On CP-frames and the Artin-Rees property

Document Type : Research Paper

Author

Esfarayen University of Technology, Esfarayen, North Khorasan, Iran

Abstract

‎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.‎

Keywords

References

1. M. Abedi, Some notes on z-ideal and d-ideal in RL, Bull. Iranian Math. Soc., 46 (2020), 593-611.
2. M. Abedi, Rings of quotients of the ring RL, Houston J. Math. 47 (2021), 271-293.
3. M. Abedi, Concerning P-frame and the Artin-Rees Property, Collect. Math. 74 (2023), 279-297.
4. E. Abu Osba, O. Alkam, M. Henriksen Combining local and von Neumann regular rings, Comm. Algebra, 32 (2004), 2639-2653.
5. R. N. Ball and J. Walters-Wayland, C- and C-quotients in pointfree topology, Dissertationes Math. (Rozprawy Mat.), 412 (2002), 1-62.
6. B. Banaschewski and C.Gilmour Stone-  Cech compacti cation and dimension theory for regular frame, J. London Math. 39 (1989), 1-8
7. B. Banaschewski, The real numbers in pointfree topology, Textos de Matematica, Serie B, Vol. 12. Departamento de Matematica da Universidade de Coimbra, Coimbra 1997.
8. T. Dube and J. Walters-Wayland, Coz-onto frame maps and some applications, Appl. Categor. Struct., 15 (2007), 119-133.
9. T. Dube, Concerning P-frames, essential P-frames and strongly zero-dimensional frames, Algebra Univers., 69 (2009), 115-138.
10. T. Dube, Notes on pointfree disconnectivity with a ring-theoretic slant, Appl. Categ. Structures, 18 (2010), 55-72.
11. T. Dube, On the ideal of functions with compact support in pointfree function rings, Acta Math. Hungar, 129 (2010), 205-226.
12. A. A. Estaji, A. Karimi Feizabadi and M. Robat Sarpoushi, zc-Ideals and prime Ideals in the Ring RcL, Filomat, 32 (2018), 6741-6752
13. A. A. Estaji, M. Robat Sarpoushi, and M. Elyasi, Further thoughts on the ring RcL in frames, Algebra Univers, (43) 80 (2019), https://doi.org/10.1007/s00012-019-0619-z
14. A. A. Estaji and M. Robat Sarpoushi, On CP-frames, Journal of Algebra and related Topics, 9 (2021), 109-119.
15. A. A. Estaji and M. Taha, Cozero part of the pointfree version of Cc(X) (to appear).
16. M. Ghaermazi, O. A. S. Karamzadeh and M. Namdari, On the functionally countable subalgebra of C(X), Rend. Sem. Mat. Univ. Padova, 129 (2013), 47-69
17. A. Karimi Feizabadi, A. A. Estaji and M. Robat Sarpoushi, Pointfree version
of image of real-valued continuous functions, Categ. Gen. Algebra. Struct. Appl.,9 (2018), 59-75.
18. G. Mason, z-Ideals and prime ideals, J. Algebra, 26 (1973) 280-297.
19. J. Picado and A. Pultr, Frames and Locales: topology without points, Front. Math., Springer, Basel, 2012.
20. D. Rees, Two classical theorems of ideal theory, Proc. Cambridge Philos. Soc., 52 (1956), 155-157.
21. M. Taha, A. A. Estaji and M. Robat Sarpoushi, On the regularity of Cc(L), 53th Annual Iranian Mathematics Conference 5-8 September, 2022, University of Science and Technology of Mazandaran, Behshahr, Iran.
22. M. Taha, A. A. Estaji and M. Robat Sarpoushi, The pointfree version of Cc(X) via the rings of functions (to appear).
Journal of Algebra and Related Topics
Volume 11, Issue 2
December 2023
Pages 37-58

Files

Share

How to cite

Statistics