How to prove $A \cong \langle a \rangle \oplus A / \langle a \rangle $ for abelian group?

Question

Let $A$ be an abelian group and $a \in A$. I am trying to prove $$ A \cong \langle a \rangle \oplus A / \langle a \rangle $$ where $\langle a \rangle $ is the subgroup generated by $a$. I would appreciate a proof of this fact. Thank you!

Answer 1

This is not true since $\mathbb{Z}$ is not isomorphic to $\langle 2 \rangle\oplus\mathbb{Z}/2$ since the second group has torsion and not $\mathbb{Z}$.