In Section $5.2$ of dummit and Foote, the fundamental theorem of finitely generated ableian group is stated as
A finitely generated abelian group $G$ is isomorphic to $\mathbb{Z}^r \times \mathbb{Z}_{n_{1}} \times .....\times \mathbb{Z}_{n_s}$ , where $a)r \ge 0$ (b) $n_{i+1}|n_i.$ Now can someone please explain $(b)$ as to why the condition is nesecarry and why do we call $r$ as the rank of the abelian group .It would be nice if an example is given to explain the rank of the group.
b) is just a condition that holds.
To explain where this condition comes from consider the direct product $C_2\times C_{9}\times C_{21}$. You can represent it first as $C_2\times C_9\times C_3\times C_7$ and then as $C_3\times C_{126}$ which satisfies b).
If you consider the Abelian group as a $ \Bbb Z$-module and take a tensor product with the ring of rational numbers to get a $\Bbb Q$-module then that module will be free of rank $r$. It is also called the $\Bbb Q$-rank of the Abelian group.