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 β ₀ X, the Banaschewski compactification of X. The Hewitt realcompactification analogue υ_K (X) of υX is defined. Several equivalent descriptions of pseudocompactness of X via υ_K (X), β ₀ X and the uniform topology and the m-topology on C(X, K) are given. The article ends after studying an intercorrelation between z-ideals and z°-ideals in the rings C(X, K) and C_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).