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## On premaximal topologies on vector spaces. (Russian)

Authors: Arnautov Vladimir, Filippov K. M.

### Abstract

Let $(R,\tau_0)$ be a topological ring with identity, $M$ -- a unitary $R$-module, $\goth T$ -- set of all Hausdorff $(R,\tau_0)$-modular topologies on $M$, $\tau (M)$ -- the finest one. A topology $\tau\in\goth T\setminus\{ \tau (M)\}$ is said to be a premaximal $(R,\tau_0)$-modular topology if it is a maximal element of the set $\goth T\setminus\{ \tau (M)\}$. The main result: Let $(R,\tau_0)$ be a topological absolute valued skew field (i.e. it's topology is determined by a real valued ring absolute value) and $M$ - an $R$-module, $dim_R\ M\leqslant\aleph_0$. Then module $M$ admits a premaximal $(R,\tau_0)$-modular topology iff the ring $(R,\tau_0)$ is not complete and $dim_R\ M\geqslant 2$ or $(R,\tau_0)$ is a discrete skew field and $M$ is infinuite.

Institutul de Matematică Academia de Ştiinţe a Moldovei
str. Academiei 5, Chişinău, MD-2028 Moldova