Noetherian ring

From Citizendium, the Citizens' Compendium

Main Article

Definition

Let $A$ be a ring. The following conditions are equivalent:

The ring $A$ satisfies an ascending chain condition on the set of its ideals: that is, there is no infinite ascending chain of ideals $I_0\subsetneq I_1\subsetneq I_2\subsetneq\ldots$ .
Every ideal of $A$ is finitely generated.
Every nonempty set of ideals of $A$ has a maximal element when considered as a partially ordered set with respect to inclusion.

When the above conditions are satisfied, $A$ is said to be Noetherian. Alternatively, the ring $A$ is Noetherian if is a Noetherian module when regarded as a module over itself.

A Noetherian domain is a Noetherian ring which is also an integral domain.

Examples

A field is Noetherian, since its only ideals are (0) and (1).
A principal ideal domain is Noetherian, since every ideal is generated by a single element.
- The ring of integers Z
- The polynomial ring over a field
The ring of continuous functions from R to R is not Noetherian. There is an ascending sequence of ideals

$\langle 0 \rangle \subset \langle x \rangle \subset \langle x,x-1 \rangle \subset \langle x,x-1,x-2 \rangle \subset \cdots .\,$

Useful Criteria

If $A$ is a Noetherian ring, then we have the following useful results:

The quotient $A/I$ is Noetherian for any ideal $I$ .
The localization of $A$ by a multiplicative subset $S$ is again Noetherian.
Hilbert's Basis Theorem: The polynomial ring $A[X]$ is Noetherian (hence so is $A[X_1,\ldots,X_n]$ ).