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 $$S$$.