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 $$P_0 \le P_1 \le \ldots P_r$$ 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.