On elements in algebras having finite number of conjugates (Q2714169)

Let \(R\) be an algebra with unity over the field \(F\), \(U\) its unit group with FC-radical \(\Delta(U)\), and let \(\nabla(R)=\{a\in R\mid[U:C_U(a)]<\infty\}\) be the FC-subring of \(R\). An infinite subgroup \(H\) of \(U\) is said to be an \(\omega\)-subgroup if the left annihilator of each Lie commutator \([x,y]\) of elements \(x,y\in R\) contains only a finite number of elements \(1-h\) with \(h\in H\). In the paper, under the assumption that \(U\) contains an \(\omega\)-subgroup, \(\Delta(U)\) and \(\nabla(R)\) are described. These results extend earlier results of the author [Can. J. Math. 47, No. 2, 274-289 (1995; Zbl 0830.16016)].