Rings and subrings of continuous functions with countable range

Intermediate rings of the functionally countable subalgebra of C(X) (i.e., the rings A(c)(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 A(c)(X) is homeomorphic to beta X-0, the Banaschewski compactification of X. From this a main result of [A. Veisi, e(c)-filters and e(c)-ideals in the function-ally 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 m(c)-topology and U-c-topology, respectively, are introduced and using these, new characterizations of P -spaces and pseudocompact spaces are found out. Moreover, X is realized to be an almost P -space when and only when each maximal ideal/z-ideal in Cm(X) become a z(0)-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 finally, a complete description of z(0)-ideals in a typical ring A(c)(X) via z(0)-ideals in C-c(X) is established.