Purity of the Ideal of Continuous Functions with Compact Support

<P>Let C(X) be the ring of all continuous real valued functions defined on a completely regular T₁-space. Let C_K(X) be the ideal of functions with compact support. Purity of C_K(X) is studied and characterized through the subspace X_L, the set of all points in X with compact neighborhoods (nbhd). It is proved that C_K(X) is pure if and only if X_L=∪f∈CK supp f. if C_K(X) and C_K(Y) are pure ideals, then C_K(X) is isomorphic to C_K(Y) if and only if X_L is homeomorphic to Y_L. It is proved that C_K(X) is pure and XL is basically disconnected if and only if for every f ∈C_K(X), the ideal (f ) is a projective C(X)-module. Finally it is proved that if C_K(X) is pure, then XL is an F'-space if and only if every principal ideal of C_K(X) is a flat C(X)-module. Concrete examples exemplifying the concepts studied are given.