$\mathbb{Z}^{\oplus \mathbb{N}} = \mathbb{Z}^{\oplus \mathbb{N}} \oplus \mathbb{Z}^{\oplus \mathbb{N}}$.

Question

I am solving the following question from Aluffi chapter 0. I constructed the following counter example.

I know by intuition that the following there exists an isomorphism $\mathbb{Z}^{\oplus \mathbb{N}} = \mathbb{Z}^{\oplus \mathbb{N}} \oplus \mathbb{Z}^{\oplus \mathbb{N}}$. I am trying to find explicit isomorphism between these two.

Answer 1

Consider the mapping \begin{align} \mathbb Z^{\oplus \mathbb N} \oplus \mathbb Z^{\oplus \mathbb N} & \longrightarrow \mathbb Z^{\oplus \mathbb N} \\ ((a_1,a_2,\dots),(b_1,b_2,\dots)) & \longmapsto (a_1,b_1,a_2,b_2,\dots) \end{align}