Proves or counterexamples in retraction and coretractions of modules

Question

Any tip for proving or counterexamples that the following morphism of $\mathbb Z$-modules ${\mathbb Z} \to {\mathbb Q}$ is not a retraction and ${\mathbb Q} \to {\mathbb Q/\mathbb Z} $ is not a coretraction?

Answer 1

Do you know the notion of splitting sequence ? Your question is equivalent to prove that $$\mathbb{Q} \not\cong \mathbb{Z} \oplus \mathbb{Q}/ \mathbb{Z}$$

This follows from the fact that in $\mathbb{Z} \oplus \mathbb{Q}/ \mathbb{Z}$ there are elements of finite order due to the factor $\mathbb{Q}/ \mathbb{Z}$, but $\mathbb{Q}$ doesn't contain elements of finite order.