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Noetherian ring

In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals. If the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noetherian respectively. Formally, every increasing sequence I 1 ⊆ I 2 ⊆ I 3 ⊆ ⋯ {\displaystyle I_{1}\subseteq…

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Implication on injective modules

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Noetherian ring

Nodes117
Edges116
Triples120
Avg. degree1.98
Density0.017094
Components1

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Noetherian ring

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related to References · 55
Noetherian ring → Addison-Wesley, Addison-Wesley-Longman, Advanced Mathematics, Algebra, American Mathematical Society, Anderson, Arun Vinayak, Birkhäuser, Cambridge Studies, Cambridge University Press, Chapter, Chapters, Commutative Algebra, Commutative Ring Theory, D-modules, David, Edward, Eisenbud, Formanek, Frank
related to Properties · 25
Noetherian ring → Akizuki, Also, Another, Artinian, Bass, By, Eakin, Every, Hilbert's, Hopkins, If, Implication, In, Levitzki, Nagata, Noetherian, Ore, R-module, R/I, See
related to Examples · 14
Noetherian ring → Any, Dedekind, Euclidean, For, Here, Hilbert, Lie, Noetherian, PBW, Rings, Sym, The, This, Weyl
related to Commutative case · 10
Noetherian ring → For, Here, It, Krull, Krull's, Noetherian, Over, Rees, The, The Artin
related to Implication on injective modules · 10
Noetherian ring → Bass, Each, Faith, Given, Namely, Noetherian, R-module, R-modules, There, Walker
related to External links · 4
Noetherian ring → EMS Press, Encyclopedia, Mathematics, Noetherian
is a · 1
Noetherian ring → ring that satisfies the ascending chain condition on left and right ideals

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Important terminology

ring noetherian commutative left displaystyle ideals rings every right ideal generated theorem finitely injective algebra chain right-noetherian group domain left-noetherian

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SubjectPredicateObjectConfidenceSrc
Noetherian ringis aring that satisfies the ascending chain condition on left and right ideals0.90text
the Krull intersection theorem.The dimension theory of commutative rings behaves poorly over non-Noetherian ringsinstance ofIt is a technical tool that is used to prove other key theorems0.80text
Noetherian ringrelated to Commutative caseOver0.60section
Noetherian ringrelated to Commutative caseNoetherian0.60section
Noetherian ringrelated to Commutative caseFor0.60section
Noetherian ringrelated to Commutative caseThe Artin0.60section
Noetherian ringrelated to Commutative caseRees0.60section
Noetherian ringrelated to Commutative caseIt0.60section
Noetherian ringrelated to Commutative caseKrull0.60section
Noetherian ringrelated to Commutative caseThe0.60section
Noetherian ringrelated to Commutative caseKrull's0.60section
Noetherian ringrelated to Commutative caseHere0.60section

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