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Let $N$ and $K$ be sub-modules of $M$ with $I=\operatorname{Ann}(N)$ and $J=\operatorname{Ann}(K)$. Show that $I+J$ is a proper subset of $\operatorname{Ann}(N \cap K)$.

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    When $N=K$, for example, there's equality $I+J= \text{Ann}(N\cap K)$. Perhaps the word "proper" is redundant?2012-12-31

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