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Let $G$ be a module over the non-trivial commutative Noetherian ring $R$. Show that if $G$ has finite length then there exist an ideal $M$, which is a product of finitely many maximal ideals of $R$, such that $MG=0$

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The result you want is contained in Corollary 2.17 of Eisebud's "Commutative Algebra with a view toward Algebraic Geometry." –  Chris Leary Jun 23 '12 at 20:53

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