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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?

Thanks in advance

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$(\mathbb Z/2\mathbb Z)^{\mathbb N}$ –  Jonas Meyer Jan 15 '13 at 18:18

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up vote 6 down vote accepted

The abelian group $\mathbb{Q}/\mathbb{Z}$ is infinite and all torsion.

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