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I asked this question on the stack exchange, and after no answers and the recommendation of someone else, I am posting it here on MO. I am looking for an example of two ideals $I$ and $J$ in a noetherian ring that are both primary to $p$, but their sum is not. There are a few things that I do know: If $p$ is maximal, there are no examples (ruling out the artinian case). If $I+J$ is unmixed, then it will be $p$-primary (because $p$ is a minimal prime of $I+J$). Also, if $I$ and $J$ are monomial ideals, then $I+J$ will be $p$-primary, because primary ideals have the same form for monomial ideals.

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Let $R$ be the commutative ring $k[x,y,z]$. Let $I$ be the ideal generated by the regular sequence $(x^2,y)$. Let $J$ be the ideal generated by the regular sequence $(x^2,y-xz)$. Then both $R/I$ and $R/J$ are Cohen-Macaulay, hence unmixed. The prime $\mathfrak{p}=\langle x,y \rangle$ is the unique associated prime of both $I$ and $J$. However, $I+J$ equals $\langle x^2,xz,y\rangle$, and this ideal also has $\mathfrak{m}=\langle x,y,z\rangle$ as an associated prime. Indeed, $\mathfrak{m}$ is the annihilator of $\overline{x}$ in the quotient ring $R/(I+J)$.

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    $\begingroup$ In fact, $\langle x^2,xz,y\rangle=\langle x,y\rangle\cap\langle x^2,y,z\rangle$ is a primary decomposition. $\endgroup$
    – user26857
    Jan 5, 2017 at 11:36
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    $\begingroup$ Geometric intuition: $I$ and $J$ are thickenings of the line $\sqrt{I}=\sqrt{J}$ to two different ribbons, that intersect transversely as $\sqrt{I}$ away from $\vec 0$, where they're tangent to one another and the intersection is big. Nice example! $\endgroup$ Jan 8, 2017 at 7:16
  • $\begingroup$ @AllenKnutson. Yes, you are correct. That is precisely how I came up with this example. $\endgroup$ Jan 8, 2017 at 11:01

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