In the Wikipedia about Larent polynomial, there is this result:
The ring of Laurent polynomials over a field is Noetherian.
I was wondering if would it be enough for us to have Laurent polynomial over a Noetherian ring? For example, I think $\mathbb Z[X,X^{-1}]$ is Noetherian.
Question: Let $R$ be a commutative Noetherian ring. Is $R[X,X^{-1}]$ Noetherian?
If $R$ is a Noetherian ring then $R[X]$ is a Noetherian ring (by Hilbert's basis theorem). Since $R[X,X^{-1}]$ is the just the localization of $R[X]$ at the multiplicatively closed set $\{X^n: n \geq 0\}$, it too must be Noetherian.