On Duo Group Rings

It is shown that if the group ring RQ(8) of the quaternion group Q(8) of order 8 over an integral domain R is duo, then R is a field for the following cases: (1) char R not equal 0, and (2) char R = 0 and S subset of R subset of K-S, where S is a ring of algebraic integers and K-S is its quotient field. Hence, we confirm that the field Q of rational numbers is the smallest integral domain R of characteristic zero such that RQ(8) is duo. A non-field integral domain R of characteristic zero for which RQ(8) is duo is also identified. Moreover, we give a description of when the group ring RG of a torsion group G is duo.