Noether normalization theorem

Statement

Let $k$ be a field and $R$ a finitely generated $k$-algebra. Let $P_{r}\geq P_{r-1}\geq \ldots \geq P_{0}$ be a chain of descending prime ideals, maximal in the sense that no further prime ideals can be inserted in the chain. Then there exists a subring $S$ of $R$ with $S\cong k[x_{1},x_{2},\ldots ,x_{r}]$ sich that $R$ is a finitely generated $S$-module and $P_{i}\cap S=(x_{1},x_{2},\ldots ,x_{i})$.

Applications

• Hilbert's nullstellensatz