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 $MG=0$
A condition which every module (over a commutative noetherian ring) having finite length should satisfy
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commutative-algebra
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2The result you want is contained in Corollary 2.17 of Eisebud's "Commutative Algebra with a view toward Algebraic Geometry." – 2012-06-23