**Authors:** Arnautov Vladimir, G., N., Ermakova

**Keywords:** Ring, ideal, lattice of ring topologies, unrefinable chain of ring topologies, supremum of ring topologies, basis of filter of neighborhoods of zero, semidirect product of rings, restriction of ring topologies to an ideal, cover of an element in a lattice

### Abstract

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.
\newline
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.

V. I. Arnautov

Vladimir Andrunachievici Institute of Mathematics

and Computer Science

5 Academiei str., MD-2028, Chisinau

Moldova

E-mail:

G. N. Ermakova

Transnistrian State University

25 October str., 128, Tiraspol, 278000

Moldova

E-mail:

### DOI

https://doi.org/10.56415/basm.y2022.i1.p66
### Fulltext

–

0.13 Mb