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.
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 NERFINISHED ⓘ |
| property |
idempotent-lifting in semiperfect rings
ⓘ
largest quasi-regular ideal of R ⓘ nil ideal in Artinian rings ⓘ two-sided ideal ⓘ |
| relatedTo |
Jacobson semisimple ring
NERFINISHED
ⓘ
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
NERFINISHED
ⓘ
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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.