Statements (28)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
|
| gptkbp:category |
gptkb:commutative_algebra
noncommutative ring theory |
| gptkbp:characterization |
A commutative Artinian ring is a finite product of local Artinian rings
|
| gptkbp:citation |
gptkb:Atiyah_&_Macdonald,_Introduction_to_Commutative_Algebra
gptkb:Jacobson,_Structure_of_Rings |
| gptkbp:contrastsWith |
gptkb:Noetherian_rings
|
| gptkbp:defines |
A ring in which the descending chain condition on ideals holds
|
| gptkbp:example |
finite rings
matrix rings over division rings |
| gptkbp:field |
abstract algebra
|
| https://www.w3.org/2000/01/rdf-schema#label |
Artinian rings
|
| gptkbp:introducedIn |
20th century
|
| gptkbp:isA |
gptkb:King
|
| gptkbp:namedAfter |
gptkb:Emil_Artin
|
| gptkbp:property |
Every simple module over an Artinian ring is finite-dimensional
Every Artinian ring has finite length as a module over itself Every descending chain of ideals terminates Every ideal is finitely generated Every prime ideal is maximal Has only finitely many maximal ideals Jacobson radical is nilpotent Every Artinian ring is Noetherian if it is also commutative Every module over an Artinian ring is of finite length |
| gptkbp:subclassOf |
rings with minimum condition on ideals
|
| gptkbp:theory |
Artin–Wedderburn theorem applies to semisimple Artinian rings
|
| gptkbp:bfsParent |
gptkb:Noetherian_rings
|
| gptkbp:bfsLayer |
5
|