Exercise 26, § 4

Let $\phi : A \to B$ be a ring homomorphism such that the mapping

is surjective. Let $N$ be a finitely generated $A$-module; show that, if $u : M \to N$ is an $A$-module homomorphism such that $u \otimes 1 : M_{(B)} \to N_{(B)}$ is surjective, then $u$ is surjective (apply Proposition 19 of no. 4 to $\operatorname{Coker}(u)$).