Prime ideal structure of rings of bounded continuous functions

Abstract

Introduction. The order structure of the family of prime ideals in the ring C of all real-valued continuous functions on a topological space has been extensively studied; in this paper we study the analogous problem in the subring C* of bounded functions. The fundamental property of prime ideals in C* is the following. MAIN THEOREM. Let M* be any maximal ideal of C* and let M be the unique maximal ideal of C such that the prime ideal Mn C* is contained in M*. Then every prime ideal contained in M* is comparable with MnGC*. The proof involves topological properties of the Stone-Cech compactification AX of a completely regular Hausdorff space X. Of special interest are the prime z-ideals of C*. When X is a locally compact, a-compact Hausdorff space, we show that the family of prime z-ideals of C*(X) contained in M* is composed of two subfamilies, order-isomorphic with naturally corresponding families of prime z-ideals in the rings C(X) and C(3X-X). 1. Preliminaries. We shall use the terminology and notation of the Gillman-Jerison text [3]. Applying [3, Theorem 3.9], we immediately reduce the problem of the prime ideal structure of C*(X), and its relation to C(X), to the case that X is a completely regular Hausdorff space. A basic property of prime ideals in rings of functions that will be used several times is a theorem of Kohls ([9, Theorem 2.4], see also [3, 14.8(a), 6.6(c)]): In the ring C(X), and also in C*(X), the prime ideals containing a given prime ideal form a chain. The proof of the main theorem is based on Kohls' result and the