Hilbert's Basis Theorem - Applications

Applications

Let be a Noetherian commutative ring. Hilbert's basis theorem has some immediate corollaries. First, by induction we see that will also be Noetherian. Second, since any affine variety over (i.e. a locus-set of a collection of polynomials) may be written as the locus of an ideal and further as the locus of its generators, it follows that every affine variety is the locus of finitely many polynomials — i.e. the intersection of finitely many hypersurfaces. Finally, if is a finitely-generated -algebra, then we know that (i.e. mod-ing out by relations), where a set of polynomials. We can assume that is an ideal and thus is finitely generated. So is a free -algebra (on generators) generated by finitely many relations .