Let $R$ be a commutative reduced ring with identity with the property that if $I$ and $J$ are two ideals of $R $ such that if $I+J$ is not contained in any minimal prime ideal, then there exist ideals $I'$ and $J'$ of $R$ such that $I'J'=0$ and the ideals $I+I'$ and $J+J'$ are not contained in any minimal prime ideal.

Is there any characterization for such a ring? Or

Is there a reduced ring that dose not have this property?

Note that clearly Noetherian rings have this property.