Noether normalization theorem

Statement

Let $k$ be a field and $R$ a finitely generated $k$-algebra. Then:

• Weak version: There exists a subring $S$ of $R$ such that $S\cong k[x_1,x_2,\dots ,x_r]$ and $R$ is a finitely generated $S$-module
• Strong version: Suppose $P_0\le P_1\le \dots P_r$ is an ascending chain of prime ideals, such that no further prime ideals can be and $P_i\cap S=(x_1,x_2,\dots ,x_i)$ Then we can choose $S$ as above, with the further constraint that $P_i\cap S=(x_1,\dots ,x_i)$.

Applications

• weak nullstellensatz for arbitrary fields follows very directly from the weak version of the Noether normalization theorem.