# Noetherian ring

From Citizendium

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

## Definition

Let be a ring. The following conditions are equivalent:

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

When the above conditions are satisfied, is said to be *Noetherian*. Alternatively, the ring 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 integers
- The ring of continuous functions from
**R**to**R**is not Noetherian. There is an ascending sequence of ideals

## Useful Criteria

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

- The quotient is Noetherian for any ideal .
- The localization of by a multiplicative subset is again Noetherian.
**Hilbert's Basis Theorem**: The polynomial ring is Noetherian (hence so is ).

## References

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