---- > [!definition] Definition. ([[Noetherian module]]) > Let $R$ be a [[ring]] An $R$-[[module]] $M$ is **Noetherian** if every [[submodule]] of $M$ is [[submodule generated by a subset|finitely generated]]. ^definition > [!specialization] > If $M=R$, then this definition specializes to that of a [[Noetherian ring]], in light of [[submodule#^basic-example|this example]]. ^specialization > [!basicexample] ^basic-example > [!basicexample] Examples. > - [[module is Noetherian (resp. Artinian) iff submodule and quotient is]] > - [[module is Noetherian (resp. Artinian) iff submodule and quotient is|finitely generated module over a Noetherian ring is Noetherian]] ^basic-example > [!basicnonexample] Warning. > In particular, $M$ itself is [[submodule generated by a subset|finitely generated]]. But the converse is false: being finitely generated does not entail being Noetherian. Otherwise, every [[ring]] would be [[Noetherian ring]], since any [[ring]] $R$ is finitely generated as an $R$-[[module]] by $\{ 1_{R} \}$. But a ring [[polynomial 4|such as]] $\mathbb{Z}[T_{1},T_{2},\dots]$ has [[ideal|ideals]] which are not finitely generated, like that generated by $T_{1},T_{2},\dots$ (i.e. the polynomials without constant term). > > If *$R$* is a [[Noetherian ring]], and $M$ is finitely generated, then $M$ *will* be [[Noetherian module|Noetherian]] as an $R$-[[module]]. [[module is Noetherian (resp. Artinian) iff submodule and quotient is|See here.]] ^nonexample > [!equivalence] > - [[short exact sequence characterization of Noetherian modules]] > - If $R$ is [[commutative ring|commutative]]: [[ascending chain - maximality characterization of Noetherian modules]] ^equivalence ---- #### ---- #### 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 > ```