Commutative algebra Basic notions Concept
Jacobson radical
Summary
The Jacobson radical of a commutative ring is the intersection of all its maximal ideals. An element lies in it exactly when is a unit for every in the ring. It contains the nilradical, and it is the ideal that appears in Nakayama's lemma. For a local ring it is the unique maximal ideal.
Where this leads
25 pages build on this one.
- Nakayama's lemma
- Regular local rings are UFDs
- Krull intersection theorem
- Jacobson ring
- Semiperfect ring
- Finitely generated projective modules over local rings are free
- Cartan matrix of an algebra
- Artinian reduced rings are semisimple
- Determinant trick
- Kaplansky's theorem on projective modules
- Perfect ring
- Bass's theorem P
- Jacobson radical of an Artinian ring is nilpotent
- Semisimple ring
- Completion is flat
- Semi-local ring
- Levitzki's theorem
- Lenagan's theorem
- Semiprimary ring
- Semiprimitive ring
- Radical of a module
- Regular local rings are domains
- Amitsur's theorem on the radical of polynomial rings
- Hopkins–Levitzki theorem
- Semilocal ring
Further reading
About this page
| Tag | XA0DCNH7WY |
|---|---|
| URI | m2://XA0DCNH7WY |
| Kind | concept |
| Writing | W0 stub |
| MSC | 13A15, 16N20 |
| Same as |
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What links here
Results about this
- Nakayama's lemma
- Jacobson's conjecture
- Jacobson radical of an Artinian ring is nilpotent
- Amitsur's theorem on the radical of polynomial rings
Used by
- Nakayama's lemma
- Krull intersection theorem
- Jacobson ring
- Semiperfect ring
- Cartan matrix of an algebra
- Artinian reduced rings are semisimple
- Determinant trick
- Kaplansky's theorem on projective modules
- Perfect ring
- Bass's theorem P
- Jacobson radical of an Artinian ring is nilpotent
- Semisimple ring
- Completion is flat
- Semi-local ring
- Semiprimary ring
- Semiprimitive ring
- Radical of a module
- Amitsur's theorem on the radical of polynomial rings
- Hopkins–Levitzki theorem
- Semilocal ring