TITLE: Alternate proofs of Hilberts Basis Theorem
QUESTION [17 upvotes]: I'm interested in proofs using ideas from outside commutative algebra of Hilbert's Basis Theorem.

If $R$ is a noetherian ring, then so is $R[X]$.

or its sister version

If $R$ is a noetherian ring, then so is $R[[X]]$.

I am very much aware of the standard non-construtive proof by contradiction given by Hilbert as well as the direct version using Groebner basis.
Most important theorems in mathematics that are old enough have several very different proofs. Comparing different ideas can be very enlightening and also give a hint to possible generalizations in different areas. For the Basis Theorem however, I am not aware of such.

REPLY [3 votes]: There is a proof of the theorem for $R[[x]]$ that uses the well-known result of I.S. Cohen that a ring is noetherian if and only if its prime ideals are finitely generated.  Such a proof is given by Kaplansky in his 1970 book Commutative Rings, Theorem 70.  
This method of proof can be generalized to noncommutative rings to show that the power series ring $R[[x]]$ over a left noetherian ring is left noetherian, as long as one is careful when putting forth a definition of "prime left ideal."  Gerhard Michler gave such a proof in a paper that's not so easy to locate (but it seems to be described here), and I gave a similar proof in Theorem 3.9 in this paper.