Going-down subring

Definition with symbols
Suppose $$S$$ is a unital subring of a commutative unital ring $$R$$. We say that $$S$$ is a going-down subring if given prime idaels $$Q \subseteq Q_1$$ of $$S$$ and a prime ideal $$P_1$$ of $$R$$ lying over $$Q_1$$ (viz $$P_1 \cap S = Q_1$$, there exists a prime $$P$$ lying over $$Q$$ (viz $$P \cap S = Q$$ and contained in $$P_1$$.

Facts
If $$S$$ is a normal domain and $$R$$ is an integral extension of $$S$$, then $$R$$ is a going-down subring.