----- Let $R$ be a [[ring]] and $M$ an $R$-[[module]]. > [!proposition] Proposition. ([[finitely presented and finitely generated modules coincide over Noetherian rings]]) > It is always true that if $M$ is [[finitely presented module|finitely presented]] then it is also [[submodule generated by a subset|finitely generated]]. If $R$ is [[Noetherian ring|Noetherian]], then the converse holds. ^proposition > [!proof]- Proof. ([[finitely presented and finitely generated modules coincide over Noetherian rings]]) > If $M$ is [[submodule generated by a subset|finitely generated]], then there is an [[exact sequence]] $R^{\oplus m} \xrightarrow{\pi} M \to 0$ for some $m \in \mathbb{N}$. Since $R$ is a [[Noetherian ring]], $R^{\oplus m}$ is [[Noetherian module|Noetherian]] as an $R$-[[module]] (as [[short exact sequence characterization of Noetherian modules|every finitely generated module over a Noetherian ring is Noetherian]]). By [[short exact sequence characterization of Noetherian modules|Noetherian module iff submodules and their quotients are noetherian]], the [[kernel of a module homomorphism|kernel]] of $\pi$ is [[submodule generated by a subset|finitely generated]], i.e., there is an [[exact sequence]] $R^{\oplus n} \to \text{ker }\pi \to 0$ for some $n$. Now put together the two [[chain complex of modules|sequences]]: $R^{\oplus n} \to R^{\oplus n} \to M \to 0$ where exactness holds at each step; this is exactly what we were looking to show. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```