Hilbert produced an innovative proof by contradiction using mathematical induction; his method does not give an algorithm to produce the finitely many basis polynomials for a given ideal: it only shows that they must exist. One can determine basis polynomials using the method of Gröbner bases.

A slightly more general statement of Hilbert's basis theorem is: if *R* is a left (respectively right) Noetherian ring, then the polynomial ring *R*[*X*] is also left (respectively right) Noetherian.

The Mizar project has completely formalized and automatically checked a proof of Hilbert's basis theorem in the HILBASIS file.