Article
| Title: | The clean elements of the ring $\mathcal R(L)$ (English) |
| Author: | Estaji, Ali Akbar |
| Author: | Taha, Maryam |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 74 |
| Issue: | 1 |
| Year: | 2024 |
| Pages: | 211-230 |
| Summary lang: | English |
| . | |
| Category: | math |
| . | |
| Summary: | We characterize clean elements of $\mathcal R(L)$ and show that $\alpha \in \mathcal {R}(L)$ is clean if and only if there exists a clopen sublocale $U$ in $L$ such that $\frak {c}_L({\rm coz} (\alpha - {\bf 1})) \subseteq U \subseteq \frak {o}_L( {\rm coz} (\alpha ))$. Also, we prove that $\mathcal R(L)$ is clean if and only if $\mathcal R(L)$ has a clean prime ideal. Then, according to the results about $\mathcal R(L),$ we immediately get results about $\mathcal C_{c}(L).$ (English) |
| Keyword: | frame |
| Keyword: | ring of real-valued continuous function |
| Keyword: | strongly zero-dimensional |
| Keyword: | clean element |
| Keyword: | sublocale |
| MSC: | 06D22 |
| MSC: | 54C05 |
| MSC: | 54C30 |
| DOI: | 10.21136/CMJ.2023.0062-23 |
| . | |
| Date available: | 2024-03-13T10:08:11Z |
| Last updated: | 2024-03-18 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152276 |
| . | |
| Reference: | [1] Aliabad, A. R., Mahmoudi, M.: Pre-image of functions in $C(L)$.Categ. Gen. Algebr. Struct. Appl. 15 (2021), 35-58. Zbl 07528806, MR 4357758, 10.52547/CGASA.15.1.35 |
| Reference: | [2] Azarpanah, F.: When is $C(X)$ a clean ring?.Acta. Math. Hung. 94 (2002), 53-58. Zbl 0996.54023, MR 1905786, 10.1023/A:1015654520481 |
| Reference: | [3] Ball, R. N., Hager, A. W.: On the localic Yosida representation of an archimedean lattice ordered group with weak order unit.J. Pure Appl. Algebra 70 (1991), 17-43. Zbl 0732.06009, MR 1100503, 10.1016/0022-4049(91)90004-L |
| Reference: | [4] Ball, R. N., Walters-Wayland, J.: $C$- and $C^*$-quotients in pointfree topology.Diss. Math. 412 (2002), 1-62. Zbl 1012.54025, MR 1952051, 10.4064/dm412-0-1 |
| Reference: | [5] Banaschewski, B.: Pointfree topology and the spectra of $f$-rings.Ordered Algebraic Structures Kluwer Academic, Dordrecht (1997), 123-148. Zbl 0870.06017, MR 1445110, 10.1007/978-94-011-5640-0_5 |
| Reference: | [6] Banaschewski, B.: The real numbers in pointfree topology.Textos de Mathemática. Series B 12. Universidade de Coimbra, Coimbra (1997). Zbl 0891.54009, MR 1621835 |
| Reference: | [7] Banaschewski, B.: Gelfand and exchange rings: Their spectra in pointfree topology.Arab. J. Sci. Eng., Sect. C, Theme Issues 25 (2000), 3-22. Zbl 1271.13052, MR 1829217 |
| Reference: | [8] Banaschewski, B.: On the pointfree counterpart of the local definition of classical continuous maps.Categ. Gen. Algebr. Struct. Appl. 8 (2018), 1-8. Zbl 1477.06028, MR 3754731 |
| Reference: | [9] Banaschewski, B., Gilmour, C.: Pseudocompactness and the cozero part of a frame.Commentat. Math. Univ. Carol. 37 (1996), 577-587. Zbl 0881.54018, MR 1426922 |
