# When is an abelian group (as $\mathbb{Z}$-module) completely reducible (semisimple)?

I know the answer when the abelian group is finitely generated. That is, the direct sum of a series of $\mathbb{Z}_p$, $p$ prime.

However, I don't know the case of infinitely generated ablian group.

I've seen an example that the direct product of $\mathbb{Z}_p$ over all p is not completely reducible. But I don't have a general idea on it.

Can you please help? Thank you!

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While the direct product has trouble, the direct sum is just fine as Chris Eagle mentions. – Jack Schmidt Nov 17 '11 at 20:00

## 1 Answer

A nontrivial abelian group is simple (as a $\mathbb{Z}$-module) iff it has no proper nontrivial $\mathbb{Z}$-submodules. A $\mathbb{Z}$-submodule is just a subgroup, so the nontrivial simple $\mathbb{Z}$-modules are exactly the cyclic groups of prime order.

By definition, a module is semisimple iff it is a direct sum of simple modules. So an abelian group is semisimple as a $\mathbb{Z}$-module iff it is a direct sum of cyclic groups of prime order. This applies just as well in the infinitely generated case as the finitely generated case.

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