Artin–Wedderburn theorem
E537779
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Artin–Wedderburn theorem canonical | 2 |
| Wedderburn–Artin theory | 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
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
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Input
Subject: Artin–Wedderburn theorem Description of subject: 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.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Wedderburn–Artin theory