Artinian ring

E621106

algebraic structure property ring-theoretic concept

An Artinian ring is a ring in which every descending chain of ideals eventually stabilizes, making it a fundamental object in commutative algebra and ring theory with strong finiteness properties.

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Statements (49)

Predicate	Object
instanceOf	algebraic structure property ⓘ ring-theoretic concept ⓘ
abbreviation	DCC on ideals ⓘ
appearsIn	the Wedderburn–Artin structure theorem NERFINISHED ⓘ
definition	a ring in which every descending chain of ideals stabilizes ⓘ
equivalentCondition	commutative Artinian ring is Noetherian and every prime ideal is maximal ⓘ commutative Artinian ring is Noetherian and has finitely many prime ideals ⓘ commutative Artinian ring is Noetherian with dimension zero ⓘ left Artinian and right Artinian are equivalent for semiprimary rings ⓘ
example	any field is an Artinian ring ⓘ any finite ring is Artinian ⓘ k[x]/(x^n) over a field k is Artinian ⓘ the ring of dual numbers k[ε]/(ε²) over a field k is Artinian ⓘ
field	commutative algebra ⓘ ring theory ⓘ
hasProperty	Jacobson radical has nilpotent elements whose powers eventually vanish ⓘ Jacobson radical is nilpotent in a commutative Artinian ring ⓘ Krull dimension zero for commutative Artinian rings ⓘ every Artinian semiprime ring is semisimple ⓘ every descending chain of left ideals stabilizes in a left Artinian ring ⓘ every descending chain of right ideals stabilizes in a right Artinian ring ⓘ every ideal is an intersection of primary ideals in a commutative Artinian ring ⓘ every ideal is finitely generated in a commutative Artinian ring ⓘ every module of finite length over a ring has an Artinian endomorphism ring under suitable conditions ⓘ every nonempty set of ideals has a minimal element ⓘ every prime ideal is maximal in a commutative Artinian ring ⓘ every simple Artinian ring is isomorphic to a matrix ring over a division ring ⓘ finite length as a module over itself ⓘ has finite length as a module over itself in the commutative case ⓘ nilradical equals Jacobson radical in a commutative Artinian ring ⓘ only finitely many maximal ideals in a commutative Artinian ring ⓘ satisfies both ACC and DCC on ideals in the commutative case ⓘ satisfies descending chain condition on ideals ⓘ
implies	Noetherian ring for commutative rings with identity ⓘ semiprimary ring in the commutative case ⓘ
namedAfter	Emil Artin NERFINISHED ⓘ
nonExample	the polynomial ring k[x] over a field k is not Artinian ⓘ the ring of integers ℤ is not Artinian ⓘ
relatedTo	Artinian module ⓘ Noetherian ring ⓘ semisimple ring ⓘ
structureTheorem	every Artinian principal ideal ring is a finite direct product of Artinian principal ideal local rings ⓘ every commutative Artinian ring is a finite direct product of Artinian local rings ⓘ
topicOf	research in noncommutative ring theory and representation theory ⓘ
usedIn	Wedderburn–Artin theory NERFINISHED ⓘ classification of semisimple modules ⓘ
variant	left Artinian ring ⓘ right Artinian ring ⓘ two-sided Artinian ring ⓘ

Referenced by (1)

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

Artinian module → generalizes → Artinian ring ⓘ