# Injection of $\mathbb{Z}$ into a p-adically complete abelian group

If $$M$$ is a p-adically complete abelian group, so that it's a $$\mathbb{Z}_p$$-module, and we have an injective homomorphism $$\phi: \mathbb{Z} \hookrightarrow M$$, is it true that the induced homomorphism $$\hat{\phi}: \mathbb{Z}_p \rightarrow M$$ is also injective? If this is not true in general, would it be true if $$\phi$$ was also continuous with respect to the p-adic topologies?

This kind of statement feels too good to be true, and I'm sure there is a super basic counterexample somewhere out there, but I just can't come up with it. Any help will be greatly appreciated!

HINT: The nonzero ideals of $$\Bbb{Z}_p$$ are of the form $$p^n\Bbb{Z}_p$$, so if the induced map is not injective then $$\hat{\phi}(p^n)=0$$ for some $$n\in\Bbb{N}$$.

• Sorry, I don't quite understand the line of argument. Where do you use that $M$ is $p$-adically complete? Why would the kernel be an ideal, as opposed to some possiby non-closed subgroup? – Torsten Schoeneberg May 9 at 19:55