ON CP-FRAMES

Let rc(l) be the pointfree version of cc(x), the subring of c(x) whose elements have countable image. we shall call a frame l a cp-frame if the ring rc(l) is regular. we give some characterizations of cp-frames and we show that l is a cp-frame if and only if each prime ideal of rc(l) is an intersection of maximal ideals if and only if every ideal of rc(l) is a zc-ideal. in particular, we prove that any pframe is a cp-frame but not conversely, in general. in addition, we study some results about cp-frames like the relation between a cp-frame l and ideals of closed quotients of l. next, we characterize cp-frames as precisely those l for which every prime ideal in the ring rc(l) is a zc-ideal. finally, we show that this characterization still holds if prime ideals are replaced by essential ideals, radical ideals, convex ideals, or absolutely convex ideals.