Example where $I,J$ are $p$-primary ideals, but $I+J$ is not $p$-primary.

Question

Assume that $R$ is a noetherian ring. I am looking for an example where $I$ and $J$ are $p$-primary $R$-ideals, but $I+J$ is not $p$-primary. I am pretty sure there are no such examples for monomial ideals over a polynomial ring (over a field), or for ideals in a PID. Also, I know that $p$ is the unique minimal prime of $I+J$, so that any other associated prime must contain $p$.