Let R and J be as in Exercise 4; let A be a finitely generated R-module and ∫ : C → A an R-module homomorphism. Then ∫ induces a homomorphism ∫̅: C/JC → A/JA in the usual way (Corollary IV.1.8). Show that if ∫̅ is an epimorphism, then ∫̅ is an epimorphism

Data from exercise 4

Let R be a commutative ring with identity, J an ideal that is contained in every maximal ideal of R, and A a finitely generated R-module. If R/J⊗_R A = 0, then A = 0.