---
tag: ZRC2S3ECFK
kind: concept
name: Semiperfect ring
msc: [16L30]
specializes: [ZDCDS6EWGX]
sources:
  - url: https://en.wikipedia.org/wiki/Perfect_ring
    locator: Semiperfect ring
---

## Summary

A semiperfect ring is a [[semilocal ring|ZDCDS6EWGX]] in which idempotents [[lift|2SRDN8EJG3]] modulo the Jacobson radical.
Equivalently, every finitely generated left module has a [[projective cover|WGYXR97NPK]].
Equivalently again, $1$ is a sum of orthogonal idempotents $e_1, \ldots, e_n$ such that each corner ring $e_i R e_i$ is local.
Left Artinian rings, [[local rings|MZ02BKPHJA]] and [[perfect rings|49Y9Q949PK]] are semiperfect.

## Explanations

### L1 human

A semiperfect ring splits, as a module over itself, into finitely many indecomposable pieces, each of which behaves like a local ring.
This makes its module theory close to that of finite-dimensional algebras.
The ring of $p$-adic integers is semiperfect, being a complete local ring.
The ring of fractions $a/b$ with $b$ prime to $6$ is semilocal but not semiperfect.
Modulo its radical it is $\mathbb{F}_2 \times \mathbb{F}_3$, which has idempotents that do not come from the ring.