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Can someone please explain to me why, given an infinite field $k$, we have that $$\left(\bigcap_{a \in k}{(x-a)}\right) [U^{-1} ] \neq \bigcap_{a \in k}{\left((x-a)[U^{-1}] \right)},$$ where $U$ is the set of nonzero elements of $k[x]$?

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    It's not immediately obvious to me what this notation means.2012-11-02
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    It's the localization of the ideals $\cap (x-a)$ and $(x-a)$, interpreted as $k[x]$-modules.2012-11-02

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