Some Properties of the Ideal of Continuous Functions with Pseudocompact Support

Let C(X) be the ring of all continuous real-valued functions defined on a completely regular T1-space. Let CΨ (X) and CK(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 CΨ (X) is a P -ideal, in particular we proved that CK(X) is a P -ideal if and only if CK(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 CΨ (X), I is an injective C(X)-module if and only if cozI is finite. Finally, we showed that CΨ (X) cannot be a proper prime ideal while CK(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. 2000 Mathematics Subject Classification. 54C30, 54C40, 13C11.