Editing Noetherian ring (section)

== Characterizations ==
For [[noncommutative ring]]s, it is necessary to distinguish between three very similar concepts:

* A ring is '''left-Noetherian''' if it satisfies the ascending chain condition on left ideals.
* A ring is '''right-Noetherian''' if it satisfies the ascending chain condition on right ideals.
* A ring is '''Noetherian''' if it is both left- and right-Noetherian.

For [[commutative ring]]s, all three concepts coincide, but in general they are different.  There are rings that are left-Noetherian and not right-Noetherian, and vice versa.

There are other, equivalent, definitions for a ring ''R'' to be left-Noetherian:

* Every left ideal ''I'' in ''R'' is [[Ideal_(ring_theory)#Types_of_ideals|finitely generated]], i.e. there exist elements <math>a_1, \ldots , a_n</math> in ''I'' such that <math>I=Ra_1 + \cdots + Ra_n</math>.<ref name=":0">Lam (2001), p. 19</ref>
* Every [[empty set|non-empty]] set of left ideals of ''R'', [[partially ordered]] by inclusion, has a [[maximal element]].<ref name=":0" />

Similar results hold for right-Noetherian rings.

The following condition is also an equivalent condition for a ring ''R'' to be left-Noetherian and it is [[David Hilbert|Hilbert]]'s original formulation:<ref>{{harvnb|Eisenbud|1995|loc=Exercise 1.1.}}</ref>
*Given a sequence <math>f_1, f_2, \dots</math> of elements in ''R'', there exists an integer <math>n</math> such that each <math>f_i</math> is a finite [[linear combination]] <math display="inline">f_i = \sum_{j=1}^n r_j f_j</math> with coefficients <math>r_j</math> in ''R''.

For a commutative ring to be Noetherian it suffices that every [[prime ideal]] of the ring is finitely generated.<ref>{{Cite journal|last=Cohen|first=Irvin S.|author-link=Irvin Cohen|date=1950|title=Commutative rings with restricted minimum condition|url=https://projecteuclid.org/euclid.dmj/1077475897| journal=[[Duke Mathematical Journal]]|language=en|volume=17|issue=1|pages=27–42|doi=10.1215/S0012-7094-50-01704-2|issn=0012-7094}}</ref> However, it is not enough to ask that all the [[maximal ideal]]s are finitely generated, as there is a non-Noetherian [[local ring]] whose maximal ideal is [[principal ideal|principal]] (see a counterexample to Krull's intersection theorem at [[Local ring#Commutative case]].)