Is a finitely generated subring of a Noetherian ring $R$ also Noetherian?

Remark: In fact I'm interested in the case $R=\mathbb C[x_1,...,x_n]$.

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    $\begingroup$ Any finitely generated commutative algebra over $\Bbb C$ is Noetherian. $\endgroup$ – Angina Seng May 15 '19 at 18:57
  • $\begingroup$ Dear Lord Shark the Unknown, this solves the problem. If you post this as an answer, I will accept it. $\endgroup$ – Blazej May 15 '19 at 19:01

Any finitely generated ring is a quotient of some noetherian ring $\mathbb{Z}[x_1,...,x_n]$ and is therefore noetherian.

More generally, if $A$ is a noetherian commutative ring, $A[x_1,...,x_n]$ is noetherian, and any finitely generated $A$-algebra is a quotient of such a ring for some $n$, and is therefore noetherian as well


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