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is a Noetherian ring, as a consequence of the Hilbert basis theorem.
* The enveloping algebra U of a finite-dimensional Lie algebra is a both left and right Noetherian ring; this follows from the fact that the associated graded ring of U is a quotient of , which is a polynomial ring over a field (the PBW theorem); thus, Noetherian. For the same reason, the Weyl algebra, and more

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