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 coz I 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.