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On CP-frames and the Artin-Rees property | ||
| Journal of Algebra and Related Topics | ||
| دوره 11، شماره 2، اسفند 2023، صفحه 37-58 اصل مقاله (372.49 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22124/jart.2023.23811.1501 | ||
| نویسنده | ||
| M. Abedi* | ||
| Esfarayen University of Technology, Esfarayen, North Khorasan, Iran | ||
| چکیده | ||
| 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. | ||
| کلیدواژه‌ها | ||
| Frame؛ CP-frame؛ P-frame؛ Artin-Rees property؛ regular ring | ||
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آمار تعداد مشاهده مقاله: 61 تعداد دریافت فایل اصل مقاله: 93 |
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