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Hilbert basis corollary

Polynomial rings (and finitely generated algebras) over a Noetherian ring are Noetherian.

Hilbert basis theorem (iterated form). Let RR be a . For every integer n≥0n\ge 0, the polynomial ring R[x1,…,xn]R[x_1,\ldots,x_n] is Noetherian. Hence every in this ring is finitely generated. Moreover, every finitely generated RR-algebra is Noetherian, because it is a quotient of some R[x1,…,xn]R[x_1,\ldots,x_n].

In particular, if kk is a , then k[x1,…,xn]k[x_1,\ldots,x_n] and each quotient k[x1,…,xn]/Ik[x_1,\ldots,x_n]/I are Noetherian.

Examples
  1. Polynomial rings over a field. For a field kk, the ring k[x,y,z]k[x,y,z] is Noetherian, so every ideal—not merely one given by a finite list—is finitely generated.
  1. Polynomial rings over the integers. Since Z\mathbb Z is Noetherian, Z[x1,…,xn]\mathbb Z[x_1,\dots,x_n] is Noetherian. In particular, ideals like (2,  x2,  xy)⊂Z[x,y](2,\; x^2,\; xy) \subset \mathbb Z[x,y] are finitely generated (here by three explicit generators).
  1. Coordinate rings are Noetherian. Over a field kk, the quotient ring k[x,y]/(x2+y2−1)k[x,y]/(x^2+y^2-1) is Noetherian, because it is a quotient of the Noetherian ring k[x,y]k[x,y].