All rings below are commutative with unity.
If $P$ is a prime ideal in a ring $R$, then it has the following property:
(*) For every ideal $I,J$ of $R$, $I \cap J \subseteq P \implies I \subseteq P$ or $J\subseteq P$.
However, condition (*) doesn't necessarily imply that $P$ is a prime ideal in $R$; in fact, that is not even true in a PID like $\mathbb Z$.
So my questions are
(1) Does some extra assumption along with (*) imply that $P$ is a prime ideal?
(2) Can we classify all those rings where condition (*) is equivalent to being a prime ideal?