Noether's isomorphism theorems

E29378

family of theorems result in abstract algebra

Noether's isomorphism theorems are fundamental results in abstract algebra that relate quotient structures and substructures of groups, rings, and modules, providing a unifying framework for understanding homomorphic images and factor structures.

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All labels observed (4)

Label	Occurrences
Noether isomorphism theorems	1
Noether's isomorphism theorems canonical	1
isomorphism theorems	1
second isomorphism theorem	1

Statements (48)

Predicate	Object
instanceOf	family of theorems ⓘ result in abstract algebra ⓘ
alsoKnownAs	Noether's isomorphism theorems ⓘ surface form: Noether isomorphism theorems Noether's isomorphism theorems ⓘ surface form: isomorphism theorems
appearsIn	graduate algebra courses ⓘ standard undergraduate algebra textbooks ⓘ
appliesTo	groups ⓘ modules ⓘ rings ⓘ
assumes	existence of kernels of homomorphisms ⓘ existence of normal subgroups or ideals ⓘ
concerns	factor structures ⓘ homomorphic images ⓘ quotient structures ⓘ substructures ⓘ
context	group theory ⓘ module theory ⓘ ring theory ⓘ
field	abstract algebra ⓘ
formalism	category of groups ⓘ category of modules ⓘ category of rings ⓘ
foundationFor	modern structural algebra ⓘ
generalizes	isomorphism theorems for groups ⓘ isomorphism theorems for modules ⓘ isomorphism theorems for rings ⓘ
hasPart	group isomorphism theorems ⓘ module isomorphism theorems ⓘ ring isomorphism theorems ⓘ
historicalPeriod	early 20th century ⓘ
implies	correspondence between ideals containing the kernel and ideals of the image ⓘ correspondence between subgroups containing the kernel and subgroups of the image ⓘ
includes	first isomorphism theorem ⓘ Noether's isomorphism theorems self-linksurface differs ⓘ surface form: second isomorphism theorem third isomorphism theorem ⓘ
influencedBy	Emmy Noether's work on ideal theory ⓘ
logicalForm	equivalence of quotient by intersection and quotient of quotient ⓘ equivalence of quotient by normal subgroup and quotient of group ⓘ
namedAfter	Emmy Noether ⓘ
prerequisiteFor	Jordan–Hölder theorem ⓘ structure theory of modules over a PID ⓘ
relatedTo	lattice of ideals ⓘ lattice of subgroups ⓘ short exact sequences ⓘ
states	homomorphic image of a structure is isomorphic to a quotient by the kernel ⓘ
usedFor	classifying algebraic objects up to isomorphism ⓘ relating subobjects and quotient objects ⓘ simplifying algebraic structures ⓘ

How these facts were elicited

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Instruction

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# Requirements
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Input

Subject: Noether's isomorphism theorems
Description of subject: Noether's isomorphism theorems are fundamental results in abstract algebra that relate quotient structures and substructures of groups, rings, and modules, providing a unifying framework for understanding homomorphic images and factor structures.

Referenced by (4)

Full triples — surface form annotated when it differs from this entity's canonical label.

Emmy Noether → notableWork → Noether's isomorphism theorems ⓘ

Noether's isomorphism theorems → includes → Noether's isomorphism theorems self-linksurface differs ⓘ

this entity surface form: second isomorphism theorem

Noether's isomorphism theorems → alsoKnownAs → Noether's isomorphism theorems ⓘ

this entity surface form: isomorphism theorems

Noether's isomorphism theorems → alsoKnownAs → Noether's isomorphism theorems ⓘ

this entity surface form: Noether isomorphism theorems