On idempotent-fine group rings

Let $ A $ be an associative ring. A nonzero element $ t \in A $ is called fine if it can be written as $ t = n + v $, where $ v $ is a unit and $ n $ is a nilpotent element. A ring $ A $ is called an idempotent-fine ring if every nonzero idempotent in $ A $ is fine. Let $ A $ be a ring (respectively, an integral domain) of characteristic $ p^m $ for some prime $ p $ and positive integer $ m $, and let $ G $ be a locally finite nilpotent group (respectively, a locally finite group). We proved that $ A[G] $ is an idempotent-fine ring if and only if $ G $ is a $ p $-group. Moreover, if $ F $ is a field of characteristic $ p $ and $ F[G] $ is an idempotent-fine ring, then every nontrivial element $ g $ in the group $ G $ of finite order is a $ p $-element. Conversely, if G is a locally finite $ p $-group, then $ F[G] $ is an idempotent-fine ring.