----- > [!proposition] Proposition. ([[ascending chain - maximality characterization of Noetherian modules]]) > Let $R$ be a [[commutative ring|commutative]] [[ring]], and let $M$ be an $R$-[[module]]. The following are equivalent: > 1. $M$ is [[Noetherian module|Noetherian]], i.e., every [[submodule]] of $M$ is [[submodule generated by a subset|finitely generated]]; > 2. $M$ satisfies the [[ascending chain condition]]: if $N_{1} \subset N_{2} \subset N_{3} \subset \dots$ is a [[poset|chain]] of [[submodule|submodules]] of $M$, then there exists $i \in \mathbb{N}$ such that $N_{i}=N_{i+1}=N_{i+2}=\cdots.$ > 3. Every nonempty family of [[submodule|submodules]] of $M$ has a [[maximal]] element with respect to inclusion. ^proposition > [!proposition] Corollary. > Specializing to $M=R$, we find that a [[commutative ring|(commutative)]] [[ring]] is [[Noetherian ring|Noetherian]] if and only if the [[ascending chain condition]] holds for its [[ideal|ideals]] (= its [[submodule|submodules]]). ^proposition-2 > [!proof]- Proof. ([[ascending chain - maximality characterization of Noetherian modules]]) > We'll do $(1) \implies (2) \implies (3) \implies (1)$. > > **$(1) \implies (2)$.** Assume $M$ is [[Noetherian module|Noetherian]] and let $N_{1} \subset N_{2} \subset N_{3} \subset\dots$ > be a [[poset|chain]] of [[submodule|submodules]] of $M$. Consider the union $N:=\bigcup_{i}N_{i};$ > note that this is a [[submodule]] too. $M$ is [[Noetherian module|Noetherian]]; by the [[short exact sequence characterization of Noetherian modules]] $N$ is also, so write $N=\langle n_{1},\dots,n_{r} \rangle$. $n_{k} \in N \implies n_{k} \in N_{i_{k}}$ for some $i_{k}$; put $\hat{i}:=\max_{}\{ i_{k} \}_{k =1}^{r}$. Then $n_{1},\dots,n_{r}$ each belong to $N_{\hat{i}}$, and therefore $\langle n_{1},\dots,n_{r} \rangle=N \subset N_{\hat{i}}$. Taking inventory, we have $\bigcup_{i} N_{i} \subset N_{\hat{i}}$ > for some $\hat{i}$, from which it follows that the chain stabilizes eventually. > > **$(2) \implies (3)$.** Contrapositively suppose $M$ admits a family $\mathscr{F}$ of [[submodule|submodules]] with no [[maximal]] element. Then we may easily construct a non-stabilizing [[ascending chain condition|ascending chain]] as follows: > - Let $N_{1}$ be any element of $\mathscr{F}$; > - Choose $N_{2}$ properly containing $N_{1}$ (exists because $N_{1}$ is not [[maximal]] ) > - Choose $N_{2}$ properly containing $N_{2}$ (exists because $N_{2}$ is not [[maximal]]) > - And so on... > > **$(3) \implies (1)$.** Assume $(3)$ holds, and let $N$ be a [[submodule]] of $M$. The family $\mathscr{F}$ of [[submodule generated by a subset|finitely generated]] [[submodule|submodules]] of $N$ is nonempty (indeed, $\{ 0 \} \in \mathscr{F}$); fix a [[maximal]] element $N'=\langle n_{1},\dots,n_{r} \rangle$ of it. If $n \in N$, then the [[submodule]] $\langle n_{1},\dots,n_{r}, n \rangle$ is [[submodule generated by a subset|finitely generated]] and therefore belongs to $\mathscr{F}$, from which it follows that $n \in N'$. So two-way inclusion holds and $N=N'$. This feels in spirit somewhat like the [[redundant vector lemma]]? ----- #### ---- #### 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 > ```