FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2002, VOLUME 8, NUMBER 1, PAGES 273-279

**On existence of unit in semicompact rings and topological rings with
finiteness conditions**

A. V. Khokhlov

Abstract

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We study quasi-unitary topological rings and modules ($m$Î
Rm $$" m Î
_{R}M) and multiplicative stabilizers of
their subsets.
We give the definition of semicompact rings.
The proved statements imply, in particular, that left
quasi-unitariness of a separable ring $R$ is equvivalent to
existence of its left unit, if $R$ has one of the following
properties: 1) $R$ is (semi-)compact,
2) $R$ is left linearly
compact, 3) $R$ is countably
semicompact (countably left linearly compact) and has a dense
countably generated right ideal, 4) $R$ is precompact and
has a left stable neighborhood of zero, 5) $R$ has a dense finitely
generated right ideal (e.
g.
$R$ satisfies the
maximum condition for closed right ideals), 6) the
module $$_{R}R is
topologically finitely generated and $\$\; \{\}^\{\backslash circ\}R\; =\; 0\; \$$.

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Last modified: July 8, 2002