Canonical Basis for Ideals in a Polynomial Domain Over a Commutative Ring with Finite Basis for Ideals

Let A be a commutative ring with unit element and A[x] denote the ring of polynomials in x with coefficients from A. Hilbert's theorem is : -

If every ideal in A has a finite basis, then every ideal in A[x] also has a finite basis.

The proof of this as given by Van der Waerden in his Moderne Algebra Bd. II gives us no information beyond the fact that every ideal in A [x] has a finite basis. If we just reverse the argument in his proof we will be able to give actually a canonical basis for every ideal in A[x] which will be found to be a powerful tool in several applications.