| Title:
|
A topological study in the set of zero-dimensional subrings of a commutative ring (English) |
| Author:
|
Mouadi, Hassan |
| Author:
|
Karim, Driss |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
150 |
| Issue:
|
3 |
| Year:
|
2025 |
| Pages:
|
331-341 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We investigate the relationship between the space $\mathcal {Z}(R,T)$, defined as the largest closed subset of a ring $T$ with respect to a countable topology, and the classical prime spectrum ${\rm Spect}(R)$ of a subring $R$. We explore the topological properties of $\mathcal {Z}(R,T)$ and establish connections with ${\rm Spect}(R)$ under certain conditions. (English) |
| Keyword:
|
zero-dimensional subring |
| Keyword:
|
filter |
| Keyword:
|
$\mathcal {F}$-topology |
| Keyword:
|
countably compact |
| MSC:
|
13A15 |
| MSC:
|
13A99 |
| MSC:
|
13B02 |
| MSC:
|
54H99 |
| DOI:
|
10.21136/MB.2024.0141-23 |
| . |
| Date available:
|
2025-09-26T13:53:25Z |
| Last updated:
|
2025-09-26 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153079 |
| . |
| Reference:
|
[1] Comfort, W. W., Negrepontis, S.: The Theory of Ultrafilters.Die Grundlehren der mathematischen Wissenschaften 211. Springer, Berlin (1974). Zbl 0298.02004, MR 0396267, 10.1007/978-3-642-65780-1 |
| Reference:
|
[2] Engelking, R.: General Topology.Sigma Series in Pure Mathematics 6. Heldermann, Berlin (1989). Zbl 0684.54001, MR 1039321 |
| Reference:
|
[3] Finocchiaro, C. A.: Spectral spaces and ultrafilters.Commun. Algebra 42 (2014), 1496-1508. Zbl 1310.14005, MR 3169645, 10.1080/00927872.2012.741875 |
| Reference:
|
[4] García-Ferreira, S., Pino-Villela, H. S.: Characterizing filters by convergence (with respect to filters) in Banach spaces.Topology Appl. 159 (2012), 1246-1257. Zbl 1245.46006, MR 2876731, 10.1016/j.topol.2011.11.004 |
| Reference:
|
[5] García-Ferreira, S., Ruza-Montilla, L. M.: The $\mathcal{F}$-limit of a sequence of prime ideals.Commun. Algebra 39 (2011), 2532-2544. Zbl 1227.13001, MR 2821730, 10.1080/00927872.2010.491492 |
| Reference:
|
[6] Gilmer, R.: Background and preliminaries on zero-dimensional rings.Zero-Dimensional Commutative Rings Lecture Notes in Pure and Applied Mathematics 171. Marcel Dekker, New York (1995), 1-13. Zbl 0882.13011, MR 1335700 |
| Reference:
|
[7] Gilmer, R.: Zero-dimensional extension rings and subrings.Zero-Dimensional Commutative Rings Lecture Notes in Pure and Applied Mathematics 171. Marcel Dekker, New York (1995), 27-39. Zbl 0882.13012, MR 1335702 |
| Reference:
|
[8] Gilmer, R., Heinzer, W.: Artinian subrings of commutative rings.Trans. Am. Math. Soc. 336 (1993), 295-310. Zbl 0778.13012, MR 1102887, 10.1090/S0002-9947-1993-1102887-7 |
| Reference:
|
[9] Hochster, M.: Prime ideal structure in commutative rings.Trans. Am. Math. Soc. 142 (1969), 43-60. Zbl 0184.29401, MR 0251026, 10.1090/S0002-9947-1969-0251026-X |
| Reference:
|
[10] Karim, D.: On the set of intermediate Artinian subrings.Homological and Combinatorial Methods in Algebra Springer Proceedings in Mathematics and Statistics 228. Springer, Cham (2018), 139-149. Zbl 1401.13026, MR 3778019, 10.1007/978-3-319-74195-6_14 |
| Reference:
|
[11] Mouadi, H., Karim, D.: Some topology on zero-dimensional subrings of product of rings.Filomat 34 (2020), 4589-4595. Zbl 1499.13028, MR 4290873, 10.2298/FIL2014589M |
| Reference:
|
[12] Saks, V.: Ultrafilter invariants in topological spaces.Trans. Am. Math. Soc. 241 (1978), 79-97. Zbl 0381.54002, MR 0492291, 10.1090/S0002-9947-1978-0492291-9 |
| . |
