# Math Help - properties of radical ring

1. ## properties of radical ring

Show that $\sqrt{I}=\cap{P}$, where the intersection is over all prime ideals of $R$ that contain $I$.

I know how to verify the one inclution $\sqrt{I}\subset\cap{p}$, but how can I show the second inclution.

The nilradical of $R$ is the intersection of all prime ideals of $R$ . Apply this result to $R/I$ .
$L(R/I)=\cap{(P/I)}$ where $P$ is prime ideal in $R$ contains $I$. But we know that $\cap{(P/I)}=\cap{P}/I$ . Thus, if $x \in \cap{P}$, then $x+I \in \cap{P} +I=L(R/I)$. It is easy to show that $L(R/I)=\sqrt{I}$.