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- Artin–Wedderburn_theorem abstract "In abstract algebra, the Artin–Wedderburn theorem is a classification theorem for semisimple rings and semisimple algebras. The theorem states that an (Artinian) semisimple ring R is isomorphic to a product of finitely many ni-by-ni matrix rings over division rings Di, for some integers ni, both of which are uniquely determined up to permutation of the index i. In particular, any simple left or right Artinian ring is isomorphic to an n-by-n matrix ring over a division ring D, where both n and D are uniquely determined.As a direct corollary, the Artin–Wedderburn theorem implies that every simple ring that is finite-dimensional over a division ring (a simple algebra) is a matrix ring. This is Joseph Wedderburn's original result. Emil Artin later generalized it to the case of Artinian rings.Note that if R is a finite-dimensional simple algebra over a division ring E, D need not be contained in E. For example, matrix rings over the complex numbers are finite-dimensional simple algebras over the real numbers.The Artin–Wedderburn theorem reduces classifying simple rings over a division ring to classifying division rings that contain a given division ring. This in turn can be simplified: The center of D must be a field K. Therefore R is a K-algebra, and itself has K as its center. A finite-dimensional simple algebra R is thus a central simple algebra over K. Thus the Artin–Wedderburn theorem reduces the problem of classifying finite-dimensional central simple algebras to the problem of classifying division rings with given center.".
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- Artin–Wedderburn_theorem wikiPageRevisionID "617890925".
- Artin–Wedderburn_theorem wikiPageWikiLink Abstract_algebra.
- Artin–Wedderburn_theorem wikiPageWikiLink Algebra.
- Artin–Wedderburn_theorem wikiPageWikiLink Artinian_ring.
- Artin–Wedderburn_theorem wikiPageWikiLink Brauer_group.
- Artin–Wedderburn_theorem wikiPageWikiLink Category:Ring_theory.
- Artin–Wedderburn_theorem wikiPageWikiLink Category:Theorems_in_abstract_algebra.
- Artin–Wedderburn_theorem wikiPageWikiLink Center_(algebra).
- Artin–Wedderburn_theorem wikiPageWikiLink Central_simple_algebra.
- Artin–Wedderburn_theorem wikiPageWikiLink Classification_theorem.
- Artin–Wedderburn_theorem wikiPageWikiLink Complex_number.
- Artin–Wedderburn_theorem wikiPageWikiLink Division_algebra.
- Artin–Wedderburn_theorem wikiPageWikiLink Division_ring.
- Artin–Wedderburn_theorem wikiPageWikiLink Emil_Artin.
- Artin–Wedderburn_theorem wikiPageWikiLink Field_(mathematics).
- Artin–Wedderburn_theorem wikiPageWikiLink Finite_field.
- Artin–Wedderburn_theorem wikiPageWikiLink Frobenius_theorem_(real_division_algebras).
- Artin–Wedderburn_theorem wikiPageWikiLink Hypercomplex_number.
- Artin–Wedderburn_theorem wikiPageWikiLink Jacobson_density_theorem.
- Artin–Wedderburn_theorem wikiPageWikiLink Joseph_Wedderburn.
- Artin–Wedderburn_theorem wikiPageWikiLink Maschkes_theorem.
- Artin–Wedderburn_theorem wikiPageWikiLink Matrix_ring.
- Artin–Wedderburn_theorem wikiPageWikiLink P._M._Cohn.
- Artin–Wedderburn_theorem wikiPageWikiLink Paul_Cohn.
- Artin–Wedderburn_theorem wikiPageWikiLink Product_of_rings.
- Artin–Wedderburn_theorem wikiPageWikiLink Quaternion.
- Artin–Wedderburn_theorem wikiPageWikiLink Real_number.
- Artin–Wedderburn_theorem wikiPageWikiLink Semisimple_algebra.
- Artin–Wedderburn_theorem wikiPageWikiLink Semisimple_module.
- Artin–Wedderburn_theorem wikiPageWikiLink Semisimple_ring.
- Artin–Wedderburn_theorem wikiPageWikiLink Simple_ring.
- Artin–Wedderburn_theorem wikiPageWikiLinkText "Artin–Wedderburn theorem".
- Artin–Wedderburn_theorem wikiPageWikiLinkText "Artin–Wedderburn theory".
- Artin–Wedderburn_theorem wikiPageWikiLinkText "Artin–Wedderburn".
- Artin–Wedderburn_theorem wikiPageWikiLinkText "Wedderburn's theorem".
- Artin–Wedderburn_theorem hasPhotoCollection Artin–Wedderburn_theorem.
- Artin–Wedderburn_theorem wikiPageUsesTemplate Template:Cite_journal.
- Artin–Wedderburn_theorem wikiPageUsesTemplate Template:Reflist.
- Artin–Wedderburn_theorem subject Category:Ring_theory.
- Artin–Wedderburn_theorem subject Category:Theorems_in_abstract_algebra.
- Artin–Wedderburn_theorem comment "In abstract algebra, the Artin–Wedderburn theorem is a classification theorem for semisimple rings and semisimple algebras. The theorem states that an (Artinian) semisimple ring R is isomorphic to a product of finitely many ni-by-ni matrix rings over division rings Di, for some integers ni, both of which are uniquely determined up to permutation of the index i.".
- Artin–Wedderburn_theorem label "Artin–Wedderburn theorem".
- Artin–Wedderburn_theorem sameAs Teorema_de_Artin-Wedderburn.
- Artin–Wedderburn_theorem sameAs Thxc3xa9orxc3xa8me_dArtin-Wedderburn.
- Artin–Wedderburn_theorem sameAs משפט_ודרברן-ארטין.
- Artin–Wedderburn_theorem sameAs Wedderburn–Artin-tétel.
- Artin–Wedderburn_theorem sameAs Teorema_di_Artin-Wedderburn.
- Artin–Wedderburn_theorem sameAs アルティン・ウェダーバーンの定理.
- Artin–Wedderburn_theorem sameAs Stelling_van_Artin-Wedderburn.
- Artin–Wedderburn_theorem sameAs Teorema_de_Artin-Wedderburn.
- Artin–Wedderburn_theorem sameAs m.023f_p.
- Artin–Wedderburn_theorem sameAs Q776578.
- Artin–Wedderburn_theorem sameAs Q776578.
- Artin–Wedderburn_theorem wasDerivedFrom Artin–Wedderburn_theorem?oldid=617890925.
- Artin–Wedderburn_theorem isPrimaryTopicOf Artin–Wedderburn_theorem.