Artinian ring
E621106
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Artinian ring canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T6833837 — 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: Artinian ring Context triple: [Artinian module, generalizes, Artinian ring]
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A.
Artinian module
An Artinian module is an algebraic structure over a ring that satisfies the descending chain condition on its submodules, meaning every decreasing sequence of submodules eventually stabilizes.
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B.
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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C.
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.
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D.
Noetherian module
A Noetherian module is an algebraic structure in which every ascending chain of submodules stabilizes, ensuring that all submodules are finitely generated and enabling powerful finiteness arguments in ring and module theory.
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E.
Henselian ring
A Henselian ring is a local ring in which Hensel’s lemma holds, allowing certain types of polynomial factorizations and root liftings from the residue field to the ring itself.
- 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: Artinian ring Target entity description: 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.
-
A.
Artinian module
An Artinian module is an algebraic structure over a ring that satisfies the descending chain condition on its submodules, meaning every decreasing sequence of submodules eventually stabilizes.
-
B.
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.
-
C.
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.
-
D.
Noetherian module
A Noetherian module is an algebraic structure in which every ascending chain of submodules stabilizes, ensuring that all submodules are finitely generated and enabling powerful finiteness arguments in ring and module theory.
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E.
Henselian ring
A Henselian ring is a local ring in which Hensel’s lemma holds, allowing certain types of polynomial factorizations and root liftings from the residue field to the ring itself.
- F. None of above. chosen
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic structure property
ⓘ
ring-theoretic concept ⓘ |
| abbreviation | DCC on ideals ⓘ |
| appearsIn |
the Wedderburn–Artin structure theorem
ⓘ
linked to:
Artin–Wedderburn theorem
|
| definition | a ring in which every descending chain of ideals stabilizes ⓘ |
| equivalentCondition |
commutative Artinian ring is Noetherian and every prime ideal is maximal
ⓘ
commutative Artinian ring is Noetherian and has finitely many prime ideals ⓘ commutative Artinian ring is Noetherian with dimension zero ⓘ left Artinian and right Artinian are equivalent for semiprimary rings ⓘ |
| example |
any field is an Artinian ring
ⓘ
any finite ring is Artinian ⓘ k[x]/(x^n) over a field k is Artinian ⓘ the ring of dual numbers k[ε]/(ε²) over a field k is Artinian ⓘ |
| field |
commutative algebra
ⓘ
ring theory ⓘ |
| hasProperty |
Jacobson radical has nilpotent elements whose powers eventually vanish
ⓘ
Jacobson radical is nilpotent in a commutative Artinian ring ⓘ Krull dimension zero for commutative Artinian rings ⓘ every Artinian semiprime ring is semisimple ⓘ every descending chain of left ideals stabilizes in a left Artinian ring ⓘ every descending chain of right ideals stabilizes in a right Artinian ring ⓘ every ideal is an intersection of primary ideals in a commutative Artinian ring ⓘ every ideal is finitely generated in a commutative Artinian ring ⓘ every module of finite length over a ring has an Artinian endomorphism ring under suitable conditions ⓘ every nonempty set of ideals has a minimal element ⓘ every prime ideal is maximal in a commutative Artinian ring ⓘ every simple Artinian ring is isomorphic to a matrix ring over a division ring ⓘ finite length as a module over itself ⓘ has finite length as a module over itself in the commutative case ⓘ nilradical equals Jacobson radical in a commutative Artinian ring ⓘ only finitely many maximal ideals in a commutative Artinian ring ⓘ satisfies both ACC and DCC on ideals in the commutative case ⓘ satisfies descending chain condition on ideals ⓘ |
| implies |
Noetherian ring for commutative rings with identity
ⓘ
semiprimary ring in the commutative case ⓘ |
| namedAfter | Emil Artin ⓘ |
| nonExample |
the polynomial ring k[x] over a field k is not Artinian
ⓘ
the ring of integers ℤ is not Artinian ⓘ |
| relatedTo |
Artinian module
ⓘ
Noetherian ring ⓘ semisimple ring ⓘ |
| structureTheorem |
every Artinian principal ideal ring is a finite direct product of Artinian principal ideal local rings
ⓘ
every commutative Artinian ring is a finite direct product of Artinian local rings ⓘ |
| topicOf | research in noncommutative ring theory and representation theory ⓘ |
| usedIn |
Wedderburn–Artin theory
ⓘ
linked to:
Artin–Wedderburn theorem
classification of semisimple modules ⓘ |
| variant |
left Artinian ring
ⓘ
right Artinian ring ⓘ two-sided Artinian ring ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Artinian ring Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.