Essential Ideals in Subrings of C(X) that Contain C(X)

Let A(X) be a subring of C(X) that contains C*(X). In Redlin and Watson (1987) and in Panman et al. (2012), correspondences ZAand ZAare 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 A(X). Moreover, the inverse map Z A − 1 A sets up a one-one correspondence between the maximal ideals of A(X) and the z-ultrafilters on X. In this paper, first, we characterize essential ideals in A(X). Afterwards, we show that Z A − 1 A maps essential (resp., free) z-filters on X to essential (resp., free) ideals in A(X) and Z A − 1 A maps essential ZA-filters to essential ideals. Similar to C(X) we observe that the intersection of all essential minimal prime ideals in A(X) is equal to the socle of A(X). Finally, we give a new characterization for the intersection of all essential maximal ideals of A(X).