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.