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]
-
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
- 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
ⓘ
Zariski–Samuel: Commutative Algebra ⓘ 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 ⓘ |
| 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
ⓘ
Emmy Noether ⓘ |
| provedBy | Emanuel Lasker ⓘ |
| relatedTo |
Krull’s principal ideal theorem
ⓘ
Noether normalization lemma ⓘ |
| relatesConcept |
Noetherian condition
ⓘ
linked to:
Noetherian rings
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
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
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 linked to:
Noetherian rings