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I was able to prove that if $R$ is an integral domain and $M$ is a finitely generated $R$-module is torsion if and only if $Ann(M) \neq 0$, but so far, I have failed to come up with an example that this fact does not hold for arbitrary modules over $R$, more specifically, I am looking for a torsion (non-finitely generated) module $M$ with $Ann(M) = 0$.

Here is a couple of definitions:

  • An element $m \in M$ is a torsion element if $\exists r \in R$ such that $rm = 0$
  • $M$ is a torsion module if every elements of M is torsion.
  • $Ann(M) = \{r \in R \mid \forall m \in M, rm = 0 \}$

Any help would be greatly appreciated.

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    $\begingroup$ Hint: Restrict to the simplest case, $R = \mathbb{Z}$. What do finitely generated torsion $\mathbb{Z}$-modules look like? What do their annihilators look like? Can you think of any simple examples of infinitely generated torsion $\mathbb{Z}$-modules? $\endgroup$ – Brandon Carter Mar 4 at 4:37
  • $\begingroup$ Also, you've slipped in your first bullet point: it is important to say $r\neq 0$. $\endgroup$ – rschwieb Mar 4 at 13:56
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Hint:

One might observe that

$ann(\mathbb Z/(2))=(2)$

$ann(\mathbb Z/(2)\oplus \mathbb Z/(3))=(6)$

$ann(\mathbb Z/(2)\oplus \mathbb Z/(3)\oplus \mathbb Z/(5))=(30)$

What might you conjecture about the annihilator of $\bigoplus_{p\in P} \mathbb Z/(p)$ where $P$ is the set of positive prime integers? Don't forget to consider the question of whether it is torsion or not...

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