# Rendiconti del Seminario Matematico della Università di Padova

Volume 129, 2013, pp. 47–69
DOI: 10.4171/RSMUP/129-4

Published online: 2013-05-15

On the Functionally Countable Subalgebra of $C(X)$

Let ${C_c{{\char 40}}X{{\char 41}}} =\{\,f\in C(X): f(X) \hbox{ is countable}\}$. Similar to $C(X)$ it is observed that the sum of any collection of semiprime (resp. prime) ideals in the ring ${C_c{{\char 40}}X{{\char 41}}}$ is either ${C_c{{\char 40}}X{{\char 41}}}$ or a semiprime (resp. prime) ideal in ${C_c{{\char 40}}X{{\char 41}}}$. For an ideal $I$ in ${C_c{{\char 40}}X{{\char 41}}}$, it is observed that $I$ and $\sqrt{I}$ have the same largest $z_c$-ideal. If $X$ is any topological space, we show that there is a zero-dimensional space $Y$ such that ${C_c{{\char 40}}X{{\char 41}}} \cong {C_c{{\char 40}}Y{{\char 41}}}$. Consequently, if $X$ has only countable number of components, then ${C_c{{\char 40}}X{{\char 41}}} \cong C(Y)$ for some zero-dimensional space $Y$. Spaces X for which ${C_c{{\char 40}}X{{\char 41}}}$ is regular (called $CP$-spaces) are characterized both algebraically and topo log ically and it is shown that $P$-spaces and $CP$-spaces coincide when $X$ is zero-dimensional. In contrast to $C^*(X)$, we observe that ${C_c{{\char 40}}X{{\char 41}}}$ enjoys the algebraic properties of regularity, $\aleph _{_0}$-selfinjectivity and some others, whenever $C(X)$ has these properties. Finally an example of a space $X$ such that ${C_c{{\char 40}}X{{\char 41}}}$ is not isomorphic to any $C(Y)$ is given.
Keywords: Functionally countable subring, socle, $z_c$-idea, regular ring, $P$-space, $CP$-space, Scattered space, $\mathcal N_0$-selfinjective, zero-dimensional space, component
Ghadermazi Mostafa, Karamzadeh O.A.S., Namdari M.: On the Functionally Countable Subalgebra of $C(X)$. Rend. Sem. Mat. Univ. Padova 129 (2013), 47-69. doi: 10.4171/RSMUP/129-4