Let $\mathbb{P}$ be the set of all prime numbers. Consider the $\mathbb{Z}$-Module $\mathbb{Z}^{(\mathbb{P})}$, that is, the external direct sum of copies of the additive abelian group $\mathbb{Z}$ indexed in the set of prime numbers.
Does anybody know any books or articles where I can read about this object? In particular, I'm looking for any characterization of it's submodules.
Thanks in advance for any help provided.
As in Servaes' comment, presumably this question was inspired by trying to understand the multiplicative group $\mathbb{Q}^{\times}$ of the rationals. Note that abstractly you don't need to know anything about $\mathbb{P}$ other than that it's countably infinite, so it's equivalent to just study the direct sum $\bigoplus_{i \in \mathbb{N}} \mathbb{Z}$ of countably many copies of $\mathbb{Z}$.
This is a free module, and submodules of free modules over a PID are free. So every submodule is the span of some linearly independent set of elements. I'm not sure what else there is to say other than this.