Noetherian ring

In algebra, a Noetherian ring is a ring with a condition on the lattice of ideals.

Definition

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

1. The ring $A$ satisfies an ascending chain condition on the set of its ideals: that is, there is no infinite ascending chain of ideals $I_{0}\subsetneq I_{1}\subsetneq I_{2}\subsetneq \ldots$.
2. Every ideal of $A$ is finitely generated.
3. 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:

1. The quotient $A/I$ is Noetherian for any ideal $I$.
2. The localization of $A$ by a multiplicative subset $S$ is again Noetherian.
3. Hilbert's Basis Theorem: The polynomial ring $A[X]$ is Noetherian (hence so is $A[X_{1},\ldots ,X_{n}]$).

References

• Serge Lang (1993). Algebra, 3rd ed. Addison-Wesley, 186-187. ISBN 0-201-55540-9.