Editing Noetherian ring (section)

== Examples ==
* Any field, including the fields of [[rational number]]s, [[real number]]s, and [[complex number]]s, is Noetherian. (A field only has two ideals &mdash; itself and (0).)
* Any [[principal ideal ring]], such as the integers, is Noetherian since every ideal is generated by a single element. This includes [[principal ideal domain]]s and [[Euclidean domain]]s.
* A [[Dedekind domain]] (e.g., [[ring of integers|rings of integers]]) is a Noetherian domain in which every ideal is generated by at most two elements.
* The [[coordinate ring]] of an [[affine variety]] is a Noetherian ring, as a consequence of the Hilbert basis theorem.
* The enveloping algebra ''U'' of a finite-dimensional [[Lie algebra]] <math>\mathfrak{g}</math> is a both left and right Noetherian ring; this follows from the fact that the [[associated graded ring]] of ''U'' is a quotient of <math>\operatorname{Sym}(\mathfrak{g})</math>, which is a polynomial ring over a field (the [[PBW theorem]]); thus, Noetherian.<ref>{{harvnb|Bourbaki|1989|loc=Ch III, §2, no. 10, Remarks at the end of the number}}</ref> For the same reason, the [[Weyl algebra]], and more general rings of [[differential operator]]s, are Noetherian.<ref>{{harvtxt|Hotta|Takeuchi|Tanisaki|2008|loc=§D.1, Proposition 1.4.6}}</ref>
* The ring of polynomials in finitely-many variables over the integers or a field is Noetherian.

Rings that are not Noetherian tend to be (in some sense) very large.  Here are some examples of non-Noetherian rings:

* The ring of polynomials in infinitely-many variables, ''X''<sub>1</sub>, ''X''<sub>2</sub>, ''X''<sub>3</sub>, etc.  The sequence of ideals (''X''<sub>1</sub>), (''X''<sub>1</sub>, ''X''<sub>2</sub>), (''X''<sub>1</sub>, ''X''<sub>2</sub>, ''X''<sub>3</sub>), etc. is ascending, and does not terminate.
* The ring of all [[algebraic integer]]s is not Noetherian. For example, it contains the infinite ascending chain of principal ideals: (2), (2<sup>1/2</sup>), (2<sup>1/4</sup>), (2<sup>1/8</sup>), ...
* The ring of [[continuous function]]s from the real numbers to the real numbers is not Noetherian:  Let ''I<sub>n</sub>'' be the ideal of all continuous functions ''f'' such that ''f''(''x'') = 0 for all ''x'' ≥ ''n''.  The sequence of ideals ''I''<sub>0</sub>, ''I''<sub>1</sub>, ''I''<sub>2</sub>, etc., is an ascending chain that does not terminate.
* The ring of [[Homotopy groups of spheres#Ring structure|stable homotopy groups of spheres]] is not Noetherian.<ref>[https://math.stackexchange.com/q/1513353 The ring of stable homotopy groups of spheres is not noetherian]</ref>

However, a non-Noetherian ring can be a subring of a Noetherian ring. Since any [[integral domain]] is a subring of a field, any integral domain that is not Noetherian provides an example. To give a less trivial example, 
* The ring of [[rational function]]s generated by ''x'' and ''y''/''x''<sup>''n''</sup> over a field ''k'' is a subring of the field ''k''(''x'',''y'') in only two variables.

Indeed, there are rings that are right Noetherian, but not left Noetherian, so that one must be careful in measuring the "size" of a ring this way. For example, if ''L'' is a [[subgroup]] of '''Q'''<sup>2</sup> [[isomorphic]] to '''Z''', let ''R'' be the ring of homomorphisms ''f'' from '''Q'''<sup>2</sup> to itself satisfying ''f''(''L'') ⊂ ''L''. Choosing a basis, we can describe the same ring ''R'' as
:<math>R=\left\{\left.\begin{bmatrix}a & \beta \\0 & \gamma \end{bmatrix} \, \right\vert\, a\in \mathbf{Z}, \beta\in \mathbf{Q},\gamma\in \mathbf{Q}\right\}.</math>
This ring is right Noetherian, but not left Noetherian; the subset ''I'' ⊂ ''R'' consisting of elements with ''a'' = 0 and ''γ'' = 0 is a left ideal that is not finitely generated as a left ''R''-module.

If ''R'' is a commutative subring of a left Noetherian ring ''S'', and ''S'' is finitely generated as a left ''R''-module, then ''R'' is Noetherian.<ref>{{harvnb|Formanek|Jategaonkar|1974|loc=Theorem 3}}</ref> (In the special case when ''S'' is commutative, this is known as [[Eakin–Nagata theorem|Eakin's theorem]].) However, this is not true if ''R'' is not commutative: the ring ''R'' of the previous paragraph is a subring of the left Noetherian ring ''S'' = Hom('''Q'''<sup>2</sup>, '''Q'''<sup>2</sup>), and ''S'' is finitely generated as a left ''R''-module, but ''R'' is not left Noetherian.

A [[unique factorization domain]] is not necessarily a Noetherian ring. It does satisfy a weaker condition: the [[ascending chain condition on principal ideals]]. A ring of polynomials in infinitely-many variables is an example of a non-Noetherian unique factorization domain.

A [[valuation ring]] is not Noetherian unless it is a principal ideal domain. It gives an example of a ring that arises naturally in [[algebraic geometry]] but is not Noetherian.

=== Noetherian group rings ===
Consider the [[group ring]] <math>R[G]</math> of a [[group (mathematics)|group]] <math>G</math> over a [[ring (mathematics)|ring]] <math>R</math>. It is a [[ring (mathematics)|ring]], and an [[associative algebra]] over <math>R</math> if <math>R</math> is [[commutative ring|commutative]]. For a group <math>G</math> and a commutative ring <math>R</math>, the following two conditions are equivalent.
* The ring <math>R[G]</math> is left-Noetherian.
* The ring <math>R[G]</math> is right-Noetherian.
This is because there is a [[bijection]] between the left and right ideals of the group ring in this case, via the <math>R</math>-[[associative algebra]] [[homomorphism]]
:<math>R[G]\to R[G]^{\operatorname{op}},</math>
:<math>g\mapsto g^{-1}\qquad(\forall g\in G).</math>
Let <math>G</math> be a group and <math>R</math> a ring. If <math>R[G]</math> is left/right/two-sided Noetherian, then <math>R</math> is left/right/two-sided Noetherian and <math>G</math> is a [[Noetherian group]]. Conversely, if <math>R</math> is a Noetherian commutative ring and <math>G</math> is an [[group extension|extension]] of a [[Noetherian group|Noetherian]] [[solvable group]] (i.e. a [[polycyclic group]]) by a [[finite group]], then <math>R[G]</math> is two-sided Noetherian. On the other hand, however, there is a [[Noetherian group]] <math>G</math> whose group ring over any Noetherian commutative ring is not two-sided Noetherian.<ref name="Ol’shanskiĭ">{{cite book
|last1=Ol’shanskiĭ
|first1=Aleksandr Yur’evich
|title=Geometry of defining relations in groups
|translator-last=Bakhturin
|translator-first=Yu. A.
|language=en
|series=Mathematics and Its Applications. Soviet Series
|volume=70
|publisher=Kluwer Academic Publishers
|location=Dordrecht
|date=1991
|isbn=978-0-7923-1394-6
|issn=0169-6378
|doi=10.1007/978-94-011-3618-1
|mr=1191619
|zbl=0732.20019
}}</ref>{{rp|423, Theorem 38.1}}