---
tag: W29AA0266Z
kind: statement
role: theorem
name: Hopkins–Levitzki theorem
msc: [16P20, 16P40, 16N20]
uses: [XVP517QRH5, XA0DCNH7WY, C2H26BYWHE, NY25HXA478, AK1ZRCXR32, YQNZTQGP1X, 0XV2A49X6Q, N3MRHDKDGC, 20QEAM8AX2, 0670T59HGE, H7RAXJV9CP, C7X657TF19, R1F271180N]
about: [YQNZTQGP1X, AK1ZRCXR32, 20QEAM8AX2]
sources:
  - url: https://en.wikipedia.org/wiki/Hopkins–Levitzki_theorem
same_as:
  wikidata: Q17027556
  wikipedia: Hopkins–Levitzki theorem
---

## Statement

Let $R$ be a [[semiprimary ring|XVP517QRH5]]: its [[Jacobson radical|XA0DCNH7WY]] $J = J(R)$ is a [[nilpotent ideal|C2H26BYWHE]] and $R/J$ is a [[semisimple ring|NY25HXA478]].
Let $M$ be a right $R$-module.
Then the following conditions are equivalent.

1. $M$ is [[Noetherian|AK1ZRCXR32]].
2. $M$ is [[Artinian|YQNZTQGP1X]].
3. $M$ has a [[composition series|0XV2A49X6Q]].

## Remarks

Over any ring, condition 3 holds exactly when $M$ is both Noetherian and Artinian, so the point is that each chain condition alone gives the other.
Choose $n$ with $J^n = 0$ and look at the filtration $M \supseteq MJ \supseteq MJ^2 \supseteq \cdots \supseteq MJ^n = 0$.
Each factor $MJ^{i}/MJ^{i+1}$ is killed by $J$, so it is a module over the semisimple ring $R/J$, hence a [[semisimple module|N3MRHDKDGC]].
A semisimple module satisfies one chain condition only when it is a finite direct sum of simple modules, and then it has finite length.
So each factor has finite length, and so does $M$.

Main corollary: a right [[Artinian ring|20QEAM8AX2]] is semiprimary, because [[its Jacobson radical is nilpotent|H7RAXJV9CP]] and $R/J$ is Artinian and semiprimitive, hence semisimple.
So every right Artinian ring is right [[Noetherian|0670T59HGE]], and the same holds on the left.

The hypothesis on $R$ cannot be dropped.
The ring $\Z$ is not semiprimary, and the Prüfer group $\Z[1/p]/\Z$ is an Artinian $\Z$-module that is not Noetherian.
Also $\Z$ itself is a Noetherian $\Z$-module that is not Artinian.

For commutative rings, the [[Akizuki–Hopkins–Levitzki theorem|C7X657TF19]] gives the sharper description of Artinian rings as the Noetherian rings in which every prime ideal is maximal.

## Explanations

### L1 human

For modules in general, "no infinite descending chains" and "no infinite ascending chains" are different conditions.
Over a semiprimary ring the two conditions agree, and both mean that the module has finite length.
The best-known case is the ring itself: a ring with the descending chain condition on right ideals also has the ascending one.

### L2 human

The proof peels the module into finitely many layers using the powers of the radical.
Each layer is a module over a semisimple ring, where both chain conditions just say "finitely many simple summands".
This argument is the reason why right Artinian rings are right Noetherian, a fact used all through the structure theory of Artinian rings, together with the [[Artin–Wedderburn theorem|R1F271180N]].
The converse fails: $\Z$ is Noetherian but not Artinian, so Artinian rings are a strictly smaller class.