A topological study in the set of zero-dimensional subrings of a commutative ring
Hassan Mouadi, Driss Karim
Received September 7, 2023. Published online December 4, 2024.
Abstract: 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.
References: [1] W. W. Comfort, S. Negrepontis: The Theory of Ultrafilters. Die Grundlehren der mathematischen Wissenschaften 211. Springer, Berlin (1974). DOI 10.1007/978-3-642-65780-1 | MR 0396267 | Zbl 0298.02004
[2] R. Engelking: General Topology. Sigma Series in Pure Mathematics 6. Heldermann, Berlin (1989). MR 1039321 | Zbl 0684.54001
[3] C. A. Finocchiaro: Spectral spaces and ultrafilters. Commun. Algebra 42 (2014), 1496-1508. DOI 10.1080/00927872.2012.741875 | MR 3169645 | Zbl 1310.14005
[4] S. García-Ferreira, H. S. Pino-Villela: Characterizing filters by convergence (with respect to filters) in Banach spaces. Topology Appl. 159 (2012), 1246-1257. DOI 10.1016/j.topol.2011.11.004 | MR 2876731 | Zbl 1245.46006
[5] S. García-Ferreira, L. M. Ruza-Montilla: The $\mathcal{F}$-limit of a sequence of prime ideals. Commun. Algebra 39 (2011), 2532-2544. DOI 10.1080/00927872.2010.491492 | MR 2821730 | Zbl 1227.13001
[6] R. Gilmer: 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. MR 1335700 | Zbl 0882.13011
[7] R. Gilmer: 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. MR 1335702 | Zbl 0882.13012
[8] R. Gilmer, W. Heinzer: Artinian subrings of commutative rings. Trans. Am. Math. Soc. 336 (1993), 295-310. DOI 10.1090/S0002-9947-1993-1102887-7 | MR 1102887 | Zbl 0778.13012
[9] M. Hochster: Prime ideal structure in commutative rings. Trans. Am. Math. Soc. 142 (1969), 43-60. DOI 10.1090/S0002-9947-1969-0251026-X | MR 0251026 | Zbl 0184.29401
[10] D. Karim: 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. DOI 10.1007/978-3-319-74195-6_14 | MR 3778019 | Zbl 1401.13026
[11] H. Mouadi, D. Karim: Some topology on zero-dimensional subrings of product of rings. Filomat 34 (2020), 4589-4595. DOI 10.2298/FIL2014589M | MR 4290873 | Zbl 1499.13028
[12] V. Saks: Ultrafilter invariants in topological spaces. Trans. Am. Math. Soc. 241 (1978), 79-97. DOI 10.1090/S0002-9947-1978-0492291-9 | MR 0492291 | Zbl 0381.54002
Affiliations: Hassan Mouadi (corresponding author), Polydisciplinary Faculty of Taroudant, Ibnou Zohr University, BP 32/S, CP 80000 Agadir, Morocco, e-mail: hassanmouadi@hotmail.com, h.mouadi@uiz.ac.ma; Driss Karim, Faculty of Sciences and Technologies of Mohammedia, Hassan 2 University, Casablanca, Morocco, e-mail: dkarim@ced.uca.ac.ma
