Unions of minimal prime ideals in rings of continuous functions on compact spaces
In the ring \(C(X)\) of real-valued continuous functions on a Tychonoff space \(X\), every prime ideal is contained in a unique maximal ideal. Using the language of Gillman, Jerison, and Henriksen, the authors study the union of all the minimal prime ideals contained in the maximal ideal \(M_p = \{f \in C(X) : f(p) = 0\}\) for some \(p \in X\). A motivation for the study is given in the abstract of the paper. A space \(X\) is called a UMP-space if every maximal ideal of \(C(X)\) is the union of the minimal prime ideals contained in it. Every P-space (in which every prime ideal of \(C(X)\) is maximal) is a UMP-space, which in turn is an almost-P-space (in which every element of a minimal prime ideal of \(C(x)\) is a zero-divisor [\textit{A. I. Veksler}, Sov. Math., Dokl. 14, 1445--1450 (1973); translation from Dokl. Akad. Nauk SSSR 212, 789--792 (1973; Zbl 0291.54046), \textit{R. Levy}, Can. J. Math. 29, 284--288 (1977; Zbl 0342.54032)]. The almost-P-space \(\beta \omega - \omega\) is not a UMP-space. If \(D\) is an uncountable discrete space, \(\alpha D = D \cup \{d\}\) is the one-point compactification of \(D\), then \(\alpha D\) is a UMP-space. A Cantor set is not UMP, being metrizable but not discrete. The paper gives an algebraic characterization of a UMP-space when \(X\) is compact.
scientific article; zbMATH DE number 5770086