Is a finitely generated subring of a Noetherian ring $R$ also Noetherian?
Remark: In fact I'm interested in the case $R=\mathbb C[x_1,...,x_n]$.
2026-03-25 15:47:28.1774453648
Is a finitely generated subring of a Noetherian ring also Noetherian?
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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