Editing Commutative algebra (section)

===Noetherian rings===
{{Main|Noetherian ring}}
A '''Noetherian ring''', named after [[Emmy Noether]], is a ring in which every  [[ideal (ring theory)|ideal]] is [[finitely generated ideal|finitely generated]]; that is, all elements of any ideal can be written as a [[linear combination]]s of a finite set of elements, with coefficients in the ring. 

Many commonly considered commutative rings are Noetherian, in particular, every [[field (mathematics)|field]], the ring of the [[integer]], and every [[polynomial ring]] in one or several indeterminates over them. The fact that polynomial rings over a field are Noetherian is called [[Hilbert's basis theorem]].

Moreover, many ring constructions preserve the Noetherian property. In particular, if a commutative ring {{math|R}} is Noetherian, the same is true for every polynomial ring over it, and for every [[quotient ring]], [[localization (commutative algebra)|localization]], or [[completion of a ring|completion]] of the ring.

The importance of the Noetherian property lies in its ubiquity and also in the fact that many important theorems of commutative algebra require that the involved rings are Noetherian, This is the case, in particular of [[Lasker–Noether theorem]], the [[Krull intersection theorem]], and [[Nakayama's lemma]].

Furthermore, if a ring is Noetherian, then it satisfies the [[descending chain condition]] on [[prime ideal]]s, which implies that every  Noetherian [[local ring]] has a finite [[Krull dimension]].