University of Guilan Journal of Algebra and Related Topics 2345-3931 9 1 2021 06 01 On CP-frames 109 119 4812 10.22124/jart.2021.18801.1252 EN A. A. Estaji Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran M. Robat Sarpoushi Faculty of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran Journal Article 2021 02 02 Let $\mathcal{R}_c( L)$ be the pointfree version of $C_c(X)$, the subring of $C(X)$ whose elements have countable image.<br /> We shall call a frame $L $ a $CP$-frame if the<br />ring $\mathcal{R}_c( L)$ is regular.<br /> % The main aim of this paper is to introduce $CP$-frames, that is $\mathcal{R}_c( L)$ is a regular ring. We give some<br /> 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. https://jart.guilan.ac.ir/article_4812_0fb07b5a618f63d62f015812ce49a533.pdf