Ring ideals and the Stone-Čech compactification in pointfree topology
There are several ways of constructing the compact completely regular coreflection, \(\beta L\), of a completely regular frame \(L\). The original construction, which is lattice-theoretic and realizes \(\beta L\) as the frame of completely regular ideals of \(L\), goes back to \textit{B. Banaschewski} and \textit{C. J. Mulvey} [Houston J. Math. 6, 301--312 (1980; Zbl 0473.54026)]. There have subsequently been other constructions in terms of \(\ell\)-ideals of the \(f\)-ring \(\mathbb RL\) of continuous real-valued functions on \(L\) [see \textit{B. Banaschewski}, The real numbers in pointfree topology. Textos de Matemática. Série B. 12. Coimbra: Universidade de Coimbra, Departamento de Matemática (1997; Zbl 0891.54009)]. In the paper under review, \(\beta L\) is realized in terms of mere ring ideals of \(\mathbb RL\). Namely, it is the frame of Jacobson radical ideals of \(\mathbb RL\). Analogously, the compact zero-dimensional coreflection of \(L\) is described in terms of mere ring ideals of the ring \(\mathbb ZL\) of continuous integer-valued functions on \(L\). As the authors point out, the constructions above can be obtained by putting together several results in some papers by the first author. However, the direct proofs in the article accentuate the clarity of the results with great elegance.
scientific article; zbMATH DE number 5775710