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 $RG$ 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 $RG$ 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].

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关于带小单位的群环的说明

令R是一个具有特征二恒等式的环，而G是一个非平凡的扭转群。我们证明如果群环 $RG$ 中的单元都是平凡的，那么G一定是二阶或三阶循环的。我们还考虑R是具有奇素数特征的交换环和G是非平凡局部有限群的情况。我们证明在这种情况下，如果 $RG$ 中的单位都是平凡的，那么G必须是二阶循环的。这些结果改进了 Herman等人的结果。['具有G适应系数环的群环的平凡单位', 加拿大。数学。公牛。 48 (1) (2005), 80–89]。