---
tag: 49Y9Q949PK
kind: concept
name: Perfect ring
msc: [16L30]
specializes: [ZRC2S3ECFK]
sources:
  - url: https://en.wikipedia.org/wiki/Perfect_ring
same_as:
  wikidata: Q7168096
  wikipedia: Perfect ring
---

## Summary

A left perfect ring is a ring $R$ such that $R/J(R)$ is semisimple and the [[Jacobson radical|XA0DCNH7WY]] $J(R)$ is left T-nilpotent.
By [[Bass's theorem P|D2M8BSW720]], these are exactly the rings over which every left module has a [[projective cover|WGYXR97NPK]].
Left perfect rings are [[semiperfect|ZRC2S3ECFK]], and [[semiprimary rings|XVP517QRH5]], in particular left Artinian rings, are left and right perfect.
Left perfect and right perfect are different conditions.

## Explanations

### L1 human

In a semiperfect ring the finitely generated modules have projective covers, a "best" projective module mapping onto them.
A perfect ring asks this of all modules, however large.
This needs a strong smallness condition on the Jacobson radical, called T-nilpotence.
It says that for every infinite sequence of elements of the radical, some initial product is already zero.