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

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, σ-compact space (e.g., the reals), then βX-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 fЄC(X), there exists kЄC(X) such that f=k|f|; (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. A necessary and sufficient condition that C(X) be an Hermite ring is that for all f, gЄC(X), there exist k, lЄC(X) such that f=k|f|, g=l|g|, and (k, l) = (1). We also construct an F-ring that is not an Hermite ring, and an Hermite ring that is not an elementary divisor ring. To produce these examples, we translate the algebraic conditions on C(X) into topological conditions on X, as indicated above. The construction of a ring having one algebraic property but not the other is then accomplished by finding a space that has the topological properties corresponding to the one, but not to the other. In §§5 and 6, we investigate some further special classes of F-rings, including regular rings and adequate rings. Appendices (§§7 and 8) touch upon various related questions. A diagram is included to show the implications that exist among the principal classes of spaces that have been considered.