Algebraic properties of rings of continuous functions

This paper is devoted to the study of algebraic properties of rings of continuous functions. Our aim is to show that these rings, even if they are highly nonnoetherian, have properties quite similar to the elementary properties of noetherian rings: we give going-up and going-down theorems, a characterization of z-ideals and of primary ideals having as radical a maximal ideal and a flatness criterion which is entirely analogous to the one for modules over principal ideal domains. Introduction. Throughout this paper, C(X) will denote the ring of realvalued continuous functions defined on a topological space X and C∗(X) will be the subring of bounded functions. The paper is divided into three sections. In the first one, we shall prove a theorem showing close relationships between topological properties of a continuous map X → S and algebraic properties of the induced morphism of rings C(S)→ C(X). Explicitly, we shall prove the following Theorem. If a continuous map X → S is open and closed (respectively , open and proper) then going-up and going-down theorems hold for the morphism C∗(S)→ C∗(X) (respectively , for C(S)→ C(X)). Using this theorem we shall prove that, under the same hypothesis, the continuous map between the prime spectra Spec(C∗(X)) → Spec(C∗(X)) is open and closed. Since the Stone–Čech compactification βX is homeomorphic to the maximal spectrum of C∗(X), this result generalizes that obtained by Isiwata [5] for the extension βX → βS. We think that this going-up and down theorem may also be used to establish other results concerning relationships between algebraic and topological properties. In fact, we have used it in [9] to characterize finite branched 1991 Mathematics Subject Classification: 54C40, 13B24, 13C11.