The ring R c L is introduced as a sub-f-ring of RL as a pointfree analogue to the subring C c (X) of C(X) consisting of elements with the countable image. We introduce z c-ideals in R c L and study their properties. We prove that for any frame L, there exists a space X such that βL OX with C c (X) R c (OX) R c βL R * c L, and from this, we conclude that if α, β ∈ R c L, |α| ≤ |β| q for some q > 1, then α is a multiple of β in R c L. Also, we show that IJ = I ∩ J whenever I and J are z c-ideals. In particular, we prove that an ideal of R c L is a z c-ideal if and only if it is a z-ideals. In addition, we study the relation between z c-ideals and prime ideals in R c L. Finally, we prove that R c L is a Gelfand ring.