Non-trivial submodules of direct sum of simple modules

Suppose $M_1$ and $M_2$ are two non-isomorphic simple, nonzero $R$ -modules.

Determine all non-trivial submodules of $M_1\oplus M_2$ .

Let $N$ be a non-trivial submodule of $M_1\oplus M_2$ . Note that $\displaystyle \{0\}\subset M_1\subset M_1\oplus M_2$ is a composition series. By Jordan-Holder theorem, all composition series are equivalent and have the same length. Hence $\displaystyle \{0\}\subset N\subset M_1\oplus M_2$ must be a composition series too.

Thus $N\cong M_1$ or $M_2$ . In particular $N$ is simple.

Let $\pi_1: M_1\oplus M_2\to M_1$ and $\pi_2:M_1\oplus M_2\to M_2$ be the canonical projections. Note that $\pi_1(N)$ is a submodule of $M_1$ , so $\pi_1(N)\cong 0$ or $M_1$ . Similarly, $\pi_2(N)\cong 0$ or $M_2$ .

By Schur’s Lemma $\pi_1|_N: N\to \pi_1(N)$ and $\pi_2|_N: N\to\pi_2(N)$ are either 0 or isomorphisms.

They cannot be both zero since $N$ is non-zero. They cannot be both isomorphisms either, as that would imply $M_1\cong\pi_1(N)\cong\pi_2(N)\cong M_2$ .