Going down extension

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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 $P_2$.

Relation with other properties

Stronger properties

• Flat extension: For full proof, refer: Going down for flat extensions
• Integral extension where both are integral domains and the base is a normal domain