arXiv subjects

Ali Akbar Estaji

Publications and source records attributed to Ali Akbar Estaji.

3 recordsLinked to original sources

arXiv2024

The Cozero part of the pointfree version of $C_c (X)$

Let $\mathcal C_{c}(L):= \{α\in \mathcal{R}(L) \mid R_α \, \text{ is a countable subset of } \, \mathbb R \}$, where $R_α:=\{r\in\mathbb R \mid {\mathrm{coz}}(α-r)\neq\top\}$ for every $α\in\mathcal R (L).$ By using idempotent elements, it is going to prove that ${\mathrm{Coz}}_c[L]:= \{\mathrm{coz}(α) \mid α\in\mathcal{C}_c (L) \}$ is a $σ$-frame for every completely regular frame $L,$ and from this, we conclude that it is regular, paracompact, perfectly normal and an Alexandroff algebra frame such that each cover of it is shrinkable. Also, we show that $L$ is a zero-dimensional frame if and only if $ L$ is a $c$-completely regular frame.

math.GN↗

arXiv2018

Maximal ideals in rings of real measurable functions

Let $ M (X)$ be the ring of all real measurable functions on a measurable space $(X, \mathscr{A})$. In this article, we show that every ideal of $M(X)$ is a $Z^{\circ}$-ideal. Also, we give several characterizations of maximal ideals of $M(X)$, mostly in terms of certain lattice-theoretic properties of $\mathscr{A}$. The notion of $T$-measurable space is introduced. Next, we show that for every measurable space $(X,\mathscr{A})$ there exists a $T$-measurable space $(Y,\mathscr{A}^{\prime})$ such that $M(X)\cong M(Y)$ as rings. The notion of compact measurable space is introduced. Next, we prove that if $(X, \mathscr{A})$ and $(Y, \mathfrak{M^{\prime}})$ are two compact $T$-measurable spaces, then $X\cong Y$ as measurable spaces if and only if $M(X)\cong M (Y)$ as rings.

math.GN↗

arXiv2018

Some ring-theoretic properties of the ring of $\mathcal{R}L_τ$

The aim of this article is to survey ring-theoretic properties of Kasch, the regularity and the injectivity of the ring of real-continuous functions on a topoframe $L_{ τ}$, i.e., $\mathcal{R}L_τ$. In order to study these properties, the concept of $P$-spaces and extremally disconnected spaces are extend to topoframes. For a $P$- topoframe $L_{ τ}$, the ring $\mathcal{R}L_τ$ is $\aleph_0$-Kasch ring. $P$- topoframes are characterized in terms of ring-theoretic properties of the regularity and injectivity of the ring of real-continuous functions on a topoframe. It follows from these characterizations that the ring $\mathcal{R}L_τ$ is regular if and only if it is $\aleph_0$-selfinjective. For a completely regular topoframe $L_τ$, we show that $\mathcal{R}L_τ$ is a Bear ring if and only if it is a $CS$-ring if and only if $L_τ$ is extremally disconnected and also prove that it is selfinjective ring if and only if $L_{ τ}$ is an extremally disconnected $P$-topoframe.

math.GN↗