An innocent theorem of Banaschewski, applied to an unsuspecting theorem of De Marco, and the aftermath thereof (Q2836446)

A commutative ring with identity is called a pm-ring if every prime ideal in it is contained in a unique maximal ideal. Typical examples are the rings \(C(X)\) of real-valued continuous functions on topological spaces \(X\). In [Rend. Sem. Mat. Univ. Padova 69, 289--304 (1983; Zbl 0543.13004)], \textit{G. De Marco} proves that in any pm-ring \(A\), the lattice of pure ideals (an ideal is pure if every member is a product of itself with a member of the ideal) is isomorphic to the frame \(\mathfrak{O}(\text{Max}(A))\), where \(\text{Max}(A)\) is endowed with the hull-kernel topology. In the (rather flamboyantly-titled) paper under review, the author proves that De Marco's theorem is an easy consequence of a theorem of Bernhard Banaschewski about compact normal frames, obtained by interpreting the latter in the frame \(\text{Rad}(A)\) of radical ideals of \(A\). If \(A\subseteq B\) is an extension of rings, not every maximal ideal of \(B\) necessarily contracts to a maximal ideal of \(A\). Indeed, for a Tychonoff space \(X\), every maximal ideal of \(C(X)\) contracts to a maximal ideal of \(C^*(X)\) if and only if \(X\) is pseudocompact. In this paper the author introduces what he calls \(s\)-maps (these are frame homomorphisms which are compatible with saturation, in some sense), and then applies them to commutative ring extensions where maximal ideals contract to maximal ideals.NEWLINENEWLINERecently, algebraic frames have, at the behest of Jorge Martínez and some of his former students (mainly Eric Zenk and Warren McGovern), found themselves performing tasks way beyond their original brief and mandate. This paper continues in that spirit by probing commutative rings through frame-theoretic techniques.