# Are torsion modules always finite?

I was just wondering...are all torsion modules finite? Because, for instance, if the annihilator is (a), where a is nonzero, then all multiples of a are zeros...so that's like a cycle, right?

$(\mathbb Z/2\mathbb Z)^{\mathbb N}$ – Jonas Meyer Jan 15 '13 at 18:18
The abelian group $\mathbb{Q}/\mathbb{Z}$ is infinite and all torsion.