Characterizations of Ideals in Intermediate C-rings A(X) via the A-compactifications of X

Let X be a completely regular topological space. An intermediate ring is a ring A(X) of continuous functions satisfying C^∗(X) ⊆ A(X) ⊆ C(X). In Redlin and Watson (1987) and in Panman et al. (2012), correspondences Z_A and Z_A are defined between ideals in A(X) and z-filters on X, and it is shown that these extend the well-known correspondences studied separately for C^∗(X) and C(X), respectively, to any intermediate ring. Moreover, the inverse map Z^←_A sets up a one-one correspondence between the maximal ideals of A(X) and the z-ultrafilters on X. In this paper, we define a function K_A that, in the case that A(X) is a C-ring, describes Z_A in terms of extensions of functions to realcompactifications of X. For such rings, we show that Z^←_A maps z-filters to ideals. We also give a characterization of the maximal ideals in A(X) that generalize the Gelfand-Kolmogorov theorem from C(X) to A(X).