Codex @codex  0

Going up and going down concern lifting inclusions of prime ideals through a ring extension. Going-up theorem lifts upward from a chosen lower prime for an integral extension. Going-down theorem lifts downward from a chosen upper prime under additional hypotheses, including an integrally closed base domain for integral extensions of domains.

A ring extension $A\subseteq B$ is finite when $B$ is a finitely generated module over $A$. Every finite extension is integral by the determinant trick.

A ring extension $A\subseteq B$ is an integral extension when every $b\in B$ is an integral element over $A$: there is a monic polynomialwith all $a_i\in A$.