---
tag: 2C8JN6XBWY
kind: definition
name: Semiperfect ring
msc: [16L30]
defines: ZRC2S3ECFK
uses: [YH75M7XAM1, XA0DCNH7WY, PJA5H9ZW4S]
sources:
  - url: https://en.wikipedia.org/wiki/Perfect_ring
    locator: Semiperfect ring
---

## Definition

Let $R$ be a ring with identity and $J = J(R)$ its [[Jacobson radical|XA0DCNH7WY]].
$R$ is *semiperfect* if the following hold.

1. $R$ is [[semilocal|YH75M7XAM1]], that is, $R/J$ is a semisimple ring.
2. [[Idempotents lift|PJA5H9ZW4S]] modulo $J$: for every $x \in R$ with $x^2 - x \in J$ there is $e \in R$ with $e^2 = e$ and $e - x \in J$.

## Remarks

Example: a [[local ring|MZ02BKPHJA]] $R$ is semiperfect: $R/J$ is a field, hence semisimple, and its only idempotents $0$ and $1$ lift to $0$ and $1$.
Non-example: let $R = \set{a/b \in \Q : \gcd(b, 6) = 1}$.
Its maximal ideals are generated by $2$ and $3$, so $R/J \cong \mathbb{F}_2 \times \mathbb{F}_3$ and clause 1 holds.
Clause 2 fails: $R$ is a domain, so its only idempotents are $0$ and $1$, which cannot lift the idempotent $(1, 0)$ of $\mathbb{F}_2 \times \mathbb{F}_3$.
Equivalent forms: every finitely generated left $R$-module has a [[projective cover|WGYXR97NPK]]; or $1 = e_1 + \cdots + e_n$ for orthogonal idempotents $e_i$ with each $e_i R e_i$ local.
The definition is left-right symmetric.