Artin–Wedderburn theorem

E537779

theorem theorem in ring theory

The Artin–Wedderburn theorem is a fundamental result in ring theory that classifies all semisimple rings as finite direct products of matrix rings over division rings.

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Observed surface forms (1)

Surface form	Occurrences
Wedderburn–Artin theory	1

Statements (43)

Predicate	Object
instanceOf	theorem ⓘ theorem in ring theory ⓘ
appliesTo	semisimple rings ⓘ
assumes	ring is semisimple ⓘ
characterizes	semisimple rings ⓘ
concerns	division rings ⓘ matrix rings ⓘ semisimple modules ⓘ semisimple rings ⓘ simple rings ⓘ
equivalentCondition	ring is semisimple if and only if it is isomorphic to a finite direct product of matrix rings over division rings ⓘ ring is semisimple if and only if its Jacobson radical is zero and it is Artinian ⓘ
equivalentConditionFor	ring being semisimple ⓘ
field	abstract algebra ⓘ ring theory ⓘ
givesClassificationOf	semisimple rings ⓘ
hasComponentResult	Artin’s refinement for Artinian rings NERFINISHED ⓘ Wedderburn’s structure theorem for semisimple rings NERFINISHED ⓘ
hasConsequence	classification of finite-dimensional semisimple algebras over a field ⓘ finite-dimensional semisimple algebras over a field are finite direct products of matrix algebras over division algebras ⓘ
hasSpecialCase	finite-dimensional semisimple algebras over an algebraically closed field are finite direct products of full matrix algebras over that field ⓘ group algebras of finite groups over fields of characteristic zero decompose as finite direct products of matrix algebras over division rings ⓘ
historicalPeriod	early 20th century ⓘ
implies	decomposition of semisimple rings into simple components ⓘ structure theorem for semisimple rings ⓘ
isUsedIn	algebraic number theory ⓘ module theory ⓘ representation theory of finite groups ⓘ theory of central simple algebras ⓘ theory of semisimple Lie algebras ⓘ
namedAfter	Emil Artin NERFINISHED ⓘ Joseph Wedderburn NERFINISHED ⓘ
relatesConcept	semisimple rings and direct products of simple rings ⓘ simple Artinian rings and matrix rings over division rings ⓘ
statesThat	every semisimple Artinian ring is isomorphic to a finite direct product of simple Artinian rings ⓘ every semisimple ring is isomorphic to a finite direct product of matrix rings over division rings ⓘ every simple Artinian ring is isomorphic to a matrix ring over a division ring ⓘ
typicalFormulation	R is semisimple Artinian if and only if R is isomorphic to a finite direct product of matrix algebras over division rings ⓘ every semisimple module is a direct sum of simple modules and endomorphism rings of semisimple modules decompose accordingly ⓘ
usesConcept	Artinian ring NERFINISHED ⓘ Jacobson radical NERFINISHED ⓘ semisimple module ⓘ simple module ⓘ

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Emil Artin → notableWork → Artin–Wedderburn theorem ⓘ

Artinian module → appearsIn → Artin–Wedderburn theorem ⓘ

this entity surface form: Wedderburn–Artin theory

ring theory → usesConcept → Artin–Wedderburn theorem ⓘ