Remarks on subrings of C(X) of the form I + C* (X)

Abstract

Let A(X) be an intermediate subring of C(X) and I be an ideal of A(X). We give some topological characterizations for the smallest z-ideal I_z (A) containing I and the largest z-ideal I_z (A) contained in I. Using this, we generalize some facts in the context of C(X) to the intermediate subrings of C(X). Furthermore, by investigating some properties of subrings of the form I + C^∗(X), where I is an ideal of C(X), we show that every intermediate C-algebra of C(X) is generated by a family of subrings of the form I + C^∗(X). Moreover, we characterize realcompactifications of X (specially, those which have compact remainder) via the subrings I + C^∗(X).