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 $S\cong k[x_{1},x_{2},\ldots ,x_{r}]$ and $R$ is a finitely generated $S$-module
• Strong version: Suppose $P_{0}\leq P_{1}\leq \ldots P_{r}$ is an ascending chain of prime ideals, such that no further prime ideals can be and $P_{i}\cap S=(x_{1},x_{2},\ldots ,x_{i})$ Then we can choose $S$ as above, with the further constraint that $P_{i}\cap S=(x_{1},\ldots ,x_{i})$.

Applications

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