Proposition 6
Tag 09V2, permanent. Cite it as [Bourbaki, Tag 09V2].
Lie Groups and Lie Algebras, chapter I, § 2. Enveloping algebra of a Lie algebra
If K is Noetherian and g is a finitely generated module, the ring U is right and left Noetherian.
S is a finitely generated algebra over K and hence a Noetherian ring (Commutative Algebra, Chapter III, § 2, no. 10, Corollary 3 to Theorem 2). Hence G, which is isomorphic to a quotient ring of S, is Noetherian. Hence U is right and left Noetherian (Commutative Algebra, Chapter III, § 2, no. 10, Remark 2).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
Cited by
Cites
- *Commutative Algebra*, Chapter III, § 2, no. 10 not yet on this site: AC III
- *Commutative Algebra*, Chapter III, § 2, no. 10, Corollary 3 to Theorem 2 not yet on this site: AC III