Properties of finite unrefinable chains of ring topologies. (Q1930190)

Rings are not necessarily associative, topological spaces not necessarily Hausdorff. It is known that the set of all ring topologies on a ring \(R\) forms a lattice. The author studies the following question related to the Schmidt-Ore theorem about principal series in modular lattices: Let \(R\) be a ring, \(\mathfrak M\) a lattice of ring topologies on \(R\), and \(\mathcal T_1,\mathcal T_2\in\mathfrak M\) such that there exists a unrefinable chain of length \(k\) of topologies from \(\mathfrak M\) joining \(\mathcal T_1\) with \(\mathcal T_2\). Under which conditions the length of any unrefinable chain of topologies from \(\mathfrak M\) joining \(\mathcal T_1\) with \(\mathcal T_2\) is \(k\)? In the paper are given some sufficient conditions in the following cases: (i) \(R\) is nilpotent and \(\mathfrak M\) is the lattice of all ring topologies on \(R\); (ii) \(R\) is nilpotent and \(\mathfrak M\) is the set of all ring topologies on \(R\) for which \(R\) has a fundamental system of neighbourhoods of zero consisting of subgroups of its additive group.