a. a. estaji

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.

In this article we consider the $m$-topology on linebreak $M(X,mathscr{A})$, the ring of all real measurable functions on a measurable space $(X, mathscr{A})$, and we denote it by $M_m(X,mathscr{A})$. We show that $M_m(X,mathscr{A})$ is a Hausdorff regular topological ring, moreover we prove that if $(X, mathscr{A})$ is a $T$-measurable space and $X$ is a finite set with $|X|=n$, then $M_m(X,mathscr{A})‎cong‎ mathbb R^n$ as topological rings. Also, we show that $M_m(X,mathscr{A})$ is never a pseudocompact space and it is also never a countably compact space. We prove that $(X,mathscr{A})$ is a pseudocompact measurable space, if and only if $ {M}_{m}(X,mathscr{A})= {M}_{u}(X,mathscr{A})$, if and only if $ M_m(X,mathscr{A}) $ is a first countable topological space, if and only if $M_m(X,mathscr{A})$ is a connected space, if and only if $M_m(X,mathscr{A})$ is a locally connected space, if and only if $M^*(X,mathscr{A})$ is a connected subset of $M_m(X,mathscr{A})$.

It is well-known that the sum of two $z$-ideals in $C(X)$ is either $C(X)$ or a $z$-ideal. The main aim of this paper is to study the sum of strongly $z$-ideals in ${mathcal{R}} L$, the ring of real-valued continuous functions on a frame $L$. For every ideal $I$ in ${mathcal{R}} L$, we introduce the biggest strongly $z$-ideal included in $I$ and the smallest strongly $z$-ideal containing $I$, denoted by $I^{sz}$ and $I_{sz}$, respectively. We study some properties of $I^{sz}$ and $I_{sz}$. Also, it is observed that the sum of any family of minimal prime ideals in the ring ${mathcal{R}} L$ is either ${mathcal{R}} L$ or a prime strongly $z$-ideal in ${mathcal{R}} L$. In particular, we show that the sum of two prime ideals in ${mathcal{R}} L$ such that are not a chain, is a prime strongly $z$-ideal.the formula is not displayed correctly!