On Structure Spaces of Ideals in Rings of Continuous Functions

A ring of continuous functions is a ring of the form C(X), the ring of all continuous real-valued functions on a completely regular Hausdorff space X. With each ideal / of C(X), we associate certain subalgebras of C(X), and discuss their structure spaces. We give necessary and sufficient conditions for two ideals in rings of continuous functions to have homeomorphic structure spaces. Introduction. For a subset A of C(X), we define rA to be {/+ c\f £ A and c £ R\. (R denotes the set of all real numbers, and we make the usual identification between the real number c and the function which maps every x £ X onto c.) We denote by A" the closure of A in the uniform topology. With each ideal / in C(X), we associate four subalgebras of C(X), I itself, I", rl, and r(lu). (in [A], rl and r(Iu) were denoted by (/) and (/"), respectively.) In this paper, we characterize the maximal ideals of /", rl, and r(lu) (the maximal ideals of I were characterized in [4]) and then endow these sets of maximal ideals with the hull-kernel topology. We then investigate the resulting structure spaces. In §1 we show that the prime and maximal ideals of rl ate the intersections of rl respectively with the prime and maximal ideals of C(X). This allows us to establish homeomorphisms between structure spaces of rl and modifications of structure spaces of C(X). We also show that the structure space of rl is (homeomorphic to) the one-point compactification of the structure space of /. In §2 we discuss the prime and maximal ideals of the algebras /" and r(lu). We show that / and /" have the same structure space and that rl and r(l") do also. In view of the fact that r(l") is (isomorphic to) a ring of continuous functions (see [4, 5.6]), it is thus established that every ideal in C(X) is a real ideal in a subalgebra of C(X) which has the same structure space as a ring of continuous functions. Results in §§1 and 2 generalize certain results in [5]. Received by the editors August 1, 1972 and, in revised form, May 23, 1973. AMS (MOS) subject classifications (1970). Primary 46E25, 54C40.