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.

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