Some remarks concerning pseudocompactness in pointfree topology (Q2376827)

In [Completeness and nearly fine uniform frames. Ottignies-Louvain-la-Neuve: University of Catholique de Louvain (Ph.D. Thesis) (1995)], \textit{J. Walters-Wayland} proved, using frame uniformities, that a completely regular frame \(L\) is pseudocompact if and only if the coreflection map \(j\) from the Stone-Čech compactification \(\beta L\) to \(L\) has the property that if \(c\) is a cozero element of \(\beta L\) such that \(j(c)=1\), then \(c=1\). In the paper under review, the author provides a purely ring-theoretic argument for that statement. He also proves that if \(L\) fails to be pseudocompact, then the ring of all bounded real-valued continuous functions on \(L\) has a free non-maximal prime ideal. The latter is a pointfree version of a 1954 result of \textit{L. Gillman} and \textit{M. Henriksen} [Trans. Am. Math. Soc. 77, 340--362 (1954; Zbl 0058.10003)].