Bulletin of the Belgian Mathematical Society - Simon Stevin

Some remarks concerning pseudocompactness in pointfree topology

Themba Dube

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Abstract

Recall that a frame $L$ is pseudocompact if $\mathcal{R}L=\mathcal{R}^*L$, where $\mathcal{R}L$ is the $f$-ring of real-valued continuous functions on $L$, and $\mathcal{R}^*L$ its bounded part. Using properties of uniform frames, Walters-Wayland proved that a completely regular frame $L$ is pseudocompact iff the frame homomorphism $\beta L\to L$ is coz-codense. In this note we give a purely ring-theoretic reaffirmation of this characterization by observing that a frame homomorphism $L\to M$ is coz-codense iff the ring homomorphism $\mathcal{R}L\to\mathcal{R}M$ it induces maps non-invertible elements to non-invertible elements, and that $L$ is pseudocompact iff every finitely generated proper ideal of $\mathcal{R}^*L$ is fixed. We also show that if $L$ is not pseudocompact, then $\mathcal{R}^*L$ has a non-maximal free prime ideal -- thus generalizing a 1954 result of Gillman and Henriksen.

Article information

Source
Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 2 (2013), 213-219.

Dates
First available in Project Euclid: 23 May 2013

Permanent link to this document
https://projecteuclid.org/euclid.bbms/1369316540

Digital Object Identifier
doi:10.36045/bbms/1369316540

Mathematical Reviews number (MathSciNet)
MR3082760

Zentralblatt MATH identifier
06180324

Subjects
Primary: 06D22: Frames, locales {For topological questions see 54-XX} 13A15: Ideals; multiplicative ideal theory 54C30: Real-valued functions [See also 26-XX]

Keywords
frame pseudocompact frame ring of real-valued continuous functions on a frame proper ideal

Citation

Dube, Themba. Some remarks concerning pseudocompactness in pointfree topology. Bull. Belg. Math. Soc. Simon Stevin 20 (2013), no. 2, 213--219. doi:10.36045/bbms/1369316540. https://projecteuclid.org/euclid.bbms/1369316540


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