Finitely generated algebra

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

Source: Wikipedia — Finitely generated algebra (CC BY-SA 4.0)