Let $R$ be a commutative reduced ring with ideantity 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 charactrization for such a ring? Or

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

Note that clearly noetherian rings have this property.