When certain prime ideals in rings of continuous functions are minimal or maximal.
Let \(A\) be a commutative ring with identity. The symbols \(\mathrm{Spec}(A)\), \(\mathrm{Max}(A)\), \(\mathrm{Min}(A)\), \(\mathrm{Spec}_d(A)\) and \(\mathrm{Spec}_z(A)\) denote the set of prime ideals, maximal prime ideals, minimal prime ideals, prime \(d\)-ideals and prime \(z\)-ideals of \(A\), respectively. The main goal of this paper is to characterize frames \(L\) such that, for each of the following conditions, the ring \(\mathcal RL\) of continuous real functions on \(L\) satisfies that condition: (dMin) \(\mathrm{Spec}_d(A)\subseteq\mathrm{Min}(A)\); (dMax) \(\mathrm{Spec}_d(A)\subseteq\mathrm{Max}(A)\); (dMM) \(\mathrm{Spec}_d(A)\subseteq\mathrm{Min}(A)\cup\mathrm{Max}(A)\); (zMin) \(\mathrm{Spec}_z(A)\subseteq\mathrm{Min}(A)\); (zMax) \(\mathrm{Spec}_z(A)\subseteq\mathrm{Max}(A)\); (zMM) \(\mathrm{Spec}_z(A)\subseteq\mathrm{Min}(A)\cup\mathrm{Max}(A)\). First, it is observed that in the class of rings with no nonzero nilpotent elements (i.e. reduced rings), (zMin), (zMax) and (dMax) are all equivalent, and are equivalent to the property of being von Neumann regular. This implies that \(\mathcal RL\) satisfies any of these three conditions if and only if \(L\) is a \(P\)-frame, a generalization of the classical notion of a \(P\)-space. Then it is proved that \(\mathcal RL\) satisfies (dMin) (resp. (dMM), resp. (zMM)) precisely if \(L\) is \textit{cozero complemented} (resp. \textit{quasi cozero complemented}, resp. a \textit{quasi \(P\)-frame}). In the last part of the article the authors examine how far the theory of quasi \(P\)-frames parallels that of quasi \(P\)-spaces as defined by \textit{M. Henriksen} et al. [Commentat. Math. Univ. Carol. 44, No. 2, 261-294 (2003; Zbl 1098.54013)].
scientific article; zbMATH DE number 6473608
scientific article