---
tag: C7X657TF19
kind: statement
role: theorem
name: Akizuki–Hopkins–Levitzki theorem
msc: ["13E10", "13E05", "16P20"]
uses: [0GYGHKX834, ZV04WWYE2Y, 0MVH4Y1CRZ, 06HC6HK0Z8, 09J6KGT65G]
about: [20QEAM8AX2, 0670T59HGE]
sources:
  - url: https://en.wikipedia.org/wiki/Hopkins–Levitzki_theorem
---

## Statement

Let $R$ be a [[commutative ring|0GYGHKX834]].
Then $R$ is [[Artinian|ZV04WWYE2Y]] if and only if $R$ is [[Noetherian|0MVH4Y1CRZ]] and every [[prime ideal|06HC6HK0Z8]] of $R$ is [[maximal|09J6KGT65G]].

## Remarks

The condition that every prime ideal is maximal says that $R$ has [[Krull dimension|MVGBS3J1P7]] $0$ when $R$ is not the zero ring.
If $R$ is Artinian, each prime $P$ gives an Artinian domain $R/P$, which is a field, so $P$ is maximal.
Then $R$ has finitely many maximal ideals $\mathfrak{m}_1, \dots, \mathfrak{m}_n$, and $(\mathfrak{m}_1 \cdots \mathfrak{m}_n)^k = 0$ for some $k$.
The chain $R \supseteq \mathfrak{m}_1 \supseteq \mathfrak{m}_1 \mathfrak{m}_2 \supseteq \cdots \supseteq 0$ has factors that are vector spaces over fields $R/\mathfrak{m}_i$, and for vector spaces the two chain conditions agree, so $R$ has finite [[length|CZSCNZH8VY]] and is Noetherian.
For example, $\Z/12\Z$ is Artinian and Noetherian, while $\Z$ is Noetherian but has the non-maximal prime $(0)$, so it is not Artinian.
For noncommutative rings, the Hopkins–Levitzki theorem says that every left Artinian ring is left Noetherian, while the converse needs extra hypotheses.

## Explanations

### L1 human

For commutative rings, the decreasing chain condition is stronger than the increasing one.
The theorem says exactly how much stronger: Artinian means Noetherian plus "no prime ideal sits strictly inside another".

### L2 human

The theorem shows that Artinian rings are the Noetherian rings of dimension $0$, and it is often used together with the [[structure theorem for Artinian rings|HE7MP4HP6Q]].
A typical use: if $R$ is a Noetherian local ring and $I$ is an ideal whose radical is the maximal ideal, then $R/I$ is Artinian, so it has finite length.
This fact is the start of dimension theory and of the [[Krull–Akizuki theorem|FSXA3CF365]].