Countability of countably generated abelian groups

A countably generated abelian group is a countable set. Given generators $g_1,g_2,\dots$, every element is ${\sum }_{j=1}^m{a}_j{g}_j$ for some $m$ and some $a_j\in \mathbb{Z}$. For each fixed $m$, this is the image of a finite cartesian product of copies of the countable set $\mathbb{Z}$; taking a countable union of countable sets proves the assertion. The same proof works for a module generated by a countable set over any countable ring.