Abstract:
Let $(R,\tau_R)$ be a topological ring and ${}_RM$, a left unitary $R$-module. The set $L(M)$ of all $(R,\tau_R)$-module topologies on ${}_RM$ is a complete modular lattice. A topology $\tau\in L(M)$ is $n$-premaximal if in $L(M)$ there exists an inclusion-maximal chain $\tau_>\tau_1>\dots>\tau_n$ such that $\tau_0$ is the largest element in $L(M)$ and $\tau_n=\tau$. Section 1 contains conditions for existence of 1-premaximal Hausdorff topologies on ${}_RM$. Section 2 contains a description of all $n$-premaximal topologies in the case when $(R,\tau_R)$ is a topological skew field whose topology is determined by a real absolute value.
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