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An interesting class of ideals in subalgebras of \(C(X)\) containing \(C^*(X)\). (English) Zbl 1199.54153
Summary: We give a duality between a special type of ideals of subalgebras of \(C(X)\) containing \(C^*(X)\) and \(z\)-filters of \(\beta X\) by generalization of the notion \(z\)-ideal of \(C(X)\). We also use it to establish some intersecting properties of prime ideals lying between \(C^*(X)\) and \(C(X)\). For instance, we may mention that such an ideal becomes prime if and only if it contains a prime ideal. Another interesting consequence is that for such an ideal the residue class ring is totally ordered if and only if it is prime.

MSC:
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54C35 Function spaces in general topology
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