Geodesic
On the number of ring topologies on countable rings
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2015), pp. 103-114 
V. I. Arnautov; G. N. Ermakova. On the number of ring topologies on countable rings. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2015), pp. 103-114. http://geodesic.mathdoc.fr/item/BASM_2015_1_a6/
@article{BASM_2015_1_a6,
     author = {V. I. Arnautov and G. N. Ermakova},
     title = {On the number of ring topologies on countable rings},
     journal = {Buletinul Academiei de \c{S}tiin\c{t}e a Republicii Moldova. Matematica},
     pages = {103--114},
     year = {2015},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BASM_2015_1_a6/}
}
TY  - JOUR
AU  - V. I. Arnautov
AU  - G. N. Ermakova
TI  - On the number of ring topologies on countable rings
JO  - Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica
PY  - 2015
SP  - 103
EP  - 114
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/BASM_2015_1_a6/
LA  - en
ID  - BASM_2015_1_a6
ER  - 
%0 Journal Article
%A V. I. Arnautov
%A G. N. Ermakova
%T On the number of ring topologies on countable rings
%J Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica
%D 2015
%P 103-114
%N 1
%U http://geodesic.mathdoc.fr/item/BASM_2015_1_a6/
%G en
%F BASM_2015_1_a6

Voir la notice de l'article provenant de la source Math-Net.Ru

For any countable ring $R$ and any non-discrete metrizable ring topology $\tau_0$, the lattice of all ring topologies admits: – Continuum of non-discrete metrizable ring topologies stronger than the given topology $\tau_0$ and such that $\sup\{\tau_1,\tau_2\}$ is the discrete topology for any different topologies; – Continuum of non-discrete metrizable ring topologies stronger than $\tau_0$ and such that any two of these topologies are comparable; – Two to the power of continuum of ring topologies stronger than $\tau_0$, each of them being a coatom in the lattice of all ring topologies.

[1] Markov A.A., “On absolutely closed sets”, Mat. Sb., 18(60):1 (1946), 3–28 (in Russian) | MR | Zbl

[2] Arnautov V. I., Glavatsky S. T., Mikhalev A. V., Introduction to the topological rings and modules, Marcel Dekker, inc., New York–Basel–Hong Kong, 1996 | MR | Zbl

[3] Arnautov V. I., “Non-discrete topologizability of countable rings”, DAN SSSR, 191 (1970), 747–750 (in Russian) | MR | Zbl

[4] Arnautov V. I., Ermakova G. N., “On the number of metrizable group topologies on countable groups”, Bul. Acad. Ştiinţe Repub. Moldova, Mat., 2013, no. 2(72)–3(73), 17–26 | MR | Zbl

[5] Arnautov V. I., Ermakova G. N., “On the number of group topologies on countable groups”, Bul. Acad. Ştiinţe Repub. Moldova, Mat., 2014, no. 1(74), 101–112 | MR | Zbl

[6] Engelking R., General topology, Moskva, 1986 (in Russian) | MR