Finitely generated algebra over a ring ($B=A[b_1,\dots ,b_n]$)

An algebra over a commutative ring $A$ is finitely generated if it is a quotient ring of a polynomial ring $A[t_1,\dots ,t_n]$ in finitely many variables. Equivalently, finitely many elements generate it under addition, multiplication, and the action of $A$. The induced map of affine schemes $\mathrm{Spec}B\to \mathrm{Spec}A$ is a morphism of finite type exactly when this holds. Finite generation as an algebra does not require finite generation as an $A$-module.