Let $G$ be a finite group and $R$ be a principal ideal domain. Is the group ring $RG$ a principal ideal ring?

Question

I think the answer is negative. I have an idea. If $G$ be a cyclic group of order $n$ then we have $RG\cong R[x]/(x^n-1)$. Now consider the ideal $J=(2,x)+(x^n-1)$. I think that we can show $J$ is not principal. This method works for abelian group $G$, because $G$ is a finite direct product of cyclic groups by fundamental theorem. In general i don't have any idea! I would appreciate if someone could explain this question.

Thank you.

Answer 1

In general $\Bbb ZG$ will have $\Bbb Z[\zeta_n]$, the ring of integers of the $n$-th cyclotomic field, as a quotient. In general (for instance for $n=23$), $\Bbb Z[\zeta_n]$ is not a principal ideal domain.