I have to prove the following: Let $I$ be an ideal of $K[G]$ ($K$ is a field, $G$ is a multiplicative group, $K[G]$ is a group ring) and let $H_1,\ldots,H_n$ be normal subgroups of $G$, each controlling $I$. Show that $H= H_1\cap H_2\cap\ldots\cap H_n$ controls $I$.
I can show that $\pi_{H}(I)\cdot K[G]\subseteq \pi_{H_i}\cdot K[G]=I$ for all $i=1,\ldots,n$. How to show $I\subseteq \pi_{H}(I)\cdot K[G]$?