Geodesic
Properties of coverings in lattices of ring topologies
Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2022), pp. 66-74 Cet article a éte moissonné depuis la source Math-Net.Ru

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When studying unrefinable chains of ring topologies, it is natural to find out how neighborhoods of zero of ring topologies in such chains are related to each other. It is proved that for any ideal the restrictions of these topologies to the ideal coincides, or the sum of any neighborhood of zero in the stronger topology with the intersection of the ideal with any neighborhood of zero in the weaker topology is a neighborhood of zero in the weaker topology. We construct a ring and two ring topologies which form an unrefinable chain in the lattice of all ring topologies that a basis of filter of neighborhoods of zero which consists of subgroups of the additive group of the ring and restriction of these topologies to some ideal of the ring is no longer a unrefinable chain. This example shows that the given in [4] conditions under which the properties of a unrefinable chain of ring topologies, are preserved under taking the supremum are essential. 
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V. I. Arnautov; G. N. Ermakova. Properties of coverings in lattices of ring topologies. Buletinul Academiei de Ştiinţe a Republicii Moldova. Matematica, no. 1 (2022), pp. 66-74. http://geodesic.mathdoc.fr/item/BASM_2022_1_a4/

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

[2] Arnautov V. I., “Properties of finite unrefinable chains of ring topologies”, Fundam. Prikl. Mat., 16:8 (2010), 5–16 (in Russian) | MR

[3] V. I. Arnautov, G. N. Ermakova, “Unrefinable chains when taking the infimum in the lattice of ring topologies for a nilpotent ring”, Buletinul Academiei de Stiinte a Republicii Moldova, Matematica, 2017, no. 2(84), 71–76 (in Russian) | MR | Zbl

[4] V. I. Arnautov, G. N. Ermakova, “Proprieties of finite unrefinable chains of ring topologies for nilpotent ring”, Buletinul Academiei de Stiinte a Republicii Moldova, 2018, no. 1(86), 67–75 | MR | Zbl