  Let $R$ be a ring with unity and $U(R)$ its group of units. Let $\Delta
U=\{a\in U(R)\mid [U(R):C_{U(R)}(a)]<\infty\}$ 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 $\omega$-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
$\omega$-subgroup, we describe its $FC$-subalgebra and the $FC$-radical. This
paper is an extension of [1].
