Noetherian ring

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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:

  1. The ring satisfies an ascending chain condition on the set of its ideals: that is, there is no infinite ascending chain of ideals .
  2. Every ideal of is finitely generated.
  3. 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

Useful Criteria

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

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

References