(FUNDAMENTAL AND APPLIED MATHEMATICS)

2010, VOLUME 16, NUMBER 8, PAGES 5-16

## Properties of finite unrefinable chains of ring topologies

V. I. Arnautov

Abstract

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Let $R\left(+,$× ) be a nilpotent ring and $\left(\mathfrak M, <\right)$ be the lattice of all ring topologies on $R\left(+,$× ) or the lattice of all such ring topologies on $R\left(+,$× ) in each of which the ring $R$ possesses a basis of neighborhoods of zero consisting of subgroups. Let t and t ' be ring topologies from $\mathfrak M$ such that $\tau =\tau_0 \prec_\left\{\mathfrak M\right\} \tau_1 \prec_\left\{\mathfrak M\right\} \ldots \prec_\left\{\mathfrak M\right\} \tau_n = \tau\text{'}$. Then $k$£ n for every chain t = t '0 < t'1 < ¼ < t 'k= t ' of topologies from $\mathfrak M$, and also $n = k$ if and only if $\tau\text{'}_i \prec_\left\{\mathfrak M\right\} \tau\text{'}_\left\{i+1\right\}$ for all $0$£ i < k.

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