# Showing $\mbox{Hom}_{\mathbb{Z}}(\mathbb{Z}/n\mathbb{Z},A)$ is isomorphic to the annihilator of $(n)$?

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!

• Hint on surjectivity: If $\phi\colon\mathbb Z/n\mathbb Z\to A$ is a module homomorphism, then $\phi(\overline{k})=k\phi(1)$, so you can let $a=\phi(1)$. – Cheerful Parsnip Jul 23 '11 at 2:00
• Jacobson is elegantly written, but in fact more terse than D&F. I suggest on first reading of D&F that you skip sections 10.4-5 on tensor products and projective modules, and move on to linear algebra in Ch 11 & 12. This is also essentially what Jacobson does, but in fewer pages and with less examples. – John M Jul 23 '11 at 3:33
• Thank you for the hint and advice! I got the problem, and I'll continue to work through D&F for a while, looking up extra things on the side. – John Lee Jul 23 '11 at 15:29

Note that what you're proving is already pretty general, since any module over any ring is a $\mathbf{Z}$-module. I like to think that this is true because into each ring there is a unique ring homomorphism from $\mathbf{Z}$.
A more general statement: Let $A$ be a ring, and let $M$ be a left $A$-module. For each $a \in A$, let $M(a)$ be the set of $x \in M$ such that $ax = 0$. If we assume now that $A$ is commutative, then $M(a)$ is a submodule of $M$ and I claim that $M(a)$ and $\operatorname{Hom}_A(A/(a), M)$ are isomorphic $A$-modules. The proof is nearly identical to the one in your case. If $A$ is not commutative, then I think we can identify these two objects as mere abelian groups.
As for books, they are always going to be a very personal matter. I've heard good things about this book of Jacobson's, and you're certainly capable of reading it if you can follow Dummit & Foote. As for motivation, many books apply the structure theorem for modules over principal rings to modules like $\mathbf{C}[X]$ in order to obtain the Jordan normal form. But this end can be reached without ever mentioning modules, so perhaps that isn't good enough. On the other hand, once you know the basics of algebra, you are not so far from beautiful theorems in algebraic number theory, algebraic geometry, and representation theory. All three rest atop the theory of modules.