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Hilbert's basis theorem

Classical affine algebraic varieties appear as sets of zeros of a set $S=\{P_{\alpha }\mid \alpha \in A\}$ of polynomials in affine $n$-dimensional space ${𝔸}_k^n$ over a field $k$. The coordinate algebra of ${𝔸}_k^n$ is the algebra of polynomials in $n$ variables, $k[x_1,\dots ,x_n]$, and the coordinate algebra of an affine algebraic variety is $R≔k[x_1,\dots ,x_n]/I$ where $I≔⟨S⟩$ is the ideal generated by $S$.

The Hilbert basis theorem asserts that this ideal $I$ is finitely generated; and consequently $R$ is a noetherian ring. For a proof see standard textbooks on commutative algebra or algebraic geometry (e.g. Atiyah, MacDonald); there is also a proof on wikipedia.

More generally, a finitely generated commutative algebra over a commutative noetherian ring $R$ is noetherian. For the case of $R$ a field, this is the case in the previous paragraphs.