Finite algebra

In abstract algebra, an associative algebra ${\displaystyle A}$ over a ring ${\displaystyle R}$ is called finite if it is finitely generated as an ${\displaystyle R}$-module. An ${\displaystyle R}$-algebra can be thought as a homomorphism of rings ${\displaystyle f\colon R\to A}$, in this case ${\displaystyle f}$ is called a finite morphism if ${\displaystyle A}$ is a finite ${\displaystyle R}$-algebra.^[1]

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 ${\displaystyle V\subseteq \mathbb {A} ^{n}}$, ${\displaystyle W\subseteq \mathbb {A} ^{m}}$ and a dominant regular map ${\displaystyle \phi \colon V\to W}$, the induced homomorphism of ${\displaystyle \Bbbk }$-algebras ${\displaystyle \phi ^{*}\colon \Gamma (W)\to \Gamma (V)}$ defined by ${\displaystyle \phi ^{*}f=f\circ \phi }$ turns ${\displaystyle \Gamma (V)}$ into a ${\displaystyle \Gamma (W)}$-algebra:

${\displaystyle \phi }$ is a finite morphism of affine varieties if ${\displaystyle \phi ^{*}\colon \Gamma (W)\to \Gamma (V)}$ is a finite morphism of ${\displaystyle \Bbbk }$-algebras.^[2]

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