When a group ring is semilocal

For a ring $R$ and a group $G$: $R[G]$ is semilocal iff the following two conditions hold: 1) $R$ is semilocal; 2) $G$ is finite OR $G=G_p\times G_p'$ where $G_p$ is an infinite $p$-group, and $G_p'$ is a finite group with no elements of order divisible by $p$, and $p$ is the characteristic of $R/J(R)$.


  • G. Renault. Sur les anneaux de groupes. (1971) @ Theorem 2