A note on ideals of C∞(X)

For a topological space X, let Σ(X) be the family of all subrings of C(X) containing C⁎(X). In 1997, S.K. Acharyya, K.C. Chattopadhyay and D.P. Ghosh showed that for an A(X)∈Σ(X), the intersection of all free maximal ideals of A(X) is an ideal of C∞(X) containing CK(X). In this paper we define z∞-ideals and nicely balanced ideals of C∞(X) and show that if I is a z∞-ideal of C∞(X) then I is the intersection of all free maximal ideals of A(X) for some A(X)∈Σ(X) if and only if I is nicely balanced. Examples related to z∞-ideals and nicely balanced ideals are given.