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

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Last modified: July 8, 2002