Suppose we have a countable, torsion-free abelian group $A$ with the property that for each element $a\neq 0$ the set $D_a=\{x\in A|\exists n\in \mathbb{Z}:nx=a\}$ is finite. Is $A$ already a free abelian group?

If one drops the condition "countable" the infinite direct product of countably many copies of $\mathbb{Z}$ is a counterexample.