cp-frame

‎The set $\mathcal{C}_{c}(L)=\Big\{\alpha\in\mathcal{R}L‎ : ‎\big\vert\{ r\in\mathbb{R}‎ : ‎\coz(\alpha-{\bf r})\ne 1\big\}\big\vert\leq\aleph_0 \Big\}$ is a sub-$f$-ring of $\mathcal{R}L$‎, ‎that is‎, ‎the ring of all continuous real-valued functions on a completely regular frame $L$.‎ ‎The main purpose of this paper is to continue our investigation begun in \cite{a} of extending ring-theoretic properties in $\mathcal{R}L$ to‎ ‎the context of completely regular frames by replacing the ring $\mathcal{R}L$ with the ring $\mathcal{C}_{c}(L)$ to the context of zero-dimensional frames.‎ ‎We show that a frame $L$ is a $CP$-frame if and only if $\mathcal{C}_{c}(L)$ is a regular ring if and only if every ideal of $\mathcal{C}_{c}(L)$ is pure if and only if $\mathcal{C}_c(L)$ is an Artin-Rees ring if and only if every ideal of $\mathcal{C}_c(L)$ with the Artin-Rees property is an Artin-Rees ideal if and only if the factor ring $\mathcal{C}_{c}(L)/\langle\alpha\rangle$ is an Artin-Rees ring for any $\alpha\in\mathcal{C}_{c}(L)$ if and only if every minimal prime ideal of $\mathcal{C}_c(L)$ is an Artin-Rees ideal.‎

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.