Exercise 4, § 4

Let M be a module that is the direct sum of a family $(M_{\alpha})_{\alpha\in I}$ of submodules. Prove that the submodules $M_{\alpha}$ are stable under all automorphisms of M if and only if every homomorphism $M_{\alpha}\rightarrow M_{\beta}$ with $\alpha \not=\beta$ is zero.