Showing that $\bigoplus_{i\in\mathbb N}\mathbb Z/2\mathbb Z$ is not a direct summand of $\prod_{i\in\mathbb N}\mathbb Z/2\mathbb Z$

Question

I know that if the direct sum of countably many copies of $\mathbb Z/2\mathbb Z$, $I$ were a direct summand of the direct product of countably many copies, $R$ (direct summand as $R$-modules), then the complement would be a direct summand isomorphic to $R/I$. But I can't think of any properties of the quotient that would preclude it from being a subgroup. Could I have a hint?

Answer 1

Let $R=\prod k$ (infinitely many copies) be considered a module over itself, $k$ a field. Then $\bigoplus k$ is a submodule of $R$. Suppose that $N$ is any other nontrivial submodule (equivalently, ideal) of $R$.

Show that $N$ and $\bigoplus k$ intersect nontrivially: take an arbitrary nonzero element of the former, and multiply it by an appropriately coordinate-annihilating element of $R$, and you remain in $N$...