Noether normalization theorem
Statement
Let be a field and a finitely generated -algebra. Then:
- Weak version: There exists a subring of such that and is a finitely generated -module
- Strong version: Suppose is an ascending chain of prime ideals, such that no further prime ideals can be and Then we can choose as above, with the further constraint that .
Applications
- weak nullstellensatz for arbitrary fields follows very directly from the weak version of the Noether normalization theorem.