= Finitely generated algebra over a ring
{title2=$B=A[b_1,\ldots,b_n]$}
An <algebra over a commutative ring> $A$ is finitely generated if it is a <quotient ring> of a <polynomial ring> $A[t_1,\ldots,t_n]$ in finitely many variables. Equivalently, finitely many elements generate it under addition, multiplication, and the action of $A$. The induced map of <affine schemes> $\operatorname{Spec}B\to\operatorname{Spec}A$ is a <morphism of finite type> exactly when this holds. Finite generation as an algebra does not require finite generation as an $A$-<module>.
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