Some properties of the ideal of continuous functions with pseudocompact support

E. A. Abu Osba and H. Al-Ezeh

International Journal of Mathematics and Mathematical Sciences, 2001, vol. 27, 1-8

Abstract:

Let C ( X ) be the ring of all continuous real-valued functions defined on a completely regular T 1 -space. Let C Ψ ( X ) and C K ( X ) be the ideal of functions with pseudocompact support and compact support, respectively. Further equivalent conditions are given to characterize when an ideal of C ( X ) is a P -ideal, a concept which was originally defined and characterized by Rudd (1975). We used this new characterization to characterize when C Ψ ( X ) is a P -ideal, in particular we proved that C K ( X ) is a P -ideal if and only if C K ( X ) = { f ∈ C ( X ) : f = 0 except on a finite set } . We also used this characterization to prove that for any ideal I contained in C Ψ ( X ) , I is an injective C ( X ) -module if and only if coz I is finite. Finally, we showed that C Ψ ( X ) cannot be a proper prime ideal while C K ( X ) is prime if and only if X is an almost compact noncompact space and ∞ is an F -point. We give concrete examples exemplifying the concepts studied.

Date: 2001
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:304840

DOI: 10.1155/S0161171201010389

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