Geodesic
Properties of finite unrefinable chains of ring topologies
Fundamentalʹnaâ i prikladnaâ matematika, Tome 16 (2010) no. 8, pp. 5-16.

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Let $R(+,\cdot)$ be a nilpotent ring and $(\mathfrak M,)$ be the lattice of all ring topologies on $R(+,\cdot)$ or the lattice of all such ring topologies on $R(+,\cdot)$ in each of which the ring $R$ possesses a basis of neighborhoods of zero consisting of subgroups. Let $\tau$ and $\tau'$ be ring topologies from $\mathfrak M$ such that $\tau=\tau_0\prec_\mathfrak M\tau_1\prec_\mathfrak M\dots\prec_\mathfrak M\tau_n=\tau'$. Then $k\leq n$ for every chain $\tau=\tau'_0\tau'_1\dots\tau'_k=\tau'$ of topologies from $\mathfrak M$, and also $n=k$ if and only if $\tau'_i\prec_\mathfrak M\tau'_{i+1}$ for all $0\leq i$. 
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V. I. Arnautov. Properties of finite unrefinable chains of ring topologies. Fundamentalʹnaâ i prikladnaâ matematika, Tome 16 (2010) no. 8, pp. 5-16. http://geodesic.mathdoc.fr/item/FPM_2010_16_8_a0/

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