2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/59853Let $R$ be a ring with unity and $U(R)$ its group of units. Let $ΔU=\{a\in U(R)\mid [U(R):C_{U(R)}(a)]<\infty\}$ be the $FC$-radical of $U(R)$ and let $\nabla(R)=\{a\in R\mid [U(R):C_{U(R)}(a)]<\infty\}$ be the $FC$-subring of $R$. An infinite subgroup $H$ of $U(R)$ is said to be an $ω$-subgroup if the left annihilator of each nonzero Lie commmutator $[x,y]$ in $R$ contains only finite number of elements of the form $1-h$, where $x,y \in R$ and $h\in H$. In the case when $R$ is an algebra over a field $F$, and $U(R)$ contains an $ω$-subgroup, we describe its $FC$-subalgebra and the $FC$-radical. This paper is an extension of [1].8 pages, AMS-TeXRings and AlgebrasGroup TheoryPrimary 16U50, 16U60, 20C05; Secondary 16N99text