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 $$P_2$$.

Stronger properties

 * Integral extension:

Incomparable properties

 * Going down extension