"3 from 2 result" ----- > [!proposition] Proposition. ([[module is Noetherian (resp. Artinian) iff submodule and quotient is]]) > Let $R$ be a [[ring]], $M$ an $R$-[[module]], and $N$ a [[submodule]] of $M$. Then $M$ is [[Noetherian module|Noetherian]] (resp. Artninian) if and only if $N$ and $M / N$ are both [[Noetherian module|Noetherian]] (resp. Noetherian). > So, given a [[short exact sequence]] of $R$-modules $0 \to M \to N \to L \to 0,$ one has that $M$ is [[Noetherian module|Noetherian]] (resp. Artinian) if and only if $N,L$ are [[Noetherian module|Noetherian]] (resp. [[Artinian ring|Artinian]]). > [!proposition] Corollary 1. Consideration of the [[short exact sequence]] of $R$-modules $0 \to A \to A \oplus B \to B \to 0$ indicates that the [[direct sum of modules|direct sum]] of [[Noetherian module|Noetherian]] modules is [[Noetherian module|Noetherian]]. By induction, $R^{\oplus n}$ is [[Noetherian module|Noetherian]] for any $n \in \mathbb{N}$ if $R$ is [[Noetherian module|Noetherian]]. > [!proposition] Corollary 2. > Let $R$ be a [[Noetherian ring]], and let $M$ be a [[submodule generated by a subset|finitely generated]] $R$-[[module]]. Then $M$ is [[Noetherian module|Noetherian]] as an $R$-[[module]]. > > Indeed, if $M$ is finitely generated as an $R$-[[module]] [[submodule generated by a subset|then]] we have a [[surjection]] $R^{\oplus n} \to M$ for some $n \in \mathbb{N}$. Since $R^{\oplus n}$ is [[Noetherian module|Noetherian]] by the first corollary, [[quotient module|quotienting]] by the kernel[^1] [[first isomorphism theorem for modules|yields an isomorphism of]] $M$ with a [[Noetherian module]]. > > A deeper (and still true) statement is [[Hilbert's basis theorem]]: every [[subalgebra generated by a subset|finitely generated algebra]] over a [[Noetherian ring]] is Noetherian. ^proposition [^1]: This preserves Noetherianity by the proposition statement. > [!proof]- Proof. ([[module is Noetherian (resp. Artinian) iff submodule and quotient is]]) > See [[short exact sequence characterization of Noetherian modules]] ----- #### See also (duplicate?): [[short exact sequence characterization of Noetherian modules]]. But I think this note is better. ---- #### 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 > ```