Corollary 3
Tag 000W, permanent. Cite it as [Bourbaki, Tag 000W].
Algebra, VIII, p. 12, § 1. Artinian Modules and Noetherian Modules
Every commutative ring is the union of a right directed family of Noetherian subrings.
Indeed, let A be a commutative ring. The subrings of A generated (as -algebras) by finitely many elements are Noetherian by Corollary 2. They form a right directed family of subrings of A, with union A.
Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.