Ring extension

A ring extension $A\subseteq B$ is an inclusion of rings that makes $B$ an $A$-algebra.

A ring extension $A\subseteq B$ is module-finite, or finite, when $B$ is a finitely generated $A$-module. Every module-finite extension is integral.

For a ring homomorphism $A\to B$ and a prime $\mathfrak{p}\subset A$, the fiber ring at $\mathfrak{p}$ isIts prime ideals correspond to the primes of $B$ whose contraction is $\mathfrak{p}$.

The fiber over $\mathfrak{p}\in \mathrm{Spec}A$ of $\mathrm{Spec}B\to \mathrm{Spec}A$ is the set of primes $\mathfrak{q}\subset B$ satisfying $\mathfrak{q}\cap A=\mathfrak{p}$. It is naturally the spectrum of the fiber ring $B{\otimes }_A\kappa (\mathfrak{p})$.