Let $R$ be a commutative ring with identity and $I$ a proper ideal of $R$. We define $L$-radical of $I$, denoted by $\sqrt[L]{I}$, the intersection of all primary ideals of $R$ containing $I$. It is clear that $\sqrt[L]{I\cap J}\subseteq \sqrt[L]{I} \cap \sqrt[L]{J}$ where $I$ and $J$ are proper ideals of $R$. I'm looking for a counterexample to show that the reverse inclusion not true, in general.
If $I$ is a decomposable ideal, i.e. finite intersection of primary ideals, (this is the case when $R$ is noetherian), then $\sqrt[L]{I} = I$. Hence you should look for a counterexample when $R$ is not noetherian. –  Andrea May 28 '12 at 9:22