On elements in algebras having finite number of conjugates

Let R be a ring with unity and U(R) its group of units. Let △U = {a ∈U(R) | [U(R) : C_U(R)(a)] < ∞} be the FC-radical of U(R) and let ▽(R) = {a ∈ R | [U(R) : C_U(R) (a)] < ∞} be the FC-subring of R. An infinite subgroup H of U(R) is said to be an w-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 ∈ R and h ∈ H. In the case when R is an algebra over a field F, and U(R) contains an w-subgroup, we describe its FC-subalgebra and the FC-radical. This paper is an extension of [1].