Corollary 3

Tag 04G3, permanent. Cite it as [Bourbaki, Tag 04G3].

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

*Let A be a commutative Noetherian ring, m an ideal of A and x an element of $A$ which is not a divisor of $O$ . There exists an integer $k > 0$ such that, for all $n \geq k$ , the relation $xy \in m^n$ implies $y \in m^{n-k}$ .

Corollary 1 applied to $E = A, F = A x$ shows that there exists $k$ such that, for all $n \geq k$ , $m^n \cap Ax = m^{n-k}(m^k \cap Ax)$ . Then, if $xy \in m^n$ ,

xy \in m^n \cap Ax \subset m^{n-k}x

and, as $x$ is not a divisor of $0$ , we deduce that $y \in m^{n-k}$ .

In the notation of transporters (Chapter I, § 2, no. 10), the conclusion of Corollary 3 reads

(2)

m^n : Ax \subset m^{n-k}.

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

Cited by

Nothing.

Cites

Chapter I, § 2, no. 10
Corollary 1