This might be very elementary but I'm not sure I see it. Given an injective morphism of R-modules $f : A \rightarrow B$ is it true that the induced morphims $Hom_R(B, F) \rightarrow Hom_R(A, F)$ is surjective, where F is the fraction field of R?
Is the same thing true if I replace F with R, at least for some rings R?