Let $\mathbb{Z}(p^\infty)$, for a given prime number $p$, be the $\mathbb{Z}$-module with generators $\{a_{i}\}_{i=1}^\infty$ and relations $pa_1=0$ and $pa_{i+1}=a_i$.
I would like to show that this module is injective. I know that it is enough to show that it is divisible, i.e. that for any $x\in \mathbb{Z}(p^\infty)$ and any nonzero $r\in\mathbb{Z}$ there exists $y\in\mathbb{Z}(p^\infty)$ such that $ry=x$.
Maybe I do not see something obvious, but have some trouble with showing divisibility in this case. Any hint?