Possible difference between $\mathbb{Z}$-modules and vector spaces Suppose $G$ is a free abelian groups, i.e. a free $\mathbb{Z}$-module; we have a set $S \subset G $ such that $S$ spans $G$.
Can we conclude that the rank of $G$ as a  $\mathbb{Z}$-module is $ \leq |S| $ as in the vector-space case ? Why ?

There are also many differences between vector spaces and $\mathbb{Z}$-modules. Every vector space has a basis, but not every $\mathbb{Z}$-module. For example, any finite abelian group is not a free $\mathbb{Z}$-module, and the $\mathbb{Z}$-module $\mathbb{Q}$ is not free.
Furthermore a free $\mathbb{Z}$-module may have a linear independent set which cannot be extended to a basis. It also may have a subset $S$ which spans it, but does not contain a basis. 
