Editing Emmy Noether (section)

====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}}