$A$ is torsion free iff $Tor(A, B) = 0$ for all $B$ .

Question

I've been reading the hatcher topology book and I find myself solving some exercises in the book. I am currently solving point $2$ (pag $267$), which says that:

I have already been the first part of this problem, what I need to prove is that $A$ is torsion free iff $Tor(A, B) = 0$ for all $B$.

For this I have done the following: If $A$ is free of torsion then $T(A)=0$ ($T(A)$ is the torsion subgroup of $A$) and as $Tor(A,B)=Tor(T(A),B)$ then $Tor(A,B)=Tor(T(A),B)=Tor(0,B)=0$ for every abelian group $B$.

I don't know how to make the other implication, could someone please help me? Thank you.

Answer 1

The $n$-torsion of $A$ is isomorphic to $\text{Tor}(A,\Bbb Z/n\Bbb Z)$. Therefore if $\text{Tor}(A,\Bbb Z/n\Bbb Z)=0$ then $A$ has no $n$-torsion.