Product of simple modules not semi-simple

Question

Consider the $\mathbb{Z}$-module $M := \prod_{n \in \mathbb{N}} \mathbb{Z}/2\mathbb{Z}$. I'm looking for an exact sequence of $\mathbb{Z}$-modules $$0 \rightarrow L \rightarrow M \rightarrow N \rightarrow 0 $$ that doesn't split. I've tried a few but every time I think I have one it turns out I did something wrong. I'm starting to doubt whether $M$ really is semi-simple.

Answer 1

There is no such example: $\prod_\mathbb{N}\mathbb{Z}/2\mathbb{Z}$ is in fact a semisimple $\mathbb{Z}$-module. Indeed, since it is a $\mathbb{Z}/2\mathbb{Z}$-vector space, it can be written as a direct sum of copies of $\mathbb{Z}/2\mathbb{Z}$, so as a direct sum of simple modules it is semisimple.