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- Artinian_module abstract "In abstract algebra, an Artinian module is a module that satisfies the descending chain condition on its poset of submodules. They are for modules what Artinian rings are for rings, and a ring is Artinian if and only if it is an Artinian module over itself (with left or right multiplication). Both concepts are named for Emil Artin.In the presence of the axiom of choice, the descending chain condition becomes equivalent to the minimum condition, and so that may be used in the definition instead.Like Noetherian modules, Artinian modules enjoy the following heredity property: If M is an Artinian R-module, then so is any submodule and any quotient of M.The converse also holds: If M is any R module and N any Artinian submodule such that M/N is Artinian, then M is Artinian.As a consequence, any finitely-generated module over an Artinian ring is Artinian. Since an Artinian ring is also a Noetherian ring, and finitely-generated modules over a Noetherian ring are Noetherian, it is true that for an Artinian ring R, any finitely-generated R-module is both Noetherian and Artinian, and is said to be of finite length; however, if R is not Artinian, or if M is not finitely generated, there are counterexamples.".
- Artinian_module wikiPageID "542336".
- Artinian_module wikiPageLength "7286".
- Artinian_module wikiPageOutDegree "25".
- Artinian_module wikiPageRevisionID "635797664".
- Artinian_module wikiPageWikiLink Abstract_algebra.
- Artinian_module wikiPageWikiLink Artinian_ring.
- Artinian_module wikiPageWikiLink Ascending_chain_condition.
- Artinian_module wikiPageWikiLink Axiom_of_choice.
- Artinian_module wikiPageWikiLink Bimodule.
- Artinian_module wikiPageWikiLink Category:Module_theory.
- Artinian_module wikiPageWikiLink Composition_series.
- Artinian_module wikiPageWikiLink Descending_chain_condition.
- Artinian_module wikiPageWikiLink Emil_Artin.
- Artinian_module wikiPageWikiLink Ideal_(ring_theory).
- Artinian_module wikiPageWikiLink Krull_dimension.
- Artinian_module wikiPageWikiLink Length_of_a_module.
- Artinian_module wikiPageWikiLink Minimum_condition.
- Artinian_module wikiPageWikiLink Module_(mathematics).
- Artinian_module wikiPageWikiLink Noetherian_module.
- Artinian_module wikiPageWikiLink Noetherian_modules.
- Artinian_module wikiPageWikiLink Noetherian_ring.
- Artinian_module wikiPageWikiLink Paul_Cohn.
- Artinian_module wikiPageWikiLink Prüfer_group.
- Artinian_module wikiPageWikiLink Quasicyclic_group.
- Artinian_module wikiPageWikiLink Semisimple_module.
- Artinian_module wikiPageWikiLink Semisimple_ring.
- Artinian_module wikiPageWikiLink Simple_ring.
- Artinian_module wikiPageWikiLinkText "Artinian module".
- Artinian_module wikiPageWikiLinkText "Artinian".
- Artinian_module wikiPageWikiLinkText "examples of Artinian modules which are not Noetherian".
- Artinian_module hasPhotoCollection Artinian_module.
- Artinian_module wikiPageUsesTemplate Template:Cite_book.
- Artinian_module wikiPageUsesTemplate Template:Cite_journal.
- Artinian_module subject Category:Module_theory.
- Artinian_module hypernym Module.
- Artinian_module type Software.
- Artinian_module comment "In abstract algebra, an Artinian module is a module that satisfies the descending chain condition on its poset of submodules. They are for modules what Artinian rings are for rings, and a ring is Artinian if and only if it is an Artinian module over itself (with left or right multiplication).".
- Artinian_module label "Artinian module".
- Artinian_module sameAs Module_artinien.
- Artinian_module sameAs מודול_ארטיני.
- Artinian_module sameAs Modulo_artiniano.
- Artinian_module sameAs アルティン加群.
- Artinian_module sameAs m.02n9ly.
- Artinian_module sameAs Артинов_модуль.
- Artinian_module sameAs Q2919100.
- Artinian_module sameAs Q2919100.
- Artinian_module sameAs 阿廷模.
- Artinian_module wasDerivedFrom Artinian_module?oldid=635797664.
- Artinian_module isPrimaryTopicOf Artinian_module.