Is a finitely generated subring of a Noetherian ring also Noetherian

Question

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

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

Answer 1

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