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If $\mathfrak{g}$ is a finite-dimensional Lie algebra, then it is very known that the Universal enveloping algebra $U(\mathfrak{g})$ of $\mathfrak{g}$ is a Noetherian ring. What is the simplest way to show this fact?

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There is a nice treatment in this book. In particular, the associated graded ring $\operatorname{gr}U(\mathfrak g)$ is shown to be a finitely generated commutative $k$-algebra, generated by at most $\dim_k\mathfrak g$ elements over $k.$ It is a basic fact that such rings are noetherian, so this implies that $U(\mathfrak g)$ is itself noetherian.

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