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Real closed rings. I: Residue rings of rings of continuous functions. (English) Zbl 0605.54014

Starting from the well-known result that if M is a maximal ideal in the ring of continuous functions C(X), then C(X)/M is a real closed field, the authors attack the problem of classifying the real closed ideals P of C(X). By a real closed ideal P they mean that P is a prime ideal with the property that C(X)/P is a real closed ring, that is to say, C(X)/P is a convex subring of a real closed field. In the sequel to the present paper [Ann. Pure Appl. Logic 25, 213-231 (1983; Zbl 0538.03028)] the authors enlarge upon the model theoretic aspects of their investigations, whereas in the present paper they make a very comprehensive study of real closed ideals in C(X). They show that for any prime z-ideal P in C(X), a necessary and sufficient condition that P should be real closed is that for every zero-set \(Z\subset X\) not belonging to Z(P) and every \(g\in C(X-Z)\) with \(0\leq g\leq 1\), we can find \(W\in Z(P)\) and \(h\in C(X)\) with \(0\leq h\leq 1\) such that \(h| (W-Z)=g| (W-Z)\). Their results characterize to some extent the real closed prime z-ideals in C(X). We recall that X is an F-space in case the ideal \(O^ p\) is prime for all points \(p\in \beta X\). The authors show that if X is an F-space, then every prime ideal of C(X) is real closed. Corresponding results for the ring \(C^*(X)\) are also established. We recall that a point p in \(\beta\) X is a \(\beta\) F-point in case the ideal \(O^ p\) is prime. By extending some work of C. W. Kohls [Ill. J. Math. 2, 505-536 (1958; Zbl 0105.094)] the present authors show that if p is a nonisolated \(G_{\delta}\)- and \(\beta\) F-point of X, then the maximal ideal \(M_ p\) has an immediate prime z-ideal predecessor which is real closed. An extensive study of the concrete spaces \(C(N^*)\) and \(C(D^*)\) is undertaken, where \(N^*\) and \(D^*\) denote the one-point compactifications of the countable and uncountable discrete spaces N and D, respectively. Among other things, the authors show that the existence of nonmaximal real closed ideals in \(C(N^*)\) is not provable within the axiomatic framework of ZFC, but that Martin’s axiom implies the existence of infinitely many nonmaximal real closed z-ideals in \(C(N^*)\). Furthermore, the existence of nonmaximal real closed ideals in \(C(D^*)\) is independent of the axiomatic framework of ZFC. Finally, the authors turn their attention to certain ideals in C(X) defined by measures on the space \(X=[0,1]\) whose existence is implied by Martin’s axiom.
Reviewer: J.V.Whittaker

MSC:

54C40 Algebraic properties of function spaces in general topology
54C50 Topology of special sets defined by functions
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54A35 Consistency and independence results in general topology
46E25 Rings and algebras of continuous, differentiable or analytic functions
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