On elements in algebras having finite number of conjugates

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Let R be a ring with unity and U(R) its group of units. Let ΔU={a∈ U(R) [U(R):Cᵤ₍ᵣ₎(a)]<∞} be the FC-radical of U(R) and let ∇(R)={a∈ R [U(R):Cᵤ₍ᵣ₎(a)]<∞} 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 ∈ R and h∈ 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].