Let $R$ be a commutative reduced ring with ideantityidentity with the property that if $I$ and $J$ are two ideals of $R $ such that if $I+J $$I+J$ is not contained in any minimal prime ideal, then there exist ideals $I' $$I'$ and $J' $$J'$ of $R$ such that $ I'J'=0 $$I'J'=0$ and the ideals $I+I' $$I+I'$ and $J+J' $$J+J'$ are not contained in any minimal prime ideal.
Is there any charactrizationcharacterization for such a ring? Or
Is there a reduced ring that dose not have this property?
Note that clearly noetherianNoetherian rings have this property.