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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All labels observed (7)

Label	Occurrences
Wedderburn–Artin theory	3
Wedderburn’s theorem	3
Artin–Wedderburn theorem canonical	2
Wedderburn decomposition	1
Wedderburn–Artin theorem	1
Wedderburn’s structure theorem for semisimple rings	1
the Wedderburn–Artin structure theorem	1

How this entity was disambiguated

This entity first appeared as the object of triple T5658020 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Artin–Wedderburn theorem
Context triple: [Emil Artin, notableWork, Artin–Wedderburn theorem]

A. Jordan–Hölder theorem
The Jordan–Hölder theorem is a fundamental result in group theory stating that any two composition series of a finite group have the same length and the same (up to order and isomorphism) simple factor groups.
B. Noether's isomorphism theorems
Noether's isomorphism theorems are fundamental results in abstract algebra that relate quotient structures and substructures of groups, rings, and modules, providing a unifying framework for understanding homomorphic images and factor structures.
C. Gelfand–Naimark theorem
The Gelfand–Naimark theorem is a foundational result in functional analysis that characterizes C*-algebras as algebras of bounded operators on a Hilbert space (and, in the commutative case, as algebras of continuous functions on a locally compact Hausdorff space).
D. Hilbert’s syzygy theorem
Hilbert’s syzygy theorem is a fundamental result in commutative algebra that describes the finite length and structure of free resolutions of modules over polynomial rings.
E. Kronecker–Weber theorem
The Kronecker–Weber theorem is a fundamental result in algebraic number theory stating that every finite abelian extension of the rational numbers is contained in a cyclotomic field generated by roots of unity.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Artin–Wedderburn theorem
Target entity description: 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.

A. Jordan–Hölder theorem
The Jordan–Hölder theorem is a fundamental result in group theory stating that any two composition series of a finite group have the same length and the same (up to order and isomorphism) simple factor groups.
B. Noether's isomorphism theorems
Noether's isomorphism theorems are fundamental results in abstract algebra that relate quotient structures and substructures of groups, rings, and modules, providing a unifying framework for understanding homomorphic images and factor structures.
C. Gelfand–Naimark theorem
The Gelfand–Naimark theorem is a foundational result in functional analysis that characterizes C*-algebras as algebras of bounded operators on a Hilbert space (and, in the commutative case, as algebras of continuous functions on a locally compact Hausdorff space).
D. Hilbert’s syzygy theorem
Hilbert’s syzygy theorem is a fundamental result in commutative algebra that describes the finite length and structure of free resolutions of modules over polynomial rings.
E. Kronecker–Weber theorem
The Kronecker–Weber theorem is a fundamental result in algebraic number theory stating that every finite abelian extension of the rational numbers is contained in a cyclotomic field generated by roots of unity.
F. None of above. chosen

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 ⓘ Wedderburn’s structure theorem for semisimple rings ⓘ linked to: Artin–Wedderburn theorem
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 ⓘ Joseph Wedderburn ⓘ
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 ⓘ Jacobson radical ⓘ semisimple module ⓘ simple module ⓘ

How these facts were elicited