Statement on an abelian group being isomorphic to a direct sum of cyclic groups

Question

Let $A$ be an abelian group such that $A$$\otimes$$\Bbb R$ $= 0$. Then $A$ is isomorphic to a direct sum of cyclic groups.

Is this statement true or fase? How can I show this with a short justification or a counterexample?

Please help me..

Answer 1

Consider $ A = \mathbb{Q} / \mathbb{Z}$, then $A \otimes \mathbb{R} = 0$, because we have $\frac{p}{q} \otimes x = \frac{p}{q} \otimes \frac{q x}{q} = p \otimes \frac{x}{q} = 0$. However, $A$ is not isomorphic to a direct sum of cyclic groups. (See here for a proof)

The statement is however true for finitely generated abelian groups, in this case it follows from the structure theorem for finitely generated abelian groups.