----- "3 from 2 property" DUPLICATE: [[module is Noetherian (resp. Artinian) iff submodule and quotient is]] > [!proposition] Proposition. ([[short exact sequence characterization of Noetherian modules]]) > Let $R$ be [[ring]], $M$ an $R$-[[module]], and > > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRGJAF9T1Nd9CKAIzkqtRizYA5LjxAZseAkQBMo6vWatEIALKzeigUQDM68VrYAFAAQAdOwGMCAcxu6bAehszuh-sooACzmmpI6HJxiMFAu8ESgAGYAThAAtkhkIDgQSEJ+ICnpedQ5SGrZdFgMbAAWEBAA1gaFqRmIFWWIZiC1MHRQbDgA7hB9AwgFRe09XUFRnEA > \begin{tikzcd} > 0 \arrow[r] & N \arrow[r, hook] & M \arrow[r, two heads] & P \cong M / N \arrow[r] & 0 > \end{tikzcd} > \end{document} > ``` > is a [[short exact sequence]] of $R$-[[module|modules]], then $M$ is [[Noetherian module|Noetherian]] if and only if the [[submodule]] $N$ and [[quotient module|quotient]] $P \cong M / N$ are. > > ^proposition > [!proposition] Corollary. > Every [[submodule generated by a subset|finitely generated]] [[module]] over a [[Noetherian ring]] is [[Noetherian module|Noetherian]]. ^proposition > [!proof] Proof of Corollary. > ^proof By hypothesis there is a [[surjection|surjective]] [[linear map]] $R^{\oplus n} \to M$ for some $n \in \mathbb{N}$. Then, by the [[first isomorphism theorem for modules]], $M$ is [[module isomorphism|isomorphic]] to a [[quotient module|quotient]] of $R^{\oplus n}$. The original proposition says that if $R^{\oplus n}$ is [[Noetherian module|Noetherian]] then so is $M$. So we just need to show (by induction) that $R^{\oplus n}$ is [[Noetherian module|Noetherian]]. (Done in [[module is Noetherian (resp. Artinian) iff submodule and quotient is]]) > [!proof]- Proof. ([[short exact sequence characterization of Noetherian modules]]) > - [ ] [[TODO]] ----- #### ---- #### 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 > ```