I am trying to show that if $M$ is a left $R$-module and $\text{End}_R (M)$ is a division ring then we cannot have $M=M_1 \oplus M_2 $ for any proper submodules $M_1, M_2 $ of $M$.
I am stuck on how to start. The fact that $\text{End}_R (M) $ is a division ring means that other than the $0 $ map, all the homomorphisms are isomorphisms. But I really don’t know how to use this or if this is even important.
Recall that the projection map from $M$ to $M_1$, say $e_1 \in \mathrm{End}_R(M)$, is an idempotent element (i.e. $e^2=e$). This forces $e(e-1)=0$, which can't be part of a division ring.