While studying modules, I wondered under what conditions the infinite direct sum commutes with the infinite direct product. I noticed that in some cases they do commute (e.g., $(\mathbb{Q}^{\oplus\mathbb{N}})^\mathbb{N} \cong (\mathbb{Q}^\mathbb{N})^{\oplus\mathbb{N}}$ as $\mathbb{Q}$-vector spaces) and in other cases they do not (e.g., $\bigoplus_{n > 0} (\mathbb{Z}/n\mathbb{Z})^\mathbb{N} \ncong (\bigoplus_{n > 0} \mathbb{Z}/n\mathbb{Z})^\mathbb{N}$ as abelian groups). I then tried to investigate what seemed to be the most natural case, $(\mathbb{Z}^{\oplus\mathbb{N}})^\mathbb{N}$ and $(\mathbb{Z}^\mathbb{N})^{\oplus\mathbb{N}}$, but I cannot determine whether these two are isomorphic. Does anyone have any ideas to solve this problem?
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https://math.stackexchange.com/a/4290802/491450 – Smiley1000 Nov 22 '24 at 09:07
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I don’t think is true, the first one is way bigger, it contains sequences like $((1){k\le n}){n \in \mathbb N } )$ but maybe you can build very complicated implicit iso – julio_es_sui_glace Nov 22 '24 at 09:12