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!