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