C and C* among intermediate rings

Given a completely regular Hausdorff space X, an intermediate ring A(X) is a ring of real valued continuous functions between C*(X) and C(X). We discuss two correspondences between ideals in A(X) and z-filters on X, both reviewing old results and introducing new results. One correspondence, ZA, extends the well-known correspondence between ideals in C*(X) and z-filters on X. The other, ζA, extends the natural correspondence between ideals in C(X) and z-filters on X. This paper highlights how these correspondences help clarify what properties of C*(X) and C(X) are shared by all intermediate rings and what properties of C*(X) and C(X) characterize those rings among intermediate rings. Using these correspondences, we introduce new classes of ideals and filters for each intermediate ring that extend the notion of z-ideals and z-filters for C(X), and with ZA, a new class of filters for each intermediate ring A(X) that extends the notion of e-filter for C*(X).