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.
Abedi, M. (2023). On CP-frames and the Artin-Rees property. Journal of Algebra and Related Topics, 11(2), 37-58. doi: 10.22124/jart.2023.23811.1501
MLA
M. Abedi. "On CP-frames and the Artin-Rees property". Journal of Algebra and Related Topics, 11, 2, 2023, 37-58. doi: 10.22124/jart.2023.23811.1501
HARVARD
Abedi, M. (2023). 'On CP-frames and the Artin-Rees property', Journal of Algebra and Related Topics, 11(2), pp. 37-58. doi: 10.22124/jart.2023.23811.1501
VANCOUVER
Abedi, M. On CP-frames and the Artin-Rees property. Journal of Algebra and Related Topics, 2023; 11(2): 37-58. doi: 10.22124/jart.2023.23811.1501
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