FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(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(+, × ) be a nilpotent ring and $ (\mathfrak M, <) $ be the lattice of all ring topologies on R(+, × ) or the lattice of all such ring topologies on R(+, × ) 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_{\mathfrak M} \tau_1 \prec_{\mathfrak M} \ldots \prec_{\mathfrak M} \tau_n = \tau' $. 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'_i \prec_{\mathfrak M} \tau'_{i+1} $ for all 0 £ i < k.

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