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Let $S$ be a noetherian ring and $M$ a finitely generated $S$-module. There exists a filtration by submodules

$$0=M_0 \subseteq M_1\subseteq \cdots \subseteq M_r=M.$$

I want to show that for any prime ideal $P$, $\mathrm{Ann}(M) \subseteq P \iff \mathrm{Ann}(M_i/M_{i-1}) \subseteq P$ for some $1\le i\le r$. I did "if", but I can't show "only if".

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  • $\begingroup$ I am a little confused, are the $M_i$'s the increasing blocks of the gradation? That is, do we have $M=\bigoplus_{i=0}^r N_i$ and $M_i=\bigoplus_{j=0}^i N_j$? $\endgroup$
    – Niccolò
    Sep 13, 2012 at 3:50

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Suppose $0 \rightarrow M^{\prime} \rightarrow M \rightarrow M^{\prime \prime} \rightarrow 0 $ is an short exact sequence of $S$ modules then show $Ann(M) \subset Ann(M^{\prime}) .Ann(M^{\prime \prime})$.

Now use the fact that (1) prime ideal is its own radical (2) the filtration can be written as short exact sequences.

I do not see why you would need the gradation though.

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  • $\begingroup$ I don't understand what $Ann(M) \subset Ann(M^{\prime}) .Ann(M^{\prime \prime})$ means. Maybe there is a comma instead of the point. $\endgroup$
    – user26857
    Oct 18, 2012 at 0:49

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