Corollary 7.7.16 An abelian group $M$ admits a $\Bbb{Z}_n$-module structure if and only if $M$ has exponent $n$ as a group.
This is something of a follow up to this post My question is, is the theorem referring to an exponent of $M$ or the exponent? I ask because I don't see how to prove that $n$ is the exponent, only that it is an exponent.
Note that every abelian group, such as $M$, has a natural $\Bbb{Z}$-module structure that is unique. Assume that $M$ admits a $\Bbb{Z}_n = \Bbb{Z}/(n)$-module structure. Then by the theorem in the link, $(n) \subseteq \mbox{Ann}(M)$ which implies $nx = 0$ for all $x \in M$. I presume this means that $n$ is an exponent of $M$. But if the theorem is referring to the exponent, then I don't know how to show that $n$ is the exponent of $M$. Isn't the exponent whatever integer generates the ideal $\mbox{Ann}(M)$?
It means "an exponent", not "the exponent". Indeed, the result is false for "the exponent": for instance, the trivial group has exponent $1$, but has a $\mathbb{Z}_n$-module structure for any $n$.