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

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 ${}^{\circ}\!R=0$.