An interesting class of ideals in subalgebras of \(C(X)\) containing \(C^*(X)\). (English) Zbl 1199.54153

Acharyya, Sudip Kumar; De, Dibyendu

Commentat. Math. Univ. Carol. 48, No. 2, 273-280 (2007).

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

Keywords:

Stone-Čech compactification; rings of continuous functions; maximal ideals; \(z^{\beta }_A\)-ideals