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