On CP-frames and the Artin-Rees property

‎The set \mathcal{C}_{c}(L)=\Big\{\alpha\in\mathcal{R}L‎ : ‎\big\vert\{ r\in\mathbb{R}‎ : ‎\coz(\alpha-{\bf r})\ne ۱\big\}\big\vert\leq\aleph_۰ \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.‎

کلیدواژه ها:

frame ، CP-frame ، P-frame ، Artin-Rees property ، regular ring