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.



$I,J$ are $p$-primary ideals, but $I+J$ is not
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 $R/(I+J)$.


