Finitely 1-convex f-rings

• Published in: Topology and its applications Vol. 158; no. 14; pp. 1888 - 1901
• Main Author: Larson, Suzanne
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.09.2011
• Subjects: 1-Convex; f-Rings; F-space; Finitely 1-convex; Finitely an F-space; SV f-ring; SV-space; SV-space; 54C40; 1-Convex; f-Rings; Finitely an F-space; F-space; 13A15; Finitely 1-convex; 06F25; SV f-ring
• ISSN: 0166-8641; 1879-3207
• DOI: 10.1016/j.topol.2011.06.025

This paper investigates f-rings that can be constructed in a finite number of steps where every step consists of taking the fibre product of two f-rings, both being either a 1-convex f-ring or a fibre product obtained in an earlier step of the construction. These are the f-rings that satisfy the algebraic property that rings of continuous functions possess when the underlying topological space is finitely an F-space (i.e. has a Stone–Čech compactification that is a finite union of compact F-spaces). These f-rings are shown to be SV f-rings with bounded inversion and finite rank and, when constructed from semisimple f-rings, their maximal ideal space under the hull-kernel topology contains a dense open set of maximal ideals containing a unique minimal prime ideal. For a large class of these rings, the sum of prime, semiprime, primary and z-ideals are shown to be prime, semiprime, primary and z-ideals respectively.