Integral extension

Summary

A ring extension $A \subseteq B$ is integral if every element of $B$ is integral over $A$. It is the ring analogue of an algebraic extension of fields. Integral extensions keep the prime ideals close to those of the base. The lying over, going-up and incomparability theorems hold, and the two rings have the same Krull dimension. Every finite ring extension is integral. In geometry an integral extension gives a closed map of spectra whose fibres have dimension zero.