Finitely Generated Modules over Noetherian Rings are Finitely Presented

Definition. A ring $R$ is Noetherian if every ideal of $R$ is finitely generated.

Definition. An $R$-module $M$ is finitely generated if there exists a surjection $R^{\oplus n}\twoheadrightarrow M$ for some $n\in \mathbb{N}$.

Definition. An $R$-module $M$ is finitely presented if there is an exact sequence

$$R^{\oplus m}⟶R^{\oplus n}⟶M⟶0$$

with $m,n\in \mathbb{N}$.

Theorem. If $R$ is a Noetherian ring and $M$ is a finitely generated $R$-module, then $M$ is finitely presented.

Proof. Let $\phi :R^{\oplus n}\twoheadrightarrow M$ be a surjection defining the finite generation of $M$. As $R$ is a Noetherian ring, the free module $R^{\oplus n}$ is a Noetherian module. The kernel, $K=\ker (\phi )$, is a submodule of $R^{\oplus n}$ and is therefore finitely generated. Let $\psi :R^{\oplus m}\to R^{\oplus n}$ be a homomorphism whose image is $K$. The sequence

$$R^{\oplus m}\stackrel{\psi }{\to }{R}^{\oplus n}\stackrel{\phi }{\to }M⟶0$$

is exact, providing a finite presentation for $M$. $\square$