Going up extension

Definition

Suppose $B$ is a commutative unital ring and $A$ is a subring of $B$. In other words, $B$ is an extension of the ring $A$. Then, we say that the extension has the going up property if it satisfies the following:

• The map is surjective on spectra
• If $P_1\subset P_2$ are prime ideals of $A$ and $Q_1$ is a prime ideal of $B$ contracting to $P_1$, then there exists a prime ideal $Q_2$ of $B$ containing $Q_1$ such that $Q_2$ contracts to $P_2$.

Relation with other properties

Stronger properties

• Integral extension: For full proof, refer: Going up theorem

Incomparable properties

• Going down extension