Some ring-theoretic properties of almost \(P\)-frames (Q1047082)

An almost \(P\)-frame is a frame \(L\) in which every cozero element is regular (that is, for any cozero element \(a\) of \(L\), if the pseudocomplement of \(a\) is the bottom element of \(L\), then \(a\) is necessarily the top element of \(L\)) [\textit{R. N. Ball} and \textit{J. Walters-Wayland}, ``\(C\)- and \(C^*\)-quotients in pointfree topology'', Diss. Math. 412, 62 p. (2002; Zbl 1012.54025)]. Almost \(P\)-frames generalize almost \(P\)-spaces in the sense that for any topological space \((X,{\mathfrak O}X)\), \((X,{\mathfrak O}X)\) is an almost \(P\)-space if and only if the frame \({\mathfrak O}X\) of its open sets is an almost \(P\)-frame. This paper provides several characterizations of almost \(P\)-frames (and \(P\)-frames). They enable the author to give, in the appendix, frame versions of the theorem of Gelfand and Kolmogorov concerning maximal ideals of the ring \(C(X)\), and of a lemma due to J. D. McKight jun., which tells us where all the ideals of \(C(X)\) lie.