I guess this is a simple question but I don't see the answer.
Definition: $M$ is Noetherian if every chain of submodules stabilize.
Theorem: $M$ is Noetherian module iff every nonempty set $S$ of submodules of M contains at least one maximal element.
Should this maximal element be in the set? The book I use says that if $S$ is finite it contains maximum element but I think that two submodules are not necessarily comparable by inclusion. (Take submodules of integers generated by 6 and 9 for example. They have nonempty intersection but non of them is included in the other. In the same time integers is Noetherian module over itself)
Thanks for any help!