Jacobson radical
E627993
The Jacobson radical is an ideal of a ring that captures elements annihilating all simple modules, playing a key role in understanding the ring’s structure and its representations.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Jacobson radical canonical | 3 |
| Jacobson semisimple ring | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6908955 — 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: Jacobson radical Context triple: [ring theory, usesConcept, Jacobson radical]
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A.
Noetherian rings
Noetherian rings are a fundamental class of rings in commutative algebra characterized by the property that every ascending chain of ideals stabilizes, ensuring that all ideals are finitely generated.
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B.
Artinian ring
An Artinian ring is a ring in which every descending chain of ideals eventually stabilizes, making it a fundamental object in commutative algebra and ring theory with strong finiteness properties.
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C.
Krull’s principal ideal theorem
Krull’s principal ideal theorem is a fundamental result in commutative algebra that relates the height of prime ideals containing a principal ideal to the Krull dimension of the ring.
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D.
ring theory
Ring theory is a branch of abstract algebra that studies rings—algebraic structures equipped with two binary operations generalizing addition and multiplication of integers—and their ideals, modules, and homomorphisms.
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E.
Artin–Wedderburn theorem
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.
- 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: Jacobson radical Target entity description: The Jacobson radical is an ideal of a ring that captures elements annihilating all simple modules, playing a key role in understanding the ring’s structure and its representations.
-
A.
Noetherian rings
Noetherian rings are a fundamental class of rings in commutative algebra characterized by the property that every ascending chain of ideals stabilizes, ensuring that all ideals are finitely generated.
-
B.
Artinian ring
An Artinian ring is a ring in which every descending chain of ideals eventually stabilizes, making it a fundamental object in commutative algebra and ring theory with strong finiteness properties.
-
C.
Krull’s principal ideal theorem
Krull’s principal ideal theorem is a fundamental result in commutative algebra that relates the height of prime ideals containing a principal ideal to the Krull dimension of the ring.
-
D.
ring theory
Ring theory is a branch of abstract algebra that studies rings—algebraic structures equipped with two binary operations generalizing addition and multiplication of integers—and their ideals, modules, and homomorphisms.
-
E.
Artin–Wedderburn theorem
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.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
concept in abstract algebra
ⓘ
concept in ring theory ⓘ ideal of a ring ⓘ |
| alsoKnownAs | Jacobson ideal ⓘ |
| appearsIn |
Wedderburn decomposition R ≅ S ⊕ J(R) for semiperfect rings
ⓘ
decomposition of Artinian rings into semisimple part and radical ⓘ |
| characterizedAs |
intersection of all maximal left ideals of R
ⓘ
intersection of all maximal right ideals of R ⓘ set of elements of R that annihilate all simple left R-modules ⓘ set of elements of R that annihilate all simple right R-modules ⓘ set of elements x such that 1-rx is left invertible for all r in R ⓘ set of elements x such that 1-xr is right invertible for all r in R ⓘ |
| conditionFor |
R/J(R) being semisimple Artinian when R is left Artinian
ⓘ
R/J(R) being semisimple when R is semiprimary ⓘ |
| containedIn |
nilradical for commutative rings
ⓘ
prime radical for Artinian rings ⓘ |
| contains | all nilpotent ideals in left Artinian rings ⓘ |
| definedFor | associative ring with identity ⓘ |
| equals |
0 for Jacobson semisimple rings
ⓘ
nilradical for Artinian commutative rings ⓘ nilradical for Noetherian commutative rings ⓘ |
| functoriality | homomorphic image of J(R) is contained in J(S) for ring homomorphism R→S ⓘ |
| generalizationOf | Jacobson radical of modules ⓘ |
| introducedIn | 20th century ring theory ⓘ |
| invariantUnder | ring isomorphisms ⓘ |
| isIdealOf | ring R ⓘ |
| namedAfter | Nathan Jacobson ⓘ |
| property |
idempotent-lifting in semiperfect rings
ⓘ
largest quasi-regular ideal of R ⓘ nil ideal in Artinian rings ⓘ two-sided ideal ⓘ |
| relatedTo |
Jacobson semisimple ring
ⓘ
linked to:
Jacobson radical
nilradical ⓘ prime radical ⓘ |
| studiedIn | graduate algebra textbooks ⓘ |
| subsetOf | R ⓘ |
| symbol |
J(R)
ⓘ
rad(J) ⓘ |
| topologicalAnalogue | Jacobson radical of a Banach algebra ⓘ |
| usedIn |
Wedderburn–Artin theory
ⓘ
linked to:
Artin–Wedderburn theorem
algebraic geometry over noncommutative rings ⓘ module theory ⓘ noncommutative algebra ⓘ representation theory of rings ⓘ structure theory of rings ⓘ theory of local rings ⓘ theory of semiperfect rings ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Jacobson radical Description of subject: The Jacobson radical is an ideal of a ring that captures elements annihilating all simple modules, playing a key role in understanding the ring’s structure and its representations.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Jacobson radical