zc-ideals and prime ideals in the ring RcL

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