Noetherian ring

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.

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 $A[X_{1},\ldots ,X_{n}]$).