Why there is no ring hom from $\Bbb{Z}_p$ to $\Bbb{Z}$?

Question

Why there is no ring hom from $\Bbb{Z}_p$ to $\Bbb{Z}$?

Let $f:\Bbb{Z}_p→ \Bbb{Z}$ be a ring hom.

My try: In the case of $p＝5$, f:$\Bbb{Z}_5→\Bbb{Z}$ should take $\sqrt -1$(it exists because of Hensel lemma) to $-1/2$, but it is not integer. Is this correct ?

In any way, I should take some good element of $\Bbb{Z}_p$ and find contradiction, how can I do that correctly ?

Answer 1

In the $p$-adic integers, there are many multiplicatively invertible elements (such as $p+1$). Where could your homomorphism possibly send them?