A question about the additive group of the $p$-adic integers Let $J_p$ be the additive group of the $p$-adic integers. I know that it is torsion-free. I'm not pretty confortable with $p$-adic. 
Is it possible to find a direct sum of infinitely many cyclic subgroup in $J_p$? (essentially I'm asking if they have finite special rank or not).
EDIT
I find out on Wikipedia that they effectively have infinite rank. But I still cannot find that direct sum.

Complete rewrite:
For this, you need to look at the fraction field $\Bbb Q_p$ of the $p$-adic integer ring $\Bbb Z_p$. It’s a vector-space over $\Bbb Q$, and according to a well-known theorem depending on the Axiom of Choice, there’s a $\Bbb Q$-basis of this vector space. Since $\Bbb Q_p$ is uncountable, the basis must be uncountable as well. For each $\beta$ in the basis, there’s a rational number $\lambda_\beta$ such that $\lambda_\beta\beta\in\Bbb Z_p$. Now take the (uncountably-indexed) direct sum
$\bigoplus_\beta\lambda_\beta\beta\,\Bbb Z$. There you are.
