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

Abstract

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 $\mathcal{Z}_{A}$ and $\mathrm{3}_{A}$ 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
$\mathcal{Z}^{-1}_{A}$ 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 $A(X)$.
After wards, we show that $\mathcal{Z}^{-1}_{A}$ and
$\mathrm{3}^{-1}_{A}$, map essential (resp., free) $z$-filters on
$X$ to essential (resp., free) ideals in $A(X)$. 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 $A(X)$.