Noetherian induction
E29920
Noetherian induction is a proof technique used in mathematics to establish statements about structures satisfying the descending chain condition, generalizing ordinary mathematical induction.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| Noetherian recursion (in some contexts) | 1 |
Statements (43)
| Predicate | Object |
|---|---|
| instanceOf |
mathematical method
ⓘ
proof technique ⓘ |
| alsoKnownAs |
Noetherian induction
ⓘ
surface form:
Noetherian recursion (in some contexts)
induction on Noetherian sets ⓘ |
| appliesTo |
Noetherian poset
ⓘ
Noetherian rings ⓘ
surface form:
Noetherian ring
Noetherian space ⓘ |
| assumes | every descending chain stabilizes ⓘ |
| basedOn | well-founded induction ⓘ |
| characterizedBy | use of descending chain condition ⓘ |
| contrastWith |
ordinary induction on natural numbers
ⓘ
transfinite induction ⓘ |
| equivalentTo |
induction on well-founded relations
ⓘ
proof by minimal counterexample on Noetherian structures ⓘ |
| field | mathematics ⓘ |
| formalizes | induction over finitely generated substructures in Noetherian settings ⓘ |
| generalizes | mathematical induction ⓘ |
| historicalContext | developed in the context of Noetherian rings and modules ⓘ |
| implies | no infinite strictly descending sequence exists ⓘ |
| logicalForm | second-order principle expressible in first-order theories with well-founded relations ⓘ |
| metaProperty | sound in any theory with a well-founded relation ⓘ |
| namedAfter | Emmy Noether ⓘ |
| oftenFormulatedOn |
partially ordered sets
ⓘ
well-founded relations ⓘ |
| relatedConcept |
Noetherian module
ⓘ
Noetherian topological space ⓘ descending chain condition ⓘ well-founded order ⓘ |
| requires |
property to be hereditary with respect to the underlying relation
ⓘ
reflexive transitive closure of a relation to be well-founded ⓘ |
| typicalForm | if a property holds for an element assuming it holds for all smaller elements, then it holds for all elements ⓘ |
| typicalPremise | every nonempty subset has a minimal element ⓘ |
| usedIn |
algebra
ⓘ
commutative algebra ⓘ computer science ⓘ order theory ⓘ ring theory ⓘ set theory ⓘ topology ⓘ |
| usedInProofStyle | minimal counterexample argument ⓘ |
| usedToProve |
finiteness properties
ⓘ
structural properties of Noetherian rings ⓘ termination of algorithms ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Noetherian recursion (in some contexts)