Lasker–Noether theorem on primary decomposition
E621103
The Lasker–Noether theorem on primary decomposition is a fundamental result in commutative algebra stating that every ideal in a Noetherian ring can be expressed as a finite intersection of primary ideals, generalizing the factorization of integers into prime powers.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Lasker–Noether theorem on primary decomposition canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6833762 — 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: Lasker–Noether theorem on primary decomposition Context triple: [Noetherian ring, hasTheorem, Lasker–Noether theorem on primary decomposition]
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A.
Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
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B.
Hilbert’s syzygy theorem
Hilbert’s syzygy theorem is a fundamental result in commutative algebra that describes the finite length and structure of free resolutions of modules over polynomial rings.
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C.
Hilbert’s fourteenth problem
Hilbert’s fourteenth problem is one of David Hilbert’s famous list of 23 problems, concerning the finite generation of certain algebras of invariants in algebraic geometry and invariant theory.
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D.
Hilbert basis theorem
The Hilbert basis theorem is a fundamental result in commutative algebra stating that if a ring is Noetherian then any polynomial ring over it is also Noetherian, ensuring that ideals in such rings are finitely generated.
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E.
Castelnuovo–Mumford regularity
Castelnuovo–Mumford regularity is an invariant in commutative algebra and algebraic geometry that measures the complexity of the minimal graded free resolution of a module or sheaf, often used to control vanishing of cohomology and bounds on generators.
- 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: Lasker–Noether theorem on primary decomposition Target entity description: The Lasker–Noether theorem on primary decomposition is a fundamental result in commutative algebra stating that every ideal in a Noetherian ring can be expressed as a finite intersection of primary ideals, generalizing the factorization of integers into prime powers.
-
A.
Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
-
B.
Hilbert’s syzygy theorem
Hilbert’s syzygy theorem is a fundamental result in commutative algebra that describes the finite length and structure of free resolutions of modules over polynomial rings.
-
C.
Hilbert’s fourteenth problem
Hilbert’s fourteenth problem is one of David Hilbert’s famous list of 23 problems, concerning the finite generation of certain algebras of invariants in algebraic geometry and invariant theory.
-
D.
Hilbert basis theorem
The Hilbert basis theorem is a fundamental result in commutative algebra stating that if a ring is Noetherian then any polynomial ring over it is also Noetherian, ensuring that ideals in such rings are finitely generated.
-
E.
Castelnuovo–Mumford regularity
Castelnuovo–Mumford regularity is an invariant in commutative algebra and algebraic geometry that measures the complexity of the minimal graded free resolution of a module or sheaf, often used to control vanishing of cohomology and bounds on generators.
- F. None of above. chosen
Statements (42)
| Predicate | Object |
|---|---|
| instanceOf | mathematical theorem ⓘ |
| appearsIn |
Atiyah–Macdonald: Introduction to Commutative Algebra
NERFINISHED
ⓘ
Zariski–Samuel: Commutative Algebra NERFINISHED ⓘ standard textbooks on commutative algebra ⓘ |
| appliesTo | Noetherian ring ⓘ |
| assumes | ring is Noetherian ⓘ |
| concerns |
ideal decomposition
ⓘ
ideals in commutative rings ⓘ primary ideals ⓘ prime ideals ⓘ |
| domain | ring theory ⓘ |
| feature |
decomposition is finite
ⓘ
primary components correspond to prime ideals ⓘ primary decomposition is not necessarily unique ⓘ set of associated primes is uniquely determined ⓘ |
| field | commutative algebra ⓘ |
| formalizes | decomposition of ideals analogous to factorization of numbers ⓘ |
| generalizedBy | Emmy Noether NERFINISHED ⓘ |
| generalizes | unique factorization of integers into prime powers ⓘ |
| historicalContext | early 20th century development of commutative algebra ⓘ |
| implies |
associated primes of an ideal in a Noetherian ring are finite
ⓘ
every ideal in a Noetherian ring has a primary decomposition ⓘ |
| involves |
Noetherian modules (in module-theoretic versions)
ⓘ
intersection of primary ideals ⓘ radical of an ideal ⓘ |
| namedAfter |
Emanuel Lasker
NERFINISHED
ⓘ
Emmy Noether NERFINISHED ⓘ |
| provedBy | Emanuel Lasker NERFINISHED ⓘ |
| relatedTo |
Krull’s principal ideal theorem
NERFINISHED
ⓘ
Noether normalization lemma NERFINISHED ⓘ |
| relatesConcept |
Noetherian condition
NERFINISHED
ⓘ
associated prime ideals ⓘ minimal prime ideals ⓘ primary decomposition of ideals ⓘ |
| requires | ascending chain condition on ideals ⓘ |
| statement | Every ideal in a Noetherian ring can be expressed as a finite intersection of primary ideals. ⓘ |
| toolFor |
analyzing structure of ideals
ⓘ
computing associated primes ⓘ studying local properties of schemes ⓘ |
| usedIn |
algebraic geometry
ⓘ
decomposition of algebraic sets into irreducible components ⓘ structure theory of modules over Noetherian rings ⓘ |
How these facts were elicited
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Input
Subject: Lasker–Noether theorem on primary decomposition Description of subject: The Lasker–Noether theorem on primary decomposition is a fundamental result in commutative algebra stating that every ideal in a Noetherian ring can be expressed as a finite intersection of primary ideals, generalizing the factorization of integers into prime powers.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.
subject surface form:
Noetherian ring