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Let $A$ be an principal ideal domain, and $M$ an $A$-module. If $p$ is irreducible in $A$, let's define $$\mathrm{Tor}_p(M):=\{m\in M\mid p^km=0\text{ for some }k\in\mathbb{N}\}.$$

I need to show that if $M$ is finitely generated, then $\mathrm{Tor}(M)$ can be written as a direct sum of the submodules $\mathrm{Tor}_p(M)$.

Through the Decomposition Theorem, I have that $M$ is a direct sum of a finite number of cyclic modules and a free module. Thus, the torsion submodule of M is the direct sum of the torsion submodules of each of these cyclic submodules (the torsion submodule of the free module is zero). But how can I show that $\mathrm{Tor}(C_{p_i^{\alpha_i}})=\mathrm{Tor}_p(M)$ or that $\mathrm{Tor}(C_{p_i^{\alpha_i}})$ is a direct sum of several $\mathrm{Tor}_p(M)$, for some irreducibles $p\in A$, where $p_i^{\alpha_i}$ is the order of the cyclic submodule?

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What's the decomposition theorem? You're not assuming that $A$ is a PID. – Dylan Moreland Jul 5 '12 at 0:12
In fact, this problem was suggested by a teacher who wrote the question wrongly; in fact it must be a PID (instead of a integral domain). – Marra Jul 5 '12 at 0:19
Done and done :) – Marra Jul 5 '12 at 2:19

You mentioned that you needed a tool to split the cyclic modules into "primary" pieces. Here's the lemma I think you're looking for:

Lemma: Let $Ax$ be a torsion module over a principal ideal domain $A$ such that $ann(x)=(pq)$ where $p$ and $q$ are coprime elements of $A$. Then $Ax=Ay\oplus Az$ where $ann(y)=(p)$ and $ann(z)=(q)$.

Proof: Set $y=qx, z=px$. Clearly $y,z\in Ax$. Notice $Ay+Az$ contains $x$, because we can find elements such that $1=ap+bq$ and then $x=(ap+bq)x=ay+bz$. Thus $Ax\subseteq Ay+Az$, proving $Ax=Ay+Az$. If $u\in Ay\cap Az$, then $u=u(ap+bq)=0$. So, the sum is direct.

Using this with induction, you can break composite cyclic torsion modules into sums of prime power cyclic modules.

Good luck!

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At first sight you seem to be working with cyclic torsion modules only. – Marc van Leeuwen Dec 28 '13 at 14:13
@MarcvanLeeuwen Sure... maybe we're looking at different parts of the question? This was written to address this specific question: But how can I show that [...] $\mathrm{Tor}(C_{p_i^{\alpha_i}})$ is a direct sum of several $\mathrm{Tor}_p(M)$, for some irreducibles $p\in A$[...]? I did, as a result of your comment, strategically add in a "torsion" in the solution to clarify the statement: thanks for helping me see that improvement. – rschwieb Dec 28 '13 at 14:23
In fact I did read the question too hastily, and did not quite get the meaning of $C_{p_i^{\alpha_i}}$; I see now that they probably mean (modules isomorphic to) $A/ p_i^{\alpha_i}A$. My interrogation was about "cyclic", even though OP mentions decomposition into cyclic modules as a strategy, I don't really think this should be needed. But your answer as it now stands is perfectly clear. – Marc van Leeuwen Dec 28 '13 at 17:51

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