---- Let $R$ be a [[ring]]. > [!definition] Definition. ([[Artinian module]]) > An $R$-[[module]] is said to be **Artinian** if it satisfies the [[ascending chain condition|descending chain condition]] for [[submodule|submodules]]. ^definition > [!basicnonexample] > $\mathbb{Z}$, as a $\mathbb{Z}$-[[module]], is [[Noetherian module|Noetherian]] (e.g. because it's a [[PID]]) but not Artinian, for the descending chain $(2) \supsetneq (4) \supsetneq (8) \supsetneq\dots$ never stabilizes. ^nonexample > [!equivalence] > Equivalently, $R$ is Artinian if every nonempty collection of [[submodule|submodules]] has a minimal element [[poset|by inclusion]] (i.e., one with no proper submodules in the collection). ^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 > ```