Going-down subring

This article defines a property that can be evaluated for a unital subring in a commutative unital ring: given any commutative unital ring and a subring thereof, the property is either true or false for the pair
View a complete list of such properties

Definition

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.