Finitely generated algebra

In mathematics, a finitely generated algebra (also called an algebra of finite type) is a commutative associative algebra ${\displaystyle A}$ over a field ${\displaystyle K}$ where there exists a finite set of elements ${\displaystyle a_{1},\dots ,a_{n}}$ of ${\displaystyle A}$ 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, ${\displaystyle A\simeq K[X_{1},\dots ,X_{n}]/{\rm {ker}}(\phi _{\bf {a}})}$.

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^[1]

${\displaystyle A}$ 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 ${\displaystyle I\subseteq K[X_{1},\dots ,X_{n}]}$.

If it is necessary to emphasize the field K then the algebra is said to be finitely generated over K. Algebras that are not finitely generated are called infinitely generated.