A $R$-module $S$ is finite, or finitely generated as a module over $R$, if there is an onto homomorphism of $R$-modules from the free $R$-module on a finite set to $S$
$$R^{\oplus n} \twoheadrightarrow S$$
$S$ is finite type, or finitely generated as an algebra over $R$, if there is an onto homomorphism of $R$-algebras from the free $R$-algebra on a finite set to $S$
$$R[x_1, \dots, x_n] \twoheadrightarrow S$$
How to prove finite $\Rightarrow$ finite type?