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- Artinian_ring abstract "In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are finite-dimensional vector spaces over fields. The definition of Artinian rings may be restated by interchanging the descending chain condition with an equivalent notion: the minimum condition.A ring is left Artinian if it satisfies the descending chain condition on left ideals, right Artinian if it satisfies the descending chain condition on right ideals, and Artinian or two-sided Artinian if it is both left and right Artinian. For commutative rings the left and right definitions coincide, but in general they are distinct from each other.The Artin–Wedderburn theorem characterizes all simple Artinian rings as the matrix rings over a division ring. This implies that a simple ring is left Artinian if and only if it is right Artinian.Although the descending chain condition appears dual to the ascending chain condition, in rings it is in fact the stronger condition. Specifically, a consequence of the Akizuki–Hopkins–Levitzki theorem is that a left (resp. right) Artinian ring is automatically a left (resp. right) Noetherian ring. This is not true for general modules, that is, an Artinian module need not be a Noetherian module.".
- Artinian_ring wikiPageID "489273".
- Artinian_ring wikiPageLength "6946".
- Artinian_ring wikiPageOutDegree "36".
- Artinian_ring wikiPageRevisionID "667080127".
- Artinian_ring wikiPageWikiLink Abstract_algebra.
- Artinian_ring wikiPageWikiLink Artin_algebra.
- Artinian_ring wikiPageWikiLink Artinian_ideal.
- Artinian_ring wikiPageWikiLink Artinian_module.
- Artinian_ring wikiPageWikiLink Artin–Wedderburn_theorem.
- Artinian_ring wikiPageWikiLink Ascending_chain_condition.
- Artinian_ring wikiPageWikiLink Cambridge_University_Press.
- Artinian_ring wikiPageWikiLink Category:Ring_theory.
- Artinian_ring wikiPageWikiLink Commutative_ring.
- Artinian_ring wikiPageWikiLink Composition_series.
- Artinian_ring wikiPageWikiLink Dedekind_domain.
- Artinian_ring wikiPageWikiLink Descending_chain_condition.
- Artinian_ring wikiPageWikiLink Division_ring.
- Artinian_ring wikiPageWikiLink Emil_Artin.
- Artinian_ring wikiPageWikiLink Field_(mathematics).
- Artinian_ring wikiPageWikiLink Finite_length.
- Artinian_ring wikiPageWikiLink Finite_ring.
- Artinian_ring wikiPageWikiLink Hopkins_theorem.
- Artinian_ring wikiPageWikiLink Hopkins–Levitzki_theorem.
- Artinian_ring wikiPageWikiLink Ideal_(ring_theory).
- Artinian_ring wikiPageWikiLink Krull_dimension.
- Artinian_ring wikiPageWikiLink Length_of_a_module.
- Artinian_ring wikiPageWikiLink Matrix_ring.
- Artinian_ring wikiPageWikiLink Minimum_condition.
- Artinian_ring wikiPageWikiLink Module_(mathematics).
- Artinian_ring wikiPageWikiLink Nilradical_of_a_ring.
- Artinian_ring wikiPageWikiLink Noetherian_module.
- Artinian_ring wikiPageWikiLink Noetherian_ring.
- Artinian_ring wikiPageWikiLink Perfect_ring.
- Artinian_ring wikiPageWikiLink Principal_ideal_ring.
- Artinian_ring wikiPageWikiLink Ring_(mathematics).
- Artinian_ring wikiPageWikiLink Semiperfect_ring.
- Artinian_ring wikiPageWikiLink Semisimple_module.
- Artinian_ring wikiPageWikiLink Semisimple_ring.
- Artinian_ring wikiPageWikiLink Serial_module.
- Artinian_ring wikiPageWikiLink Simple_ring.
- Artinian_ring wikiPageWikiLink Vector_space.
- Artinian_ring wikiPageWikiLinkText "Artinian ''ring''".
- Artinian_ring wikiPageWikiLinkText "Artinian ring".
- Artinian_ring wikiPageWikiLinkText "Artinian".
- Artinian_ring wikiPageWikiLinkText "artinian ring".
- Artinian_ring wikiPageWikiLinkText "artinian".
- Artinian_ring wikiPageWikiLinkText "right Artinian rings".
- Artinian_ring wikiPageWikiLinkText "right artinian ring".
- Artinian_ring hasPhotoCollection Artinian_ring.
- Artinian_ring wikiPageUsesTemplate Template:Citation.
- Artinian_ring wikiPageUsesTemplate Template:Citation_needed.
- Artinian_ring wikiPageUsesTemplate Template:Cite_book.
- Artinian_ring subject Category:Ring_theory.
- Artinian_ring hypernym Ring.
- Artinian_ring type AnatomicalStructure.
- Artinian_ring type Article.
- Artinian_ring type Article.
- Artinian_ring comment "In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are finite-dimensional vector spaces over fields.".
- Artinian_ring label "Artinian ring".
- Artinian_ring sameAs Artinovský_okruh.
- Artinian_ring sameAs Artinscher_Modul.
- Artinian_ring sameAs Anillo_artiniano.
- Artinian_ring sameAs Anneau_artinien.
- Artinian_ring sameAs חוג_ארטיני.
- Artinian_ring sameAs Anello_artiniano.
- Artinian_ring sameAs アルティン環.
- Artinian_ring sameAs 아르틴_환.
- Artinian_ring sameAs Artiniaanse_ring.
- Artinian_ring sameAs Mòdol_artinian.
- Artinian_ring sameAs Anel_artiniano.
- Artinian_ring sameAs m.02gmfk.
- Artinian_ring sameAs Артиново_кольцо.
- Artinian_ring sameAs Кільце_Артіна.
- Artinian_ring sameAs Q713084.
- Artinian_ring sameAs Q713084.
- Artinian_ring sameAs 阿廷環.
- Artinian_ring wasDerivedFrom Artinian_ring?oldid=667080127.
- Artinian_ring isPrimaryTopicOf Artinian_ring.