Noetherian Ring - Hilbert's Basis Theorem

Hilbert's Basis Theorem

If R is a ring, let R denote the ring of polynomials in the indeterminate X over R. Hilbert proved that if R is "not too large", in the sense that if R is Noetherian, the same must be true for R. Formally,

Theorem

If R is a Noetherian ring, then R is a Noetherian ring.

Corollary

If R is a Noetherian ring, then is a Noetherian ring.

For a proof of this result, see the corresponding section on the Hilbert's basis theorem page. Geometrically, the result asserts that any infinite set of polynomial equations may be associated to a finite set of polynomial equations with precisely the same solution set (the solution set of a collection of polynomials in n variables is generally a geometric object (such as a curve or a surface) in n-space).