Characterizations of Ideals in Intermediate C-Rings A(X) via the A-Compactifications of X

Mappings to Realcompactifications

In this paper, we introduce and study  a mapping from the collection of all  intermediate rings of $C(X)$ to the collection of all  realcompactifications of $X$ contained in $beta X$. By establishing the relations between this mapping and its converse,  we give a different approach to the main statements of De et. al. Using these, we provide different answers to the   four basic questions...

z_R-Ideals and z^0_R -Ideals in Subrings of R^X

Let X be a topological space and R be a subring of RX. By determining some special topologies on X associated with the subring R, characterizations of maximal fixxed and maximal growing ideals in R of the form Mx(R) are given. Moreover, the classes of zR-ideals and z0R-ideals are introduced in R which are topological generalizations of z-ideals and z0-ideals of C(X), respectively. Various c...

z-Ideals and zº-Ideals in the Factor Rings of C(X)

Connections between C(X) and C(Y), where Y is a subspace of X

In this paper, we introduce a method by which we can find a close connection between the set of prime $z$-ideals of $C(X)$ and the same of $C(Y)$, for some special subset $Y$ of $X$. For instance, if $Y=Coz(f)$ for some $fin C(X)$, then there exists a one-to-one correspondence between the set of prime $z$-ideals of $C(Y)$ and the set of prime $z$-ideals of $C(X)$ not containing $f$. Moreover, c...

Topology Proceedings 45 (2015) pp. 301-313: Characterizing C(X) among intermediate C-rings on 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). We give a characterization of C(X) in terms of extensions of functions in A(X) to real-compactifications of X. We also give equivalences of properties involving the closure in the real-compactifications of X of zero-sets in X; we use these equivalences to ...