Hilbert's finiteness theorem

Statement

Let $R=\mathbb{C}[x_1,\dots ,x_n]$ be a polynomial algebra graded by degree and $S\subset R$ be a subalgebra for which there exists a surjective homomorphism

$$\phi :R⟶S$$

which preserves degree and is the identity on $S$. That is

$${\text{id}}_S=\phi {|}_S:S\to S.$$

Then $S$ is a finitely generated algebra (and thus Noetherian).

Proof