| Reference: | [10] Dube, T.: Concerning $P$-frames, essential $P$-frames, and strongly zero-dimensional frames.Algebra Univers. 61 (2009), 115-138. Zbl 1190.06007, MR 2551788, 10.1007/s00012-009-0006-2 |
| Reference: | [11] Elyasi, M., Estaji, A. A., Sarpoushi, M. Robat: Locally functionally countable subalgebra of $\mathcal{R}(L)$.Arch. Math., Brno 56 (2020), 127-140. Zbl 07250674, MR 4156440, 10.5817/AM2020-3-127 |
| Reference: | [12] Estaji, A. A., Feizabadi, A. Karimi, Sarpoushi, M. Robat: $\mathcal z_c$-ideals and prime ideals in ring $\mathcal{R}_c(L)$.Filomat 32 (2018), 6741-6752. Zbl 07554263, MR 3899307, 10.2298/FIL1819741E |
| Reference: | [13] Estaji, A. A., Sarpoushi, M. Robat, Elyasi, M.: Further thoughts on the ring $\mathcal{R}_c(L)$ in frames.Algebra Univers. 80 (2019), Article ID 43, 14 pages. Zbl 1477.06030, MR 4027118, 10.1007/s00012-019-0619-z |
| Reference: | [14] Ferreira, M. J., Picado, J., Pinto, S. M.: Remainders in pointfree topology.Topology Appl. 245 (2018), 21-45. Zbl 1473.06008, MR 3823988, 10.1016/j.topol.2018.06.007 |
| Reference: | [15] Johnstone, P. T.: Stone Spaces.Cambridge Studies in Advanced Mathematics 3. Cambridge University Press, Cambridge (1982). Zbl 0499.54001, MR 0698074 |
| Reference: | [16] Johnstone, P. T.: Cartesian monads on toposes.J. Pure Appl. Algebra 116 (1997), 199-220. Zbl 0881.18001, MR 1437621, 10.1016/S0022-4049(96)00165-X |
| Reference: | [17] Johnstone, P. T.: Topos Theory.London Mathematical Society Monographs 10. Academic Press, New York (1997). Zbl 0368.18001, MR 0470019 |
| Reference: | [18] Karamzadeh, O. A. S., Namdari, M., Soltanpour, S.: On the locally functionally countable subalgebra of $C(X)$.Appl. Gen. Topol. 16 (2015), 183-207. Zbl 1397.54032, MR 3411461, 10.4995/agt.2015.3445 |
| Reference: | [19] Feizabadi, A. Karimi, Estaji, A. A., Sarpoushi, M. Robat: Pointfree topology version of image of real-valued continuous functions.Categ. Gen. Algebr. Struct. Appl. 9 (2018), 59-75. Zbl 1452.06007, MR 3833111, 10.29252/CGASA.9.1.59 |
| Reference: | [20] Kou, H., Luo, M. K.: Strongly zero-dimensional locales.Acta Math. Sin., Engl. Ser. 18 (2002), 47-54. Zbl 0996.06006, MR 1894837, 10.1007/s101140000072 |
| Reference: | [21] Mehri, R., Mohamadian, R.: On the locally countable subalgebra of $C(X)$ whose local domain is cocountable.Hacet. J. Math. Stat. 46 (2017), 1053-1068. Zbl 1396.54021, MR 3751773, 10.15672/HJMS.2017.435 |
| Reference: | [22] Picado, J., Pultr, A.: Frames and Locales: Topology Without Points.Frontiers in Mathematics. Springer, Berlin (2012). Zbl 1231.06018, MR 2868166, 10.1007/978-3-0348-0154-6 |
| Reference: | [23] Sarpoushi, M. Robat: Pointfree Topology Version of Continuous Functions with Countable Image: Ph.D. Thesis.Hakim Sabzevari University, Sabzevar (2017). |
| Reference: | [24] Taha, M., Estaji, A. A., Sarpoushi, M. Robat: On the regularity of $\mathcal{C}_c(\rm L)$.$53^{nd}$ Annual Iranian Mathematics Confrence, University of Science & Technology of Mazandaran, September 5-8, 2022 1323-1326. |
| . |
Fulltext not available (moving wall 24 months)
Search
Partner of
