---
tag: D2M8BSW720
kind: statement
role: theorem
name: Bass's theorem P
msc: [16L30, 16D40]
uses: [0894J980RK, XA0DCNH7WY, BJMEXPKRYM, WGYXR97NPK, CRE6VEKT32]
about: [49Y9Q949PK]
sources:
  - url: https://en.wikipedia.org/wiki/Perfect_ring
---

## Statement

Let $R$ be a [[ring|0894J980RK]] with identity and $J = J(R)$ its [[Jacobson radical|XA0DCNH7WY]].
The following are equivalent.

1. $R/J$ is a semisimple ring and $J$ is left T-nilpotent, that is, $R$ is [[left perfect|BJMEXPKRYM]].
2. Every left $R$-module has a [[projective cover|WGYXR97NPK]].
3. $R$ satisfies the descending chain condition on principal right ideals.
4. Every flat left $R$-module is [[projective|CRE6VEKT32]].

## Remarks

The link between 1 and 2 goes through Nakayama-type lemmas.
Left T-nilpotence of $J$ is exactly what makes $JM \ne M$ for every nonzero left module $M$, finitely generated or not.
Then a projective cover of $M$ can be built by lifting a projective cover of the semisimple module $M/JM$.
Condition 3 is notable for its side switch: a condition on left modules is equivalent to a chain condition on right ideals.
Example: left Artinian rings satisfy all four conditions, so over them every left module has a projective cover and flat left modules are projective.
Over $\Z$, which is not perfect, the flat module $\Q$ is not projective, and the module $\Z/2\Z$ has no projective cover.

## Explanations

### L1 human

The theorem gives four very different descriptions of the same class of rings.
One is about the structure of the ring, one about covers of modules, one about chains of ideals, and one about flat modules.
That they all agree is a sign that perfect rings are a natural class.

### L2 human

The theorem is the main characterization of [[perfect rings|49Y9Q949PK]] and appears in the study of projective covers and flat modules.
Its finitely generated analogue characterizes [[semiperfect rings|ZRC2S3ECFK]]: every finitely generated left module has a projective cover exactly when $R$ is semiperfect.
Because [[semiprimary rings|XVP517QRH5]] have nilpotent radical, they satisfy condition 1 on both sides.