Finite type algebra

Summary

An algebra $B$ over a commutative ring $A$ is of finite type if finitely many elements of $B$ generate it as an $A$-algebra. Equivalently, $B$ is a quotient of a polynomial ring $A[x_1, \ldots, x_n]$ in finitely many variables. Finitely generated algebras over a field are the coordinate rings of affine varieties, the main objects of classical algebraic geometry. Over a Noetherian base they are Noetherian, by Hilbert's basis theorem.