Corollary

Tag 04GQ, permanent. Cite it as [Bourbaki, Tag 04GQ].

Commutative Algebra, chapter III, § 3. m-adic topologies on Noetherian rings

Let $A$ be a Noetherian local (resp. semi-local) ring and $m$ its Jacobson radical. Then $\hat{A}$ is a Noetherian local (resp. semi-local) ring whose Jacobson radical is $\hat{m}$ .

$\hat{A}$ is Noetherian by Proposition 8 (i) and the rest follows from Proposition 8 (ii) and the third formula in Corollary 1 to Theorem 3.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Proposition 8
Corollary 1 to Theorem 3