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, Cc(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 Cc(X,F). Qc(X,F) (resp. qc(X,F)), the maximal (resp. classical) ring of quotients of Cc(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 Qc(S), the maximal ring of quotients of the subring S of Cc(X,F), is a subring of Qc(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 Cc(X, F) is investigated. We show that qc(X,F)=Cc(X, F) if and only if the set of non-units and zero-divisors in Cc(X,F) coincide if and only if X is almost CPF-space. Finally, it is shown that the fixed ring of quotients and the cofinite ring of quotients of Cc(X) coincide if and only if Hom(Mcp) = Cc(Xp) for every p ? X.