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.