Rings of quotients of the ring R(L) by coz-filters

Document Type : Dedicated to Prof. O. A. S. Karamzadeh


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.


‎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.


Main Subjects

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