Finite morphism of rings

Summary

A ring homomorphism $f \colon A \to B$ is finite if $B$, viewed as an $A$-module through $f$, is finitely generated. Finite homomorphisms are exactly the integral homomorphisms that are also of finite type. They are closed under composition and base change. In algebraic geometry they correspond to finite morphisms of affine schemes, such as the projection of a curve onto a line in general position.