If $B$ is an abelian group, then is $B{\otimes}_{\mathbb Z}{\mathbb Z}_p$ isomorphic to ${\varprojlim}B/p^{n}B$?

Question

I could get the easy map from $B{\otimes}_{\mathbb Z}{\mathbb Z}_p$ to ${\varprojlim}B/p^{n}B$ but I could not find the map in the opposite direction. Please help me. Thank you!!

Answer 1

For any module B, the isomorphism you ask for holds if and only if B is noetherian. "Only if":see Artem's counter-example. "If": see Bourbaki, Algèbre Commutative, chap. 10, III, §3, n°4, propos.8

Answer 2

This is false in general: consider $B = \mathbb{Q}$. Then $\mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Z}_p \cong \mathbb{Q}_p$, but $\mathbb{Q} / p^n \mathbb{Q} = 0$ for every $n$, so $\varprojlim_n \mathbb{Q} / p^n \mathbb{Q} = 0$.