Exercise 4, § 3

Let A be a local ring (not necessarily commutative) and m its Jacobson radical. Let M and N be two free A-modules (finitely generated or not) and $u : M \to N$ a homomorphism such that $1 \otimes u : M/mM \to N/mN$ is bijective. Show that u is bijective. (Prove first that u is surjective and then that u is injective, reducing it to the case where M and N are finitely generated and using respectively Corollary 1 to Proposition 4 and Proposition 6.)