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.

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