Rings of Quotients of the Ring Consisting of Ordered Field Valued Continuous Functions with Countable Range

Abstract

For a zero-dimensional topological space, $X$ and a totally ordered field $F$ with interval topology, $C_c(X, F)$ denotes the ring consisting of ordered field-valued continuous functions with countable range on $X$. This article aims to study and investigate the rings of quotients of $C_c(X, F)$. $Q_c(X, F)$ (resp. $q_c(X, F)$), the maximal (resp. classical) ring of quotients of $C_c(X, F)$ as a modified countable analogue of $Q(X)$ (resp. $q(X)$), the maximal (resp. classical) ring of quotients of $C(X)$ are characterized. It is proved that $Q_c(S)$, the maximal ring of quotients of the subring $S$ of $C_c(X, F)$, is a subring of $Q_c(X, F)$ if and only if every dense ideal in $S$ has dense cozero-set in $X$. Also, the coincidence of rings of quotients of $C_c(X, F)$ is investigated. We show that $q_c(X, F)=C_c(X, F)$ if and only if the set of non-units and zero-divisors in $C_c(X, F)$ coincide if and only if $X$ is almost $CP_F$-space. Finally, it is shown that the fixed ring of quotients and the cofinite ring of quotients of $C_c(X)$ coincide if and only if $Hom(M_{p}^c)=C_c(X_p)$ for every  $p$ in $X$.