A NOTE ON GROUP RINGS WITH TRIVIAL UNITS

Abstract

\n Let R be a ring with identity of characteristic two and G a nontrivial torsion group. We show that if the units in the group ring \n \n \n \n$RG$\n\n \n are all trivial, then G must be cyclic of order two or three. We also consider the case where R is a commutative ring with identity of odd prime characteristic and G is a nontrivial locally finite group. We show that in this case, if the units in \n \n \n \n$RG$\n\n \n are all trivial, then G must be cyclic of order two. These results improve on a result of Herman et al. [‘Trivial units for group rings with G-adapted coefficient rings’, Canad. Math. Bull.48(1) (2005), 80–89].