For $R$-modules $M,M'$, is every quotient $Q$ of a submodule of $M\oplus M'$ isomorphic to a quotient of a submodule of $M$ or $M'$? Does one have to assume that $Q$ is simple?
2026-03-27 17:04:30.1774631070
Quotients of submodules of direct sums
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There is a trivial counterexample when $Q$ is not simple. Consider the $\mathbb{F}_p$-modules $M = \mathbb{F}_p$ and $M' = \mathbb{F}_p$. Then $Q = M \oplus M'$ is a quotient of itself (which is in turn a submodule of itself). But it clearly is not a quotient of $M$ or $M'$.