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.