Editing Emmy Noether (section)

===Second epoch (1920–1926)===
====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>

====Commutative rings, ideals, and modules====
Noether's paper, ''Idealtheorie in Ringbereichen'' (''Theory of Ideals in Ring Domains'', 1921),{{sfn| Noether|1921}} is the foundation of general commutative [[ring theory]], and gives one of the first general definitions of a [[commutative ring]].{{efn|The first definition of an abstract ring was given by [[Abraham Fraenkel]] in 1914, but the definition in current use was initially formulated by Masazo Sono in a 1917 paper.{{sfn|Gilmer|1981|p=133}}}}{{sfn|Gilmer|1981|p=133}} Before her paper, most results in commutative algebra were restricted to special examples of commutative rings, such as polynomial rings over fields or rings of algebraic integers. Noether proved that in a ring which satisfies the ascending chain condition on [[ideal (ring theory)|ideals]], every ideal is finitely generated. In 1943, French mathematician [[Claude Chevalley]] coined the term ''[[Noetherian ring]]'' to describe this property.{{sfn|Gilmer|1981|p=133}} A major result in Noether's 1921 paper is the [[Lasker–Noether theorem]], which extends Lasker's theorem on the primary decomposition of ideals of polynomial rings to all Noetherian rings.{{sfn|Rowe|Koreuber|2020|p=27}}{{sfn|Rowe|2021|p=xvi}} The Lasker–Noether theorem can be viewed as a generalization of the [[fundamental theorem of arithmetic]] which states that any positive integer can be expressed as a product of [[prime number]]s, and that this decomposition is unique.{{sfn|Osofsky|1994}}

Noether's work ''Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern'' (''Abstract Structure of the Theory of Ideals in Algebraic Number and Function Fields'', 1927)<ref>{{harvnb|Noether|1927}}.</ref> characterized the rings in which the ideals have unique factorization into prime ideals (now called [[Dedekind domain]]s).{{sfn|Noether|1983|p=13}} Noether showed that these rings were characterized by five conditions: they must satisfy the ascending and descending chain conditions, they must possess a unit element but no [[zero divisor]]s, and they must be [[integrally closed domain|integrally closed]] in their associated field of fractions.{{sfn|Noether|1983|p=13}}{{sfn|Rowe|2021|p=96}} This paper also contains what now are called the [[isomorphism theorems]],{{sfn|Rowe|2021|pp=286–287}} which describe some fundamental [[natural isomorphism]]s, and some other basic results on Noetherian and [[Artinian module]]s.{{sfn|Noether|1983|p=14}}

====Elimination theory====
In 1923–1924, Noether applied her ideal theory to [[elimination theory]] in a formulation that she attributed to her student, Kurt Hentzelt. She showed that fundamental theorems about the [[polynomial factorization|factorization of polynomials]] could be carried over directly.{{sfn|Noether|1923}}{{sfn|Noether|1923b}}{{sfn|Noether|1924}} 

Traditionally, elimination theory is concerned with eliminating one or more variables from a system of polynomial equations, often by the method of [[resultant]]s.{{sfn|Cox|Little|O'Shea|2015|p=121}}
For illustration, a system of equations often can be written in the form
: {{math|1= Mv = 0 }}
where a matrix (or [[linear transform]]) {{math|M}} (without the variable {{math|x}}) times a vector {{math|v}} (that only has non-zero powers of {{math|x}}) is equal to the zero vector, {{math|0}}. Hence, the [[determinant]] of the matrix {{math|M}} must be zero, providing a new equation in which the variable {{math|x}} has been eliminated.

====Invariant theory of finite groups====
Techniques such as Hilbert's original non-constructive solution to the finite basis problem could not be used to get quantitative information about the invariants of a group action, and furthermore, they did not apply to all group actions. In her 1915 paper,{{Sfn | Noether| 1915}} Noether found a solution to the finite basis problem for a finite group of transformations {{math|''G''}} acting on a finite-dimensional vector space over a field of characteristic zero. Her solution shows that the ring of invariants is generated by homogeneous invariants whose degree is less than, or equal to, the order of the finite group; this is called '''Noether's bound'''. Her paper gave two proofs of Noether's bound, both of which also work when the characteristic of the field is [[coprime]] to <math>\left|G\right|!</math> (the [[factorial]] of the order <math>\left|G\right|</math> of the group {{math|''G''}}). The degrees of generators need not satisfy Noether's bound when the characteristic of the field divides the number <math>\left|G\right|</math>,{{Sfn |Fleischmann | 2000 |p = 24}} but Noether was not able to determine whether this bound was correct when the characteristic of the field divides <math>\left|G\right|!</math> but not <math>\left|G\right|</math>. For many years, determining the truth or falsehood of this bound for this particular case was an open problem, called "Noether's gap". It was finally solved independently by Fleischmann in 2000 and Fogarty in 2001, who both showed that the bound remains true.{{Sfn |Fleischmann|2000|p=25}}{{Sfn | Fogarty |2001|p=5}}

In her 1926 paper,{{Sfn |Noether|1926}} Noether extended Hilbert's theorem to representations of a finite group over any field; the new case that did not follow from Hilbert's work is when the characteristic of the field divides the order of the group. Noether's result was later extended by [[William Haboush]] to all reductive groups by his proof of the [[Haboush's theorem|Mumford conjecture]].{{sfn|Haboush|1975}} In this paper Noether also introduced the ''[[Noether normalization lemma]]'', showing that a finitely generated [[integral domain|domain]] {{math|''A''}} over a field {{math|''k''}} has a set {{math|1={''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>}}} of [[algebraic independence|algebraically independent]] elements such that {{math|''A''}} is [[integrality|integral]] over {{math|1=''k''[''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>]}}.

====Topology====
[[File:Mug and Torus morph.gif|thumb|right|250px|A continuous deformation ([[homotopy]]) of a coffee cup into a doughnut ([[torus]]) and back]]
As noted by [[Hermann Weyl]] in his obituary, Noether's contributions to [[topology]] illustrate her generosity with ideas and how her insights could transform entire fields of mathematics.{{sfn|Weyl|1935}} In topology, mathematicians study the properties of objects that remain invariant even under deformation, properties such as their [[connected space|connectedness]]. An old joke is that "''a topologist cannot distinguish a donut from a coffee mug''", since they can be [[Homeomorphism|continuously deformed]] into one another.{{sfn|Hubbard|West|1991|p=204}}

Noether is credited with fundamental ideas that led to the development of [[algebraic topology]] from the earlier [[combinatorial topology]], specifically, the idea of [[Homology theory#Towards algebraic topology|homology groups]].{{Sfn |Hilton|1988|p=284}} According to Alexandrov, Noether attended lectures given by him and [[Heinz Hopf]] in the summers of 1926 and 1927, where "she continually made observations which were often deep and subtle"{{Sfn |Dick|1981|p=173}} and he continues that,
{{blockquote |When ... she first became acquainted with a systematic construction of combinatorial topology, she immediately observed that it would be worthwhile to study directly the [[group (mathematics)|groups]] of algebraic complexes and cycles of a given polyhedron and the [[subgroup]] of the cycle group consisting of cycles homologous to zero; instead of the usual definition of [[Betti number]]s, she suggested immediately defining the Betti group as the [[quotient group|complementary (quotient) group]] of the group of all cycles by the subgroup of cycles homologous to zero. This observation now seems self-evident. But in those years (1925–1928) this was a completely new point of view.{{Sfn | Dick | 1981|p= 174}}}}

Noether's suggestion that topology be studied algebraically was adopted immediately by Hopf, Alexandrov, and others,{{Sfn | Dick | 1981|p= 174}} and it became a frequent topic of discussion among the mathematicians of Göttingen.<ref>{{citation |last=Hirzebruch |first=Friedrich |author-link=Friedrich Hirzebruch |title=Emmy Noether and Topology}} in {{Harvnb | Teicher|1999|pp= 57–61}}</ref> Noether observed that her idea of a [[Betti group]] makes the [[Euler characteristic|Euler–Poincaré formula]] simpler to understand, and Hopf's own work on this subject{{Sfn |Hopf|1928}} "bears the imprint of these remarks of Emmy Noether".{{Sfn |Dick|1981|pp = 174–175}} Noether mentions her own topology ideas only as an aside in a 1926 publication,{{Sfn |Noether | 1926b}} where she cites it as an application of [[group theory]].<ref>{{citation |last=Hirzebruch |first=Friedrich |author-link=Friedrich Hirzebruch |title=Emmy Noether and Topology}} in {{Harvnb | Teicher|1999|p= 63}}</ref>

This algebraic approach to topology was also developed independently in [[Austria]]. In a 1926–1927 course given in [[Vienna]], [[Leopold Vietoris]] defined a [[homology group]], which was developed by [[Walther Mayer]] into an axiomatic definition in 1928.<ref>{{citation |last=Hirzebruch |first=Friedrich |author-link=Friedrich Hirzebruch |title=Emmy Noether and Topology}} in {{Harvnb | Teicher|1999|pp= 61–63}}</ref>

[[File:Helmut Hasse.jpg|thumb|upright|right|[[Helmut Hasse]] worked with Noether and others to found the theory of [[central simple algebra]]s.]]