Understanding Direct Limits: $\mathbb{Z}\sqcup\coprod(\mathbb{Z}-2\mathbb{Z})$

[cohomology]

I'm trying to understand direct limits so consider the direct limit

\lim_\rightarrow (\mathbb{Z} \rightarrow \mathbb{Z} \rightarrow \mathbb{Z} \rightarrow \cdots)
where each map is multiplication by 2.
I concluded that the solution is \mathbb{Z}\sqcup\coprod(\mathbb{Z}-2\mathbb{Z}). Is this correct?

 

[yyat]

OK, note first that the direct limit of groups should again be a group.
In this particular example it is isomorphic to \mathbb{Z}[1/2], i.e. the multiplicative group of rational numbers of the form a/2^b, where a,b are integers, b\geq0.