# Example of torsion modules over integral domain that has zero annihilator

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.

• 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? – Brandon Carter Mar 4 at 4:37
• Also, you've slipped in your first bullet point: it is important to say $r\neq 0$. – rschwieb Mar 4 at 13:56

## 1 Answer

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...