Let $A=\mathbb{Z}\oplus\mathbb{Z}_3.$ I want to find all subgroups of $A$ as well as all direct summand.
For the subgroup, let $H=H_{free}\oplus H_{tor}$. And consider the inclusion: $\iota_1:H_{free}\to A$ and $\iota_2:H_{tor}\to A$. I conclude that $H_{free}=m\mathbb{Z}$ and $H_{tor}=\mathbb{Z}_3,\{0\}$. Are these all the subgroups?
For the summand, I guess that there is none apart from two trivial ones: $\mathbb{Z}_3$ and $\mathbb{Z}.$ How to give a rigorous proof?
Thanks!