---
tag: 2SRDN8EJG3
kind: concept
name: Lifting idempotents
msc: [16U40, 16L30]
sources:
  - url: https://en.wikipedia.org/wiki/Idempotent_(ring_theory)
---

## Summary

Lifting idempotents is the question of whether an [[idempotent|0N5NJZKKN3]] of a quotient ring $R/I$ comes from an