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.


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