Finite algebra

An formula_1-algebra formula_2 is finite if it is finitely generated as an formula_1-module. An formula_1-algebra can be thought as a homomorphism of rings formula_5, in this case formula_6 is called a finite morphism if formula_2 is a finite formula_1-algebra.
The definition of finite algebra is related to that of algebras 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 formula_9, formula_10 and a dominant regular map formula_11, the induced homomorphism of formula_12-algebras formula_13 defined by formula_14 turns formula_15 into a formula_16-algebra: 
The generalisation to schemes can be found in the article on finite morphisms.