Let $R$ be a commutative ring. I am interested in the following two questions:
1) If $R$ is Noetherian, then is $R[u,u^{-1}]$ Noetherian? In fact, I know that this is true (the Hilbert basis theorem tells us $R[u]$ is Noetherian and then $R[u,u^{-1}]$ is a localisation of $R[u]$, which preserves the Noetherian property).
Is there a more insightful proof? I feel like just the above two facts, whilst telling us it is true, don't really offer much insight into why this is true.
2) If $R$ is local, then is $R[u,u^{-1}]$ still local? We are localising $R[u]$ at the multiplicative subset $S=\{ u^i \}_{i \ge 0}.$ The result would follow if I knew $S$ was a maximal multiplicative subset of $R[u]$, but is this true?