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.
- 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 continuous functions from R to R is not Noetherian. There is an ascending sequence of ideals
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 ).