# A question on modules over Noetherian ring

If $G$ is a module over the non-trivial commutative Noetherian ring $R$ then is it possible that for all maximal ideal $M$ of $R$ we have $MG=G$ ?

I guess the answer is no.

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I assume you also want the assumption that $G\neq 0$? –  Ben Blum-Smith Jun 23 '12 at 19:50

Actually the answer is yes: $R=\mathbb Z, G=\mathbb Q$ .