Editing Noetherian ring (section)

{{short description|Mathematical ring with well-behaved ideals}}
In [[mathematics]], a '''Noetherian ring''' is a [[ring (mathematics)|ring]] that satisfies the [[ascending chain condition]] on left and right [[Ideal (ring theory)|ideals]]; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said '''left-Noetherian''' or '''right-Noetherian''' respectively. That is, every increasing sequence <math>I_1\subseteq I_2 \subseteq I_3 \subseteq \cdots</math> of left (or right) ideals has a largest element; that is, there exists an {{math|''n''}} such that:
<math>I_{n}=I_{n+1}=\cdots.</math>

Equivalently, a ring is left-Noetherian (respectively right-Noetherian) if every left ideal (respectively right-ideal) is [[finitely generated ideal|finitely generated]]. A ring is Noetherian if it is both left- and right-Noetherian.

Noetherian rings are fundamental in both [[commutative ring|commutative]] and [[noncommutative ring|noncommutative]] ring theory since many rings that are encountered in mathematics are Noetherian (in particular the [[ring of integers]], [[polynomial ring]]s, and [[ring of algebraic integers|rings of algebraic integers]] in [[number field]]s), and many general theorems on rings rely heavily on the Noetherian property (for example, the [[Lasker–Noether theorem]] and the [[Krull intersection theorem]]).

Noetherian rings are named after [[Emmy Noether]], but the importance of the concept was recognized earlier by [[David Hilbert]], with the proof of [[Hilbert's basis theorem]] (which asserts that polynomial rings are Noetherian) and [[Hilbert's syzygy theorem]].
{{Algebraic structures |Ring}}