Is $p^n\mathbb Z_p\cong \mathbb Z_p$ as additive groups?

Question

Is it true that $p^n\mathbb{Z}_p\cong \mathbb Z_p$ as additive groups?

Here $\mathbb Z_p$ is the ring of $p$-adic integers for $p$ prime and $n$ is any positive integer.

Thanks

Answer 1

If $R$ is any integral domain (unitary commutative ring without zero divisors) and $x\in R$ is any non-zero element, the additive groups of $R$ and of the principal ideal $Rx$ are naturally isomorphic.

Once you have clear a proof in your case, it would be clear how to prove this more general fact and how to generalize it further.

Answer 2

Yes, they are isomorphic as topological groups. Can you think of a natural homomorphism between them?