Let RG denote the group ring of a group G over a commutative ring R with unity. We recall that a group is said to be an FC-group if all its conjugacy classes are finite.
In , S. K. Sehgal and H. Zassenhaus gave necessary and sufficient conditions for U(RG) to be an FC-group when R is either Z, the ring of rational integers, or a field of characteristic 0.
One of the authors considered this problem for group rings over infinite fields of characteristic p ≠ 2 in  and G. Cliffs and S. K. Sehgal  completed the study for arbitrary fields. Also, group rings of finite groups over commutative rings containing Z(p), a localization of Z over a prime ideal (p) were studied in .