If $G$ is a finitely generated non-zero module over the non-trivial commutative Noetherian ring $R$ then is it possible that for all maximal ideal $M$ of $R$ we have $MG=G$ ?
If $R$ is semi-local then by Nakayama's lemma the answer is no (although we don't need to use the fact that $R$ is Noetherian). What about arbitrary Noetherian ring $R$ ?