Corollary 1

Tag 047Y, permanent. Cite it as [Bourbaki, Tag 047Y].

Commutative Algebra, chapter II, § 3. Local rings. Passage from the local to the global

Let $A$ be a ring which is not necessarily commutative, $m$ a right ideal of $A$ contained in the Jacobson radical of $A$ , $M$ and $N$ two left $A$ -modules and $u : M \to N$ an $A$ -linear mapping. If $N$ is finitely generated or $m$ is nilpotent and

1 \otimes u : (A_d/m) \otimes_A M \to (A_d/m) \otimes_A N

is surjective, then

u

is surjective.

$(A_d/m) \otimes_A (N/u(M))$ is canonically isomorphic to

((A_d/m) \otimes_A N)/\operatorname{Im}(1 \otimes u)

(Algebra, Chapter II, , no. 6, Corollary 1 to Proposition 6); then the hypothesis implies $(A_d/m) \otimes_A (N/u(M)) = 0$ , hence $N/u(M) = 0$ by Proposition 4

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

Cited by

Exercise 4, § 3
Corollary 2, § 3
Proposition 6, § 3
Proposition 13, § 2
Theorem 2, § 8

Cites

Proposition 4
Proposition 6