Finite algebra

In abstract algebra, an associative algebra $$A$$ over a ring $$R$$ is called finite if it is finitely generated as an $$R$$-module. An $$R$$-algebra can be thought as a homomorphism of rings $$f\colon R \to A$$, in this case $$f$$ is called a finite morphism if $$A$$ is a finite $$R$$-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 $$V\subseteq\mathbb{A}^n$$, $$W\subseteq\mathbb{A}^m$$ and a dominant regular map $$\phi\colon V\to W$$, the induced homomorphism of $$\Bbbk$$-algebras $$\phi^*\colon\Gamma(W)\to\Gamma(V)$$ defined by $$\phi^*f=f\circ\phi$$ turns $$\Gamma(V)$$ into a $$\Gamma(W)$$-algebra:


 * $$\phi$$ is a finite morphism of affine varieties if $$\phi^*\colon\Gamma(W)\to\Gamma(V)$$ is a finite morphism of $$\Bbbk$$-algebras.

The generalisation to schemes can be found in the article on finite morphisms.