On CP-frames

Let \mathcal{R}_c( L) be the pointfree version of C_c(X), the subring of C(X) whose elements have countable image. We shall call a frame L a CP-frame if thering \mathcal{R}_c( L) is regular. % The main aim of this paper is to introduce CP-frames, that is \mathcal{R}_c( L) is a regular ring. We give some We give some characterizations of CP-frames and we show that L is a CP-frame if and only if each prime ideal of \mathcal{R}_c ( L) is an intersection of maximal ideals if and only if every ideal of \mathcal{R}_c ( L) is a z_c-ideal. In particular, we prove that any P-frame 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 \mathcal{R}_c ( L) is a z_c-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.

کلیدواژه ها:

P-frame ، CP-frame ، regular ring ، z-ideal ، z-good ring