I'm an undergrad going into my third year, and I'm studying module theory and field theory this summer out of Dummit and Foote, as I'm working to fill in the gaps of my algebra knowledge so that I can attempt some graduate-level courses next year.
This is Exercise 10.2.4, and in the first part, it asks you to prove that for $\mathbb{Z}$-modules $A$ and $\mathbb{Z}/n\mathbb{Z}$, $\varphi_{a}\colon \mathbb{Z}/n\mathbb{Z}\to A$ such that $\varphi_{a}(\overline{k})=ka$ is a well-defined module homomorphism if and only if $na=0$. I did this, and now it's asking me to prove that the group of $\mathbb{Z}$-module homomorphisms from $\mathbb{Z}/n\mathbb{Z}$ to $A$ is isomorphic to the set $A_{n}=\{a\in A\,|\,na=0\}$, namely the annihilator of the ideal $(n)$ in $\mathbb{Z}$. I'm stuck on this part, and I'm guessing that the desired isomorphism is $\gamma\colon A_{n}\to \mathbb{Z}/n\mathbb{Z}$ such that $\gamma(a)=\varphi_{a}$ as defined earlier, but I'm having a hard time proving that the function is surjective, namely that every module homomorphism from $\mathbb{Z}/n\mathbb{Z}$ to $A$ is of the form $\phi_{a}$. Any suggestions or hints? Furthermore, is this a property of $\mathbb{Z}$, or is it true in general for rings as modules over themselves?
Finally, I've been having a hard time going through this material, because it doesn't feel like I'm understanding the utility of modules other than as a generalization of vector spaces to rings. Are there any other sources that you would recommend to get a more motivated treatment of modules? I've heard some good things about Jacobson's Basic Algebra I, would that be too difficult for me right now? Thanks so much for your help!