I'm trying to understand modules. And I want to know, by definition of a module $M$, the underlying group structure on $M$ is abelian. Does that mean that every $R$ module $M$ is also a $\mathbb{Z}$ module because of the underlying structure?
Is this why we can define the character module $M^* = Hom_\mathbb{Z}(M,\mathbb{Q}/\mathbb{Z})$?
I'm really trying to prove that a module map $M \to N$ is injective iff the dual map $N^* \to M^*$ is surjective. Now I know in general the reverse is not true if $M^* = Hom_R(M,R)$. The question being so ominous makes me think that the "dual" map should be a map between the character modules. So I want to use the fact that $\mathbb{Q}/\mathbb{Z}$ is an injective $\mathbb{Z}$ module, but I'm not sure if I can apply this functor $Hom_\mathbb{Z}({-},\mathbb{Q}/\mathbb{Z})$ to any module.