Some properties of the ideal of continuous functions with pseudocompact support (Q1599714)

Summary: Let \(C(X)\) be the ring of all continuous real-valued functions defined on a completely regular \(T_1\)-space. Let \(C_\Psi(X)\) and \(C_K(X)\) be the ideal of functions with pseudocomact 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 \textit{D. Rudd} [Fundam. Math. 88, No. 1, 53-59 (1975; Zbl 0307.54014)]. We use this new characterization to characterize when \(C_\Psi (X)\) is a \(P\)-ideal, in particular we prove that \(C_K(X)\) is a \(P\)-ideal if and only if \(C_K(X)= \{f\in C(X): f=0\) except on a finite set\}. We also use this characterization to prove that for any ideal \(I\) contained in \(C_\Psi(X)\), \(I\) is an injective \(C(X)\)-module if and only if \(\text{coz} I\) is finite. Finally, we show that \(C_\Psi(X)\) cannot be a proper prime ideal while \(C_K (X)\) is prime if and only if \(X\) is an almost compact noncompact space and \(\infty\) is an \(F\)-point. We give concrete examples exemplifying the concepts studied.