Let $R$ be a Noetherian integral domain of characteristic $0$, and let $R\langle x_1,\ldots,x_n\rangle$ be the free $R$-ring. Is $R\langle x_1,\ldots,x_n\rangle$ necessarily Noetherian?
1) The proof of Hilbert Basis Theorem fails. Also, even if it works, induction on the number of indeterminate would fail.
2) Another approach that I tried is constructing a short exact sequence, for example $0 \to (x_1x_2-x_2x_1) \to R\langle x_1,x_2\rangle \to R[x_1,x_2] \to 0$. However, I am not sure how to prove $(x_1x_2-x_2x_1)$ is Noetherian as an $R\langle x_1,x_2\rangle$-module (like I cannot construct an explicit ideal that is not finitely generated and I cannot prove it is Noetherian).
Can anyone give me some suggestions? Thank you!