Commutative algebra counterexample

Question

Let $M$ be an $R[x]$-module, such that $M$ is finitely generated as an $R$-module.

Does there exist one such $M$, such that $M\otimes_{R[x]}R[x,x^{-1}]$ is not finitely generated as an $R$-module?

Answer 1

Let $R=\mathbb{Z}$ and let $M=\mathbb{Z}$ with $x$ acting by $2$. Then $M\otimes_{R[x]}R[x,x^{-1}]\cong \mathbb{Z}[1/2]$ is not finitely generated over $R$.