Hilbert basis corollary - Imported Knowlpedia Content

The basic finiteness engine in commutative algebra is that “Noetherian stays Noetherian after adjoining finitely many indeterminates.” This is often invoked as a corollary whenever one wants to know that ideals in polynomial rings are finitely generated.

Corollary (Hilbert basis)

Let $R$ be a Noetherian ring. Then for every $n \ge 1$ , the polynomial ring $R[x_1,\dots,x_n]$ is Noetherian.

Consequently:

Every ideal of $R[x_1,\dots,x_n]$ is finitely generated.
Every quotient $R[x_1,\dots,x_n]/I$ is Noetherian (so coordinate rings of affine varieties over a field inherit Noetherianity).
In particular, if $k$ is a field, then $k[x_1,\dots,x_n]$ is Noetherian.

Examples

Polynomial rings over a field. For a field $k$ , the ring $k[x,y,z]$ is Noetherian. For instance, the ideal $(x^2+yz,\; y^3-z)$ is automatically finitely generated (indeed, generated by the displayed two elements).

Polynomial rings over the integers. Since $\mathbb Z$ is Noetherian, $\mathbb Z[x_1,\dots,x_n]$ is Noetherian. In particular, ideals like $(2,\; x^2,\; xy) \subset \mathbb Z[x,y]$ are finitely generated (here by three explicit generators).

Coordinate rings are Noetherian. Over a field $k$ , the quotient ring $k[x,y]/(x^2+y^2-1)$ is Noetherian, because it is a quotient of the Noetherian ring $k[x,y]$ .