Finitely generated algebra

In mathematics, a finitely generated algebra (also called an algebra of finite type) over a (commutative) ring ${\displaystyle R}$, or a finitely generated ${\displaystyle R}$-algebra for short, is a commutative associative algebra ${\displaystyle A}$ defined by ring homomorphism ${\displaystyle f:R\to A}$, such that every element of ${\displaystyle A}$ can be expressed as a polynomial in a finite number of generators ${\displaystyle a_{1},\dots ,a_{n}\in A}$ with coefficients in ${\displaystyle f(R)}$. Put another way, there is a surjective ${\displaystyle R}$-algebra homomorphism from the polynomial ring ${\displaystyle R[X_{1},\dots ,X_{n}]}$ to ${\displaystyle A}$.

If ${\displaystyle K}$ is a field, regarded as a subalgebra of ${\displaystyle A}$, and ${\displaystyle f}$ is the natural injection ${\displaystyle K\hookrightarrow A}$, then a ${\displaystyle K}$-algebra of finite type is a commutative associative algebra ${\displaystyle A}$ where there exists a finite set of elements ${\displaystyle a_{1},\dots ,a_{n}\in A}$ such that every element of ${\displaystyle A}$ can be expressed as a polynomial in ${\displaystyle a_{1},\dots ,a_{n}}$, with coefficients in ${\displaystyle K}$.

Equivalently, there exist elements ${\displaystyle a_{1},\dots ,a_{n}\in A}$ such that the evaluation homomorphism at ${\displaystyle {\bf {a}}=(a_{1},\dots ,a_{n})}$

${\displaystyle \phi _{\bf {a}}\colon K[X_{1},\dots ,X_{n}]\twoheadrightarrow A}$

is surjective; thus, by applying the first isomorphism theorem, ${\displaystyle A\cong K[X_{1},\dots ,X_{n}]/{\rm {ker}}(\phi _{\bf {a}})}$.

Conversely, ${\displaystyle A:=K[X_{1},\dots ,X_{n}]/I}$ for any ideal ${\displaystyle I\subseteq K[X_{1},\dots ,X_{n}]}$ is a ${\displaystyle K}$-algebra of finite type, indeed any element of ${\displaystyle A}$ is a polynomial in the cosets ${\displaystyle a_{i}:=X_{i}+I,i=1,\dots ,n}$ with coefficients in ${\displaystyle K}$. Therefore, we obtain the following characterisation of finitely generated ${\displaystyle K}$-algebras:

${\displaystyle A}$ is a finitely generated ${\displaystyle K}$-algebra if and only if it is isomorphic as a ${\displaystyle K}$-algebra to a quotient ring of the type ${\displaystyle K[X_{1},\dots ,X_{n}]/I}$ by an ideal ${\displaystyle I\subseteq K[X_{1},\dots ,X_{n}].}$

Algebras that are not finitely generated are called infinitely generated.

A finitely generated ring refers to a ring that is finitely generated when it is regarded as a ${\displaystyle \mathbb {Z} }$-algebra.

An algebra being finitely generated (of finite type) should not be confused with an algebra being finite (see below). A finite algebra over ${\displaystyle R}$ is a commutative associative algebra ${\displaystyle A}$ that is finitely generated as a module; that is, an ${\displaystyle R}$-algebra defined by ring homomorphism ${\displaystyle f:R\to A}$, such that every element of ${\displaystyle A}$ can be expressed as a linear combination of a finite number of generators ${\textstyle a_{1},\dots ,a_{n}\in A}$ with coefficients in ${\displaystyle f(R)}$. This is a stronger condition than ${\displaystyle A}$ being expressible as a polynomial in a finite set of generators in the case of the algebra being finitely generated.