Some remarks concerning pseudocompactness in pointfree topology

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.