Let a module $V$ be local if it has a maximal submodule containing all proper submodules of $V$, hence it must be unique. Now, I have to prove the converse is not true, i.e, I have to find a module $V$ with a unique maximal submodule but $V$ is not local.
At first, I have thought that I must look for some module of infinite length, in particular, some module which is not finitely generated since otherwise all proper submodules are contained in a maximal one. However, I have not worked too much with these kind of modules and I would appreciate any hint!