Rings and subrings of continuous functions with countable range

Abstract

Intermediate rings of the functionally countable subalgebra of C(X) (i.e., the rings Ac(X) lying between C_c (X) and C_c(X)), where X is a Hausdorff zero-dimensional space, are studied in this article. It is shown that the structure space of each Ac(X) is homeomorphic to 0X, the Banaschewski compactication of X. From this a main result of [A. Veisi, ec-lters and ec-ideals in the functionally countable subalgebra C (X), Appl. Gen. Topol. 20(2) (2019), 395{405] easily follows. The countable counterpart of the m-topology and U-topology on C(X), namely mc-topology and Uc-topology, respectively, are introduced and using these, new characterizations of P-spaces and pseudocompact spaces are found out. More- over, X is realized to be an almost P-space when and only when each maximal ideal/z-ideal in C_c(X) become a z0-ideal. This leads to a characterization of C_c(X) among its intermediate rings for the case that X is an almost P-space.  Noetherianness/Artinianness of C_c(X) and a few chosen subrings of C_c(X) are examined and nally, a complete description of z0-ideals in a typical ring Ac(X) via z0-ideals in C_c(X) is established.