How do you prove that $\Bbb{Z}_p$ is an integral domain? Let $\Bbb{Z}_p$ be the $p$-adic integers given by formal series $\sum_{i\geq 0} a_i p^i$.  I'm having trouble proving that it's an integral domain.

The ring of $p$-adic integers $\mathbb{Z}_p$ is the endomorphism ring of $\mathbb{Z}(p^\infty)$ (the Prüfer $p$-group).
Since $\mathbb{Z}(p^\infty)$ is divisible, the image of an endomorphism is divisible, but the only divisible subgroups of $\mathbb{Z}(p^\infty)$ are $\{0\}$ and $\mathbb{Z}(p^\infty)$ itself, because all proper subgroups are finite. Therefore every nonzero endomorphism is surjective. Thus $fg=0$, with $g\ne0$ implies $f=0$.
