Rings of quotients of the ring R(L) by coz-filters
نویسندگان
1 Faculty of Mathematics and Computer Science, Hakim Sabzevari University, P.O. Box 397, Sabzevar, Iran.
2 Department of Basic Science, Birjand University of Technology, P.O.Box 226, Birjand, Iran.
doi
10.30504/jims.2023.388284.1096چکیده
In this article, we first introduce the concept of $z$-sets in the ring $ \mathcal{R}(L)$ of real-valued continuous functions on a completely regular frame $L$, and give some properties of them. Let $S^{-1}_{\mathcal{F}}\mathcal{R}(L)$ denote the ring of fractions of the ring $ \mathcal{R}(L)$, where ${\mathcal{F}}$ is a ${\mathrm{coz}}$-filter on $L$ and $S_{\mathcal{F}}$ is a multiplicatively closed subset related to ${\mathcal{F}}$. We show that $S^{-1}_{\mathcal{F}}\mathcal{R}(L)$ may be realized as the direct limits of the subrings $\mathcal{R}(A)$, where $A\in \{\mathfrak{o}_L\big({\mathrm{coz}}(\alpha)\big) \colon \alpha\in S_{\mathcal{F}}\}$. Also, we show that ${\mathrm{Q_{cl} }}\mathcal{R}(L)=\mathcal{R}(L)$, if and only if $\mathcal{R}(L)$ is a special saturated ring.
