---
tag: 0MVH4Y1CRZ
kind: definition
name: Noetherian ring
msc: ["13E05"]
defines: 0670T59HGE
uses: [0ZCAXY9563, 030TZ0DQWT, 0GYGHKX834]
sources:
  - url: https://en.wikipedia.org/wiki/Noetherian_ring
    locator: Characterizations
---

## Definition

Let $R$ be a [[commutative ring|0GYGHKX834]].
The ring $R$ is Noetherian if the set of its ideals, ordered by inclusion, satisfies the [[ascending chain condition|0ZCAXY9563]].
That is, for every chain $I_1 \subseteq I_2 \subseteq \dots$ of ideals of $R$, there is $N$ with $I_n = I_N$ for all $n \ge N$.

## Remarks

The ring $R$ is Noetherian if and only if every ideal of $R$ is [[finitely generated|030TZ0DQWT]].
Every field and every [[principal ideal domain|0RZYDVZ475]] is Noetherian.
By [[Hilbert's basis theorem|0YKA9J3XEE]], $R[x_1, \dots, x_n]$ is Noetherian when $R$ is.
The polynomial ring in infinitely many variables $x_1, x_2, \dots$ over a field is not Noetherian: $(x_1) \subset (x_1, x_2) \subset \dots$ never stops.
For a noncommutative ring, one defines left and right Noetherian rings with left and right ideals.

## Explanations

### L1 human

In a Noetherian ring, each ideal is generated by a finite list of elements.
This makes many arguments work by induction, because a growing chain of ideals must stop.
Rings met in practice, like $\mathbb{Z}[x, y]$, are Noetherian.