Going-down subring

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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.