On rings of real valued clopen continuous functions

Among variant kinds of strong continuity in the literature, the clopen continuity or cl-supercontinuity (i.e., inverse image of every open set is a union of clopen sets) is considered in this paper. We investigate and study the ring C-s(X) of all real valued clopen continuous functions on a topological space X . It is shown that every f is an element of C-s (X) is constant on each quasi-component in X and using this fact we show that C-s (X) congruent to C(Y), where Y is a zero-dimensional s-quotient space of X . Whenever X is locally connected, we observe that C-s(X) congruent to C(Y), where Y is a discrete space. Maximal ideals of C-s(X) are characterized in terms of quasi-components in X and it turns out that X is mildly compact (every clopen cover has a finite subcover) if and only if every maximal ideal of C-s(X) is fixed. It is shown that the socle of C-s(X) is an essential ideal if and only if the union of all open quasi-components in X is s-dense. Finally the counterparts of some familiar spaces, such as P-s-spaces, almost P-s-spaces, s-basically and s-extremally disconnected spaces are defined and some algebraic characterizations of them are given via the ring C-s(X).