The clean elements of the ring ℛ ( L )

Ali Akbar Estaji; Maryam Taha

Czechoslovak Mathematical Journal (2024)

  • Issue: 1, page 211-230
  • ISSN: 0011-4642

Abstract

top
We characterize clean elements of ℛ ( L ) and show that α ∈ ℛ ( L ) is clean if and only if there exists a clopen sublocale U in L such that 𝔠 L ( coz ( α - 1 ) ) ⊆ U ⊆ 𝔬 L ( coz ( α ) ) . Also, we prove that ℛ ( L ) is clean if and only if ℛ ( L ) has a clean prime ideal. Then, according to the results about ℛ ( L ) , we immediately get results about 𝒞 c ( L ) .

How to cite

top

Estaji, Ali Akbar, and Taha, Maryam. "The clean elements of the ring $\mathcal {R}(L)$." Czechoslovak Mathematical Journal (2024): 211-230. <http://eudml.org/doc/299243>.

@article{Estaji2024,
abstract = {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 $\mathfrak \{c\}_L(\{\rm coz\} (\alpha - \{\bf 1\})) \subseteq U \subseteq \mathfrak \{o\}_L( \{\rm coz\} (\alpha ))$. 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 $\mathcal \{C\}_\{c\}(L).$},
author = {Estaji, Ali Akbar, Taha, Maryam},
journal = {Czechoslovak Mathematical Journal},
keywords = {frame; ring of real-valued continuous function; strongly zero-dimensional; clean element; sublocale},
language = {eng},
number = {1},
pages = {211-230},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The clean elements of the ring $\mathcal \{R\}(L)$},
url = {http://eudml.org/doc/299243},
year = {2024},
}

TY - JOUR
AU - Estaji, Ali Akbar
AU - Taha, Maryam
TI - The clean elements of the ring $\mathcal {R}(L)$
JO - Czechoslovak Mathematical Journal
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
IS - 1
SP - 211
EP - 230
AB - 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 $\mathfrak {c}_L({\rm coz} (\alpha - {\bf 1})) \subseteq U \subseteq \mathfrak {o}_L( {\rm coz} (\alpha ))$. 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 $\mathcal {C}_{c}(L).$
LA - eng
KW - frame; ring of real-valued continuous function; strongly zero-dimensional; clean element; sublocale
UR - http://eudml.org/doc/299243
ER -

References

top
  1. Aliabad, A. R., Mahmoudi, M., 10.52547/CGASA.15.1.35, Categ. Gen. Algebr. Struct. Appl. 15 (2021), 35-58. (2021) Zbl07528806MR4357758DOI10.52547/CGASA.15.1.35
  2. Azarpanah, F., 10.1023/A:1015654520481, Acta. Math. Hung. 94 (2002), 53-58. (2002) Zbl0996.54023MR1905786DOI10.1023/A:1015654520481
  3. Ball, R. N., Hager, A. W., 10.1016/0022-4049(91)90004-L, J. Pure Appl. Algebra 70 (1991), 17-43. (1991) Zbl0732.06009MR1100503DOI10.1016/0022-4049(91)90004-L
  4. Ball, R. N., Walters-Wayland, J., 10.4064/dm412-0-1, Diss. Math. 412 (2002), 1-62. (2002) Zbl1012.54025MR1952051DOI10.4064/dm412-0-1
  5. Banaschewski, B., 10.1007/978-94-011-5640-0_5, Ordered Algebraic Structures Kluwer Academic, Dordrecht (1997), 123-148. (1997) Zbl0870.06017MR1445110DOI10.1007/978-94-011-5640-0_5
  6. Banaschewski, B., The real numbers in pointfree topology, Textos de Mathemática. Series B 12. Universidade de Coimbra, Coimbra (1997). (1997) Zbl0891.54009MR1621835
  7. Banaschewski, B., Gelfand and exchange rings: Their spectra in pointfree topology, Arab. J. Sci. Eng., Sect. C, Theme Issues 25 (2000), 3-22. (2000) Zbl1271.13052MR1829217
  8. Banaschewski, B., On the pointfree counterpart of the local definition of classical continuous maps, Categ. Gen. Algebr. Struct. Appl. 8 (2018), 1-8. (2018) Zbl1477.06028MR3754731
  9. Banaschewski, B., Gilmour, C., Pseudocompactness and the cozero part of a frame, Commentat. Math. Univ. Carol. 37 (1996), 577-587. (1996) Zbl0881.54018MR1426922
  10. Dube, T., 10.1007/s00012-009-0006-2, Algebra Univers. 61 (2009), 115-138. (2009) Zbl1190.06007MR2551788DOI10.1007/s00012-009-0006-2
  11. Elyasi, M., Estaji, A. A., Sarpoushi, M. Robat, 10.5817/AM2020-3-127, Arch. Math., Brno 56 (2020), 127-140. (2020) Zbl07250674MR4156440DOI10.5817/AM2020-3-127
  12. Estaji, A. A., Feizabadi, A. Karimi, Sarpoushi, M. Robat, 10.2298/FIL1819741E, Filomat 32 (2018), 6741-6752. (2018) Zbl07554263MR3899307DOI10.2298/FIL1819741E
  13. Estaji, A. A., Sarpoushi, M. Robat, Elyasi, M., 10.1007/s00012-019-0619-z, Algebra Univers. 80 (2019), Article ID 43, 14 pages. (2019) Zbl1477.06030MR4027118DOI10.1007/s00012-019-0619-z
  14. Ferreira, M. J., Picado, J., Pinto, S. M., 10.1016/j.topol.2018.06.007, Topology Appl. 245 (2018), 21-45. (2018) Zbl1473.06008MR3823988DOI10.1016/j.topol.2018.06.007
  15. Johnstone, P. T., Stone Spaces, Cambridge Studies in Advanced Mathematics 3. Cambridge University Press, Cambridge (1982). (1982) Zbl0499.54001MR0698074
  16. Johnstone, P. T., 10.1016/S0022-4049(96)00165-X, J. Pure Appl. Algebra 116 (1997), 199-220. (1997) Zbl0881.18001MR1437621DOI10.1016/S0022-4049(96)00165-X
  17. Johnstone, P. T., Topos Theory, London Mathematical Society Monographs 10. Academic Press, New York (1997). (1997) Zbl0368.18001MR0470019
  18. Karamzadeh, O. A. S., Namdari, M., Soltanpour, S., 10.4995/agt.2015.3445, Appl. Gen. Topol. 16 (2015), 183-207. (2015) Zbl1397.54032MR3411461DOI10.4995/agt.2015.3445
  19. Feizabadi, A. Karimi, Estaji, A. A., Sarpoushi, M. Robat, 10.29252/CGASA.9.1.59, Categ. Gen. Algebr. Struct. Appl. 9 (2018), 59-75. (2018) Zbl1452.06007MR3833111DOI10.29252/CGASA.9.1.59
  20. Kou, H., Luo, M. K., 10.1007/s101140000072, Acta Math. Sin., Engl. Ser. 18 (2002), 47-54. (2002) Zbl0996.06006MR1894837DOI10.1007/s101140000072
  21. Mehri, R., Mohamadian, R., 10.15672/HJMS.2017.435, Hacet. J. Math. Stat. 46 (2017), 1053-1068. (2017) Zbl1396.54021MR3751773DOI10.15672/HJMS.2017.435
  22. Picado, J., Pultr, A., 10.1007/978-3-0348-0154-6, Frontiers in Mathematics. Springer, Berlin (2012). (2012) Zbl1231.06018MR2868166DOI10.1007/978-3-0348-0154-6
  23. Sarpoushi, M. Robat, Pointfree Topology Version of Continuous Functions with Countable Image: Ph.D. Thesis, Hakim Sabzevari University, Sabzevar (2017). (2017) 
  24. Taha, M., Estaji, A. A., Sarpoushi, M. Robat, On the regularity of 𝒞 c ( L ) , Annual Iranian Mathematics Confrence, University of Science Technology of Mazandaran, September 5-8, 2022 1323-1326. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.