On the intrinsic topology and some related ideals of 𝐶(𝑋)

The above topology is defined and studied on C ( X ) C(X) , the ring of real-valued continuous functions on a completely regular Hausdorff space X X . The minimal ideals and the socle of C ( X ) C(X) are characterized via their corresponding z z -filters. We observe that these ideals are z z -ideals and X X is discrete if and only if the socle of C ( X ) C(X) is a free ideal. It is also shown that for a class of topological spaces, containing all P P -spaces, the family C k ( X ) {C_k}(X) of functions with compact support is identical with the intersection of the free maximal ideals of C ( X ) C(X) .