Example of noetherian module non-uniquely expressible as sum of indecomposable submodules

Question

If $M$ is a noetherian module then it can be written as a finite sum of indecomposable submodules of $M$. The same can be concluded if we assume instead $M$ to be artinian. If we ask for both $M$ to be noetherian and artinian, then we can conclude that such a decomposition is unique in the sense of Krull-Schmidt theorem.

My question is:

What is an example of a noetherian module expressible as a finite sum of indecomposable modules in at least two nonequivalent ways?

Answer 1

I don't think there are simple examples. You can find one given by R.G. Swan here.