Commutative algebra Chain conditions Concept
Artinian ring
Summary
An Artinian ring is a ring in which every decreasing chain of ideals stops after finitely many steps. For commutative rings, Artinian rings are exactly the Noetherian rings in which every prime ideal is maximal, by the Akizuki–Hopkins–Levitzki theorem. By the structure theorem for Artinian rings, a commutative Artinian ring is a finite product of Artinian local rings. Finite rings, fields and finite-dimensional algebras over a field are Artinian.
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| Tag | 20QEAM8AX2 |
|---|---|
| URI | m2://20QEAM8AX2 |
| Kind | concept |
| Writing | W0 stub |
| MSC | 13E10, 16P20 |
| Specializes | Commutative ring |
| Same as |
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Results about this
- Artinian reduced rings are semisimple
- Akizuki–Hopkins–Levitzki theorem
- Jacobson radical of an Artinian ring is nilpotent
- Structure theorem for Artinian rings
- Hopkins–Levitzki theorem
Used by
- Artinian reduced rings are semisimple
- Goldie's theorem
- Semisimple ring
- Hopkins–Levitzki theorem
- Descending chain condition
- Wedderburn's theorem on simple Artinian rings