Intermediate rings of a class of ordered field valued continuous functions

Abstract

Intermediate rings A(X,K) of K-valued continuous functions lying between B(X,K) and C(X,K), defined over a zero dimensional space X, are investigated and studied in this article, here K stands for a countable subfield of ℝ. It is realized that the structure space of A(X,K) is β₀X, the Banaschewski compact-ification of X. The Hewitt realcompactification analogue u_K(X) of uX is defined. Several equivalent descriptions of pseudocompactness of X via u_K(X), β₀X and the
uniform topology and the m-topology on C(X,K) are given. The article ends after ter studying an intercorrelation between z-ideals and z^◦-ideals in the rings C(X,K) and C(X), the functionally countable subalgebra of C_◦(X). This study eventually leads to a characterization of P-spaces and almost P-spaces in terms of z-ideals and z^◦-ideals in C(X,K).