Geodesic
On existence of unit in semicompact rings and topological rings with finiteness conditions
Fundamentalʹnaâ i prikladnaâ matematika, Tome 8 (2002) no. 1, pp. 273-279.

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We study quasi-unitary topological rings and modules ($m\in Rm$ $\forall m\in {}_RM$) 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 ${}^{\circ}\!R=0$. 
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A. V. Khokhlov. On existence of unit in semicompact rings and topological rings with finiteness conditions. Fundamentalʹnaâ i prikladnaâ matematika, Tome 8 (2002) no. 1, pp. 273-279. http://geodesic.mathdoc.fr/item/FPM_2002_8_1_a19/

[1] Ramamurthi V. S., “Weakly regular rings”, Canadian Math. Bulletin, 16:3 (1973), 317–321 | DOI | MR | Zbl

[2] Tominaga H., “On s-initial rings”, Math. J. Okayama Univ., 18:2 (1976), 117–134 | MR | Zbl

[3] Khokhlov A. V., “O stabilnykh podmnozhestvakh modulei i suschestvovanii edinitsy v assotsiativnykh koltsakh”, Mat. zametki, 61:4 (1997), 596–611 | MR | Zbl

[4] Arnautov V. I., Vodinchar M. I., Mikhalev A. V., Vvedenie v teoriyu topologicheskikh kolets i modulei, Kishinev, 1988

[5] Ursul M. I., Kompaktnye koltsa i ikh obobscheniya, Kishinev, 1991 | MR

[6] Zelinsky D., “Linearly compact modules and rings”, Amer. J. Math., 75 (1953), 73–90 | DOI | MR