Not primary ideal having a prime radical
The Wikipedia article on primary ideals has the following example: Let $R=k[x,y,z]/(xy-z^2)$, $k$ a field. Take $P=(x,z)$, which is prime. Then $I:=P^2$ is not primary, as $xy\in P^2$, but $x\not\in P^2$ and $y^n\not\in P^2$ for all $n$. However, for all $r_1,r_2\in R$, $r_1 r_2\in P^2\Rightarrow r_1r_2\in P\Rightarrow r_1\in P\text{ or }r_2 \in P\Rightarrow r_1^2\in P^2\text{ or }r_2^2\in P^2$.