Hilbert basis corollary

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

1. 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).

2. 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).

3. 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 $k[x,y]$.