# Is it true that an ideal is primary iff its radical is prime?

Is it true that an ideal $I$ in a commutative ring is primary iff $Rad(I)$ is prime?

If not, what are some nice counterexamples?

Here is a simple example with $I$ not primary such that $Rad(I)$ is prime. Consider the ideal $(y^2,xy)\subset K[x,y]$. The radical of this ideal is $(y)$, i.e., it is prime.
At the same time, clearly $x\cdot y\in I$. Also, $y\notin I$. Nevertheless $x^n\notin I$ for any $n$. Hence $I$ is not primary!
Comment. As user26857 noted in a remark, one can even construct a prime ideal $p$ such that $p^2$ is not primary, though the example is a bit more complicated: