Rings of continuous functions in which every finitely generated ideal is principal

Abstract

An abstract ring in which all finitely generated ideals are principal will be called an F-ring. Let C(X) denote the ring of all continuous real-valued functions defined on a completely regular (Hausdorff) space X. This paper is devoted to an investigation of those spaces X for which C(X) is an F-ring. In any such study, one of the problems that arises naturally is to determine the algebraic properties and implications that result from the fact that the given ring is a ring of functions. Investigation of this problem leads directly to two others: to determine how specified algebraic conditions on the ring are reflected in topological properties of the space, and, conversely, how specified topological conditions on the space are reflected in algebraic properties of the ring. Our study is motivated in part by some purely algebraic questions concerning an arbitrary F-ring S-in particular, by some problems involving matrices over S. Continual application will be made of the results obtained in the preceding paper [4]. This paper will be referred to throughout the sequel as GH. We wish to thank the referee for the extreme care with which he read both this and the preceding paper, and for making a number of valuable suggestions. The outline of our present paper is as follows. In ?1, we collect some preliminary definitions and results. ?2 inaugurates the study of F-rings and F-spaces (i.e., those spaces X for which C(X) is an F-ring). The space of reals is not an F-space; in fact, a metric space is an F-space if and only if it is discrete. On the other hand, if X is any locally compact, a-compact space (e.g., the reals), then fX-X is an F-space. Examples of necessary and sufficient conditions for an arbitrary completely regular space to be an F-space are: (i) for every f C(X), there exists k E C(X) such that f k Jf J; (ii) for every maximal ideal M of C(X), the intersection of all the prime ideals of C(X) contained in M is a prime ideal. In ??3 and 4, we study Hermite rings and elementary divisor rings(2).