Noetherian module implies finite direct sum of indecomposables?
Let R be a ring and let M be a Noetherian R-module.
If M is indecomposable we are done. Otherwise, M is a direct sum of two proper and non-trivial submodules.
If M were also Artinian, I could use induction on the (finite) length of M and prove the result in the title.
I don't know how to proceed in the general case. Thanks in advance for any ideas!