# Noetherian ring

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

## Definition

Let ${\displaystyle A}$ be a ring. The following conditions are equivalent:

1. The ring ${\displaystyle A}$ satisfies an ascending chain condition on the set of its ideals: that is, there is no infinite ascending chain of ideals ${\displaystyle I_{0}\subsetneq I_{1}\subsetneq I_{2}\subsetneq \ldots }$.
2. Every ideal of ${\displaystyle A}$ is finitely generated.
3. Every nonempty set of ideals of ${\displaystyle A}$ has a maximal element when considered as a partially ordered set with respect to inclusion.

When the above conditions are satisfied, ${\displaystyle A}$ is said to be Noetherian. Alternatively, the ring ${\displaystyle A}$ 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

${\displaystyle \langle 0\rangle \subset \langle x\rangle \subset \langle x,x-1\rangle \subset \langle x,x-1,x-2\rangle \subset \cdots .\,}$

## Useful Criteria

If ${\displaystyle A}$ is a Noetherian ring, then we have the following useful results:

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