Let $G$ be finite group, $R$ is a commutative ring and $G$ acts on $R$ by ring automorphisms. It is well known and easy to show that extension $R^G \subset R$ is integral. It also not hard to find examples when such extension is not finite. But in all examples I know $R$ is not noetherian.
Is $R$ finite over $R^G$ for a noetherian ring $R$?