Editing Emmy Noether (section)

====Ascending and descending chain conditions====
In this epoch, Noether became famous for her deft use of ascending (''Teilerkettensatz'') or descending (''Vielfachenkettensatz'') chain conditions.<ref name="ACC"/> A sequence of [[empty set|non-empty]] [[subset]]s {{math|''A''<sub>1</sub>, ''A''<sub>2</sub>, ''A''<sub>3</sub>}}, ... of a [[Set (mathematics)|set]] {{math|''S''}} is usually said to be ''ascending'' if each is a subset of the next:
:<math>A_{1} \subseteq A_{2} \subseteq A_{3} \subseteq \cdots.</math>

Conversely, a sequence of subsets of {{math|''S''}} is called ''descending'' if each contains the next subset:
:<math>A_{1} \supseteq A_{2} \supseteq A_{3} \supseteq \cdots.</math>

A chain ''becomes constant after a finite number of steps'' if there is an {{math|''n''}} such that <math>A_n = A_m</math> for all {{math|''m'' ≥ ''n''}}. A collection of subsets of a given set satisfies the [[ascending chain condition]] if every ascending sequence becomes constant after a finite number of steps. It satisfies the descending chain condition if any descending sequence becomes constant after a finite number of steps.{{sfn|Atiyah|MacDonald|1994|p=74}} Chain conditions can be used to show that every set of sub-objects has a maximal/minimal element, or that a complex object can be generated by a smaller number of elements.{{sfn|Atiyah|MacDonald|1994|pp=74–75}}

Many types of objects in [[abstract algebra]] can satisfy chain conditions, and usually if they satisfy an ascending chain condition, they are called ''[[Noetherian (disambiguation)|Noetherian]]'' in her honor.{{sfn|Gray|2018|p=294}} By definition, a [[Noetherian ring]] satisfies an ascending chain condition on its left and right ideals, whereas a [[Noetherian group]] is defined as a group in which every strictly ascending chain of subgroups is finite. A [[Noetherian module]] is a [[module (mathematics)|module]] in which every strictly ascending chain of submodules becomes constant after a finite number of steps.{{sfn|Goodearl|Warfield Jr.|2004|pp=1–3}}{{sfn|Lang|2002|pp=413–415}} A [[Noetherian space]] is a [[topological space]] whose open subsets satisfy the ascending chain condition;{{efn|Or whose closed subsets satisfy the descending chain condition.{{sfn|Hartshorne|1977|p=5}}}} this definition makes the [[spectrum of a ring|spectrum]] of a Noetherian ring a Noetherian topological space.{{sfn|Hartshorne|1977|p=5}}{{sfn|Atiyah|MacDonald|1994|p=79}}

The chain condition often is "inherited" by sub-objects. For example, all subspaces of a Noetherian space are Noetherian themselves; all subgroups and quotient groups of a Noetherian group are Noetherian; and, ''[[mutatis mutandis]]'', the same holds for submodules and quotient modules of a Noetherian module.{{sfn|Lang|2002|p=414}} The chain condition also may be inherited by combinations or extensions of a Noetherian object. For example, finite direct sums of Noetherian rings are Noetherian, as is the [[ring of formal power series]] over a Noetherian ring.{{sfn|Lang|2002|p=415–416}}

Another application of such chain conditions is in [[Noetherian induction]]{{snd}}also known as [[well-founded induction]]{{snd}}which is a generalization of [[mathematical induction]]. It frequently is used to reduce general statements about collections of objects to statements about specific objects in that collection. Suppose that {{math|''S''}} is a [[partially ordered set]]. One way of proving a statement about the objects of {{math|''S''}} is to assume the existence of a [[counterexample]] and deduce a contradiction, thereby proving the [[contrapositive]] of the original statement. The basic premise of Noetherian induction is that every non-empty subset of {{math|''S''}} contains a minimal element. In particular, the set of all counterexamples contains a minimal element, the ''minimal counterexample''. In order to prove the original statement, therefore, it suffices to prove something seemingly much weaker: For any counter-example, there is a smaller counter-example.<ref>{{cite web|url=https://people.engr.tamu.edu/andreas-klappenecker/cpsc289-f08/noetherian_induction.pdf|title=Noetherian induction|first=Andreas|last=Klappenecker|work=CPSC 289 Special Topics on Discrete Structures for Computing|date=Fall 2008|type=Lecture notes|publisher=[[Texas A&M University]]|access-date=14 January 2025|archive-date=4 July 2024|archive-url=https://web.archive.org/web/20240704182844/https://people.engr.tamu.edu/andreas-klappenecker/cpsc289-f08/noetherian_induction.pdf|url-status=live}}</ref>