# Going-down subring

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 $P_1$.
If $S$ is a normal domain and $R$ is an integral extension of $S$, then $R$ is a going-down subring.