## 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 $M_1\cong\pi_1(N)\cong\pi_2(N)\cong M_2$.

Hence, exactly one of $\pi_1$, $\pi_2$ are zero. So $N=M_1\oplus\{0\}$ or $\{0\}\oplus M_2$.