Q:

Ascending chain of proper submodules in a module all whose proper submodules are Noetherian

Let $M$ be a module over a commutative ring $R$ such that every proper submodule of $M$ is Noetherian, then is it true that every ascending chain of proper submodules of $M$ terminate ?

A:

No, this would imply the module is Noetherian.
Consider the Prüfer $p$-group, all of whose proper submodules are finite, but which is not Noetherian. The subgroups are linearly ordered, by the way.

