regular if and only if L is cozero-complemented, see [9], Proposition 3.6.…”

Section: Some Properties Of Z(rl)mentioning

confidence: 99%

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When spectra of lattices of $z$-ideals are Stone-Čech compactifications

Dube¹

2017

Math.Boh.

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Let X be a completely regular Hausdorff space and, as usual, let C(X) denote the ring of real-valued continuous functions on X. The lattice of z-ideals of C(X) has been shown by Martínez and Zenk (2005) to be a frame. We show that the spectrum of this lattice is (homeomorphic to) βX precisely when X is a P-space. This we actually show to be true not only in spaces, but in locales as well. Recall that an ideal of a commutative ring is called a d-ideal if whenever two elements have the same annihilator and one of the elements belongs to the ideal, then so does the other. We characterize when the spectrum of the lattice of d-ideals of C(X) is the Stone-Čech compactification of the largest dense sublocale of the locale determined by X. It is precisely when the closure of every open set of X is the closure of some cozero-set of X.

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Section: Some Properties Of Z(rl)mentioning

confidence: 99%

“…The lattice D(RL) of d-ideals of RL is also a frame. Some categorical connections between Z(RL) and D(RL) are studied in [9].…”

Section: Introductionmentioning

confidence: 99%