Classify Artinian $\Bbb Z$-modules How can I classify Artinian $\mathbb{Z}$-modules as Noetherian $\mathbb{Z}$-module? (A $\mathbb{Z}$-module is Noetherian iff it is finitely generated). Any hint will be helpful.
I have seen the statement that if group $G$ is artinian then the three following 
conditions are satisfied: 
1 ) $G$ is a torsion group 
2 ) Its $p$-torsion $\{ x \in G : px=0\}$ is finite for all primes $p$
3 ) $G$ has non-trivial $p$-torsion for only finitely many primes $p$
How can i deduce from these 3 conditions that $G$ is a finite direct sum of finite cyclic groups and Prüfer groups?
 A: Any abelian group can be split as $G=D\oplus M$, where $D$ is divisible and $M$ is reduced (that is, $\bigcap_{n>0}nM=\{0\}$). Of course, $G$ is artinian if and only if both $D$ and $M$ are artinian.
Now, suppose $D$ is divisible artinian: then $D=t(D)\oplus D/t(D)$, where $t(D)$ is the torsion part of $D$. If $t(D)\ne D$, then $D$ contains an isomorphic copy of $\mathbb{Q}$, because $D/t(D)$ is divisible and torsion free. Since $\mathbb{Q}$ is not artinian, we conclude that $D=t(D)$.
A divisible torsion group $D$ can be split as
$$
D=\bigoplus_{p\text{ prime}}D_p
$$
where $D_p$ is a torsion $p$-group. Since $D$ is artinian, for only finitely many $p$ we have $D_p\ne\{0\}$. Now let $D$ be a divisible torsion artinian $p$-group. Then its socle is a finite dimensional vector space over $\mathbb{Z}/p\mathbb{Z}$, so it's of the form $(\mathbb{Z}/p\mathbb{Z})^n$. Therefore
$$
D=\mathbb{Z}(p^{\infty})^n
$$
(a finite direct sum of copies of the Prüfer $p$-group).
Suppose now $M$ is reduced artinian. Since $\bigcap_{n>0}nM=\{0\}$ and the family of subgroups $nM$ has a minimal element, we have $nM=0$ for some $n>0$. So $M$ is an artinian module over $\mathbb{Z}/n\mathbb{Z}$, hence it is finite.
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