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 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.