Let given torsion free abelian group $A$ of finite rank. Let for prime number $p$, given that $\cap\_i p^iA =\{0\}$. Is it true that for any $p$- torsion abelian group $B$, $\text{Hom}\_{\mathbb{Z}}(A, B)$ is torsion $\mathbb{Z}$ module.



Abelian group of finite rank
Let $p$ be any prime. There exists a subgroup $A$ of $\mathbf{Z}[1/p]^2$ containing $\mathbf{Z}^2$ such that $\bigcap\_n p^nA=\{0\}$ and $A/\mathbf{Z}^2$ is infinite (isomorphic to the quasi-cyclic group $P\_p=\mathbf{Z}[1/p]/\mathbf{Z}$). Then $\mathrm{Hom}(A,P\_p)$ contains $\mathrm{Hom}(P\_p,P\_p)\simeq\mathbf{Z}\_p$ (the $p$-adics), so is not torsion.


