Finite algebra

In abstract algebra, an associative algebra over a ring is called finite if it is finitely generated as an -module. An -algebra can be thought as a homomorphism of rings, in this case is called a finite morphism if is a finite -algebra.
Being a finite algebra is a stronger condition than being an algebra of finite type.

Finite morphisms in algebraic geometry

This concept is closely related to that of finite morphism in algebraic geometry; in the simplest case of affine varieties, given two affine varieties, and a dominant regular map, the induced homomorphism of -algebras defined by turns into a -algebra:
The generalisation to schemes can be found in the article on finite morphisms.