$\DeclareMathOperator\rad{rad}$I am searching an example of ideal $I$ of a ring $R$ such that $\rad(I)$ is irreducible but not prime ideal. 


In case $R$ is Noetherian, the radical of $I$ being irreducible implies $\rad(I)$ is primary. Then it is straight forward to see that $\rad(I)$ is prime. So we want to look at non Noetherian rings. But I am not able to find such examples. Also, $I$ cannot be primary as $\rad(I)$ is prime if $I$ is primary.