Finitely generated algebra

In mathematics, a finitely generated algebra (also called an algebra of finite type) is an associative algebra A over a field K where there exists a finite set of elements a₁,…,a_n of A such that every element of A can be expressed as a polynomial in a₁,…,a_n, with coefficients in K. 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. Finitely generated reduced commutative algebras are basic objects of consideration in modern algebraic geometry, where they correspond to affine algebraic varieties; for this reason, these algebras are also referred to as (commutative) affine algebras.