# Noetherian Ring

In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element. Equivalently, a ring is Noetherian if it satisfies the ascending chain condition on ideals; that is, given any chain:

there exists a positive integer n such that:

There are other equivalent formulations of the definition of a Noetherian ring and these are outlined later in the article.

The notion of a Noetherian ring is of fundamental importance in both commutative and noncommutative ring theory, due to the role it plays in simplifying the ideal structure of a ring. For instance, the ring of integers and the polynomial ring over a field are both Noetherian rings, and consequently, such theorems as the Lasker–Noether theorem, the Krull intersection theorem, and the Hilbert's basis theorem hold for them. Furthermore, if a ring is Noetherian, then it satisfies the descending chain condition on prime ideals. This property suggests a deep theory of dimension for Noetherian rings beginning with the notion of the Krull dimension.

### Other articles related to "noetherian ring, noetherian, ring":

Noetherian Ring - Properties
... If R is a Noetherian ring, then R is Noetherian by the Hilbert basis theorem ... Also, R], the power series ring is a Noetherian ring ... If R is a Noetherian ring and I is a two-sided ideal, then the factor ring R/I is also Noetherian ...
Emmy Noether - Contributions To Mathematics and Physics - Second Epoch (1920–26) - Ascending and Descending Chain Conditions
... chain conditions, and usually if they satisfy an ascending chain condition, they are called Noetherian in her honor ... By definition, a Noetherian ring satisfies an ascending chain condition on its left and right ideals, whereas a Noetherian group is defined as a group in which every strictly ascending chain of subgroups is ... A Noetherian module is a module in which every strictly ascending chain of submodules breaks off after a finite number ...