Exercise 16, § 2

Let A be a ring, M an A-module of finite length, and $(M_i)_{i\in I}$ a family of indecomposable submodules of M with direct sum M. Let L be an indecomposable A-module of finite length and $I_L$ the set of indices $i\in I$ such that $M_i$ is isomorphic to L. Use an example to show that the submodule $\sum_{i\in I_L}M_i$ of M is not necessarily stable under the automorphisms of M.