The group ring $R[G]$ is right Noetherian IF $R$ is right Noetherian and $G$ is finite. The converse does not hold. If $R[G]$ is right Noetherian, then $R$ is right Noetherian and $G$ has the maximum condition on subgroups. if $G$ is additionally abelian, $R[G]$ is right Noetherian iff $R$ is right Noetherian and $G$ is finitely generated.