Editing Noetherian

In mathematics, the [[adjective]] '''Noetherian''' is used to describe [[Category_theory#Categories.2C_objects.2C_and_morphisms|objects]] that satisfy an [[ascending chain condition|ascending or descending chain condition]] on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite length. Noetherian objects are named after [[Emmy Noether]], who was the first to study the ascending and descending chain conditions for rings. Specifically:
* [[Noetherian group]], a [[Group (mathematics)|group]] that satisfies the ascending chain condition on [[subgroup]]s.
* [[Noetherian ring]], a [[Ring (mathematics)|ring]] that satisfies the ascending chain condition on [[ideal of a ring|ideal]]s.
* [[Noetherian module]], a [[Module (mathematics)|module]] that satisfies the ascending chain condition on submodules.
* More generally, an object in a [[Category (mathematics)|category]] is said to be Noetherian if there is no infinitely increasing filtration of it by subobjects. A category is Noetherian if every object in it is Noetherian.
* [[Noetherian relation]], a [[binary relation]] that satisfies the ascending chain condition on its elements.
* [[Noetherian topological space]], a [[topological space]] that satisfies the descending chain condition on [[closed set]]s.
* [[Noetherian induction]], also called well-founded induction, a proof method for binary relations that satisfy the descending chain condition.
* Noetherian rewriting system, an [[abstract rewriting system]] that has no infinite chains.
* [[Noetherian scheme]], a [[scheme (mathematics)|scheme]] in [[algebraic geometry]] that admits a finite covering by open [[Spectrum of a ring|spectra]] of Noetherian rings.

== See also ==
* [[Artinian ring]], a ring that satisfies the descending chain condition on ideals.

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[[Category:Mathematical analysis]]