Hilbert basis theorem corollary

Corollary (Hilbert basis, finitely generated algebras are Noetherian).
Let $R$ be a Noetherian ring and let $A$ be a commutative $R$-algebra generated by finitely many elements over $R$, i.e. $A = R[a_1,\dots,a_n]$. Then $A$ is Noetherian.

Equivalently: if $A$ is a quotient of a polynomial ring

for some ideal $I$, then $A$ is Noetherian.

Idea: by the Hilbert basis theorem , $R[x_1,\dots,x_n]$ is Noetherian; and any quotient ring of a Noetherian ring is Noetherian.

Related knowls.

• Hilbert basis theorem
• Polynomial ring
• Quotient ring
• Noetherian ring

Examples

1. Polynomial rings over a field.
   If $k$ is a field, then $k[x_1,\dots,x_n]$ is Noetherian. Concretely, every ideal of $k[x_1,\dots,x_n]$ is finitely generated.

2. A finitely generated $\mathbb{Z}$-algebra.
   $A=\mathbb{Z}[x]/(x^2-2)$ is Noetherian (it is a quotient of the Noetherian ring $\mathbb{Z}[x]$).

3. Coordinate rings of affine algebraic sets.
   Over a field $k$, rings like

   $$k[x,y]/(y^2-x^3-x) \quad\text{or}\quad k[x,y,z]/(xz-y^2)$$

   are Noetherian: they are quotients of the Noetherian ring $k[x,y]$ or $k[x,y,z]$.