TITLE: Germs at infinity of sequence of integers
QUESTION [6 upvotes]: Consider the  $\mathbb Z$-module $\mathcal Z$ obtained as the set of sequences of integers $\mathbb Z ^ \mathbb N$ modulo the relation that two sequences are deemed equivalent when their difference is $0$ almost everywhere. This space is the space of germs at $+\infty$ of sequences of integers. I think this is a quite natural object to study, as for instance it appears naturally when you try to generalize the notion of fundamental group of the circle when dealing with paths with a non-compact source space: imagine the value of $z_n$ as being the "number of turns" done at step $n$, the classical case being recovered from the stationary sequences. In that sense it is a completion of $\mathbb Z$ with a rich structure in the non-finite part. Yet I have not been able to find any literature regarding this object. This may be so because either

nothing specific/interesting can be said about it, or

it is an obvious/trivial example of some algebraic concept I'm unaware of (I'm no algebraist myself).
So my question is: are the basic properties of this module known/interesting? For instance, is it a free module? Does anybody know a classical reference which may help me in studying this object?
Thanks in advance for your contributions.

REPLY [11 votes]: This abelian group, which can also be described as the quotient of the direct product $\prod_{\mathbb N}\mathbb Z$ by the direct sum $\sum_{\mathbb N}\mathbb Z$, is isomorphic to the direct sum of the following two pieces.  The first piece is a torsion-free, divisible abelian group (so you can view it as a vector space over $\mathbb Q$) of rank (i.e., dimension over $\mathbb Q$) equal to the cardinality of the continuum.  The second piece is the direct product, over all primes $p$, of the direct product groups $\prod_{\mathbb N}\mathbb Z_p$, where $\mathbb Z_p$ is the additive group of $p$-adic integers.  I believe this result is due to Balcerzyk.
Edit: I believe the Balcerzyk reference is "On factor groups of some subgroups of a complete direct sum of infinite cyclic groups" [Bull. Acad. Polon. Sci., Sér. Sci. Math., Astron., Phys. 7 (1959) 141-142.