For a ring with unity $R$ and a group $G$ let $R[G]$ denote the group ring. Now let $R$ be a commutative Noetherian ring with unity such that $R[\mathbb Z_m] \cong R[\mathbb Z_n]$ (isomorphic as rings); then is it true that $m=n$ ? If not true for commutative Noetherian rings, is it at least true for commutative artinian rings ?
UPDATE: The counterexample by @anon in the comment shows that the claim is false in general for commutative rings (not Noetherian). On the other hand the claim is true for $R=\mathbb Z$, and in fact a result of Graham Higman states that if $G,H$ are finite abelian groups such that $\mathbb Z [G] \cong \mathbb Z[H]$ then $G \cong H$ .