Intermediate subring condition for ideals

Definition

Let $p$ be a property of ideals in commutative unital rings. We say that $p$ satisfies the intermediate subgroup condition for ideals if the following is true. Suppose $I$ is an ideal inside a commutative unital ring $R$, and $S$ is a subring of $R$ containing $I$. Then, if $I$ satisfies property $p$ as an ideal in $R$, $I$ also satisfies property $p$ as an ideal in $S$.