For a homomorphism A → B of commutative rings, B is called an A-algebra of finite type if B is a finitely generated as an A-algebra. It is much stronger for B to be a finite A-algebra, which means that B is finitely generated as an A-module. For example, for any commutative ring A and natural number n, the polynomial ring A[x1, ..., xn] is an A-algebra of finite type, but it is not a finite A-module unless A = 0 or n = 0. Another example of a finite-type morphism which is not finite is \mathbb{C}[t] \to \mathbb{C}[t][x,y]/(y^2 - x^3 - t).
