Let $R$ be a commutative ring with $1$ and let $I,J\!\unlhd\!R$. The radical of an ideal $I$ is defined as $\mathrm{rad}(I):=\sqrt{I}:=\{r\!\in\!R;\;\exists n\!\in\!\mathbb{N}: r^n\!\in\!I\}=\bigcap\{P;\; P\text{ is a prime ideal with }I\!\subseteq\!P\}.$
I've managed to show that there holds$\sqrt{I}\sqrt{J}\subseteq\sqrt{IJ}=\sqrt{I\cap J}=\sqrt{I}\cap\sqrt{J}.$
How can I prove the first $\supseteq$?
NOTE: I guess there has been made a mistake in Ralf Froberg's An Introduction to Grabner Bases, page 19: