Decomposing a module into a direct sum of torsion and free part?

Question

I know by the fundamental theorem of finitely generated abelian groups that we can write any $\mathbb{Z}$-module of the form

$$A\cong \mathbb{Z}^r\oplus Tor(A) $$

Can we not also decompose any general $\mathbb{Z}$-module to such a form of

$$ A\cong F(A)\oplus Tor(A) $$

where $F(A)$ is the free part of $A$? If not, is there any generalization to the setting of infinitely generated $\mathbb{Z}$-modules? I was not able to find any answers on this site dealing with removing the assumption of being finitely generated.

Answer 1

If it were true for infinitely generated $\mathbb Z$ modules, then $\mathbb Q_\mathbb Z$ would be a free $\mathbb Z$ module, but it is not.