Going down extension

Statement
Suppose $$A$$ is a subring of a commutative unital ring $$B$$ i.e. $$B$$ is an extension of $$A$$. We say that the extension has the going down property if, whenever $$P_1 \supset P_2$$ are prime ideals of $$A$$, and there exists a prime ideal $$Q_1$$ of $$B$$ contracting to $$P_1$$, then there exists a prime ideal $$Q_2 \subset Q_1$$ in $$B$$, such that $$Q_2$$ contracts to $$P_2$$.

Stronger properties

 * Flat extension:
 * Integral extension where both are integral domains and the base is a normal domain