# Is a finitely generated subring of a Noetherian ring also Noetherian?

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

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

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