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Does there exist a noetherian domain $A$ and a principal ideal $I = (x)$ in it having an embedded component?

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    In a more geometric term, if $X=\mathrm{Spec}(A)$ is an integral surface with only one non-normal point $p$, then $O_{X,p}$ is not $(S_2)$ (hence has depth 1) by Serre's criterion. Take any non-zero $x \in A$ belonging to the maximal ideal corresponding to $p$, then $A/xA$ has depth $0$ at $p$ and Krull dimension $1$, so $p$ is an embedded point of $A/xA$.2012-10-02

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