regular ring

Let $ M(X, \mathscr{A})$ be the ring of all real measurable functions on a measurable space $(X, \mathscr{A})$. We show that for every measurable space $(X,\mathscr{A})$, there exists a $T$-measurable space $(Y,\mathscr{A}^{\prime})$ such that $M_K(X, \mathscr{A})\cong M_K(Y,\mathscr{A}^{\prime})$ and $M_{\infty}(X,\mathscr{A})\cong M_{\infty}(Y,\mathscr{A}^{\prime})$, where $M_K(X,\mathscr{A})$ is the ring of real measurable functions $f\in M(X, \mathscr{A})$ for which $coz(f)$ is a compact element of $\mathscr{A}$, and $M_{\infty}(X,\mathscr{A})$ is the ring of real measurable functions vanishing at infinity on $(X, \mathscr{A})$. Then, we introduce $\sigma$-compact and locally compact measurable spaces. We prove that a $T$-measurable space $(X, \mathscr{A})$ is compact ($\sigma$-compact) if and only if the set $X$ is finite (at most countable) and $\mathscr{A}= \mathcal{P}(X) $. Next, we obtain several equivalent conditions for $ M_{\infty}(X, \mathscr{A})$ to be a regular ring. Finally, we show that if $(X, \mathscr{A})$ is a $T$-measurable space and $ M_{\infty}(X, \mathscr{A})\not=\{0\}$, then there exists a locally compact measurable space $(Y, \mathscr{A}')$ such that $ M_{\infty}(X,\mathscr{A})\cong M_{\infty}(Y,\mathscr{A}^{\prime})$ and $M_K(X,\mathscr{A})\cong M_K(Y,\mathscr{A}^{\prime})$.

‎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.‎

This article aims to introduce the concept of $p-mathrm{clean ring}mathrm{s}$ as a generalization of some concepts such as $mathrm{clean rings}$ and $r-mathrm{clean rings}$. As the first result, we prove that every $mathrm{clean ring}$ is a $p-mathrm{clean ring}$ and every $r-mathrm{clean ring}$ is a $p-mathrm{clean ring}$. Furthermore, we give the relation between von Neumann $mathrm{local ring}$ and $p-mathrm{clean ring}$. Finally,  we investigate many properties of $p-mathrm{clean ring}mathrm{s}$.

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 paper we define a new type of rings”almost powerhermitian rings” (a generalization of almost hermitian rings) and establish several sufficient conditions over a ring R such that, every regular matrix admits a diagonal power-reduction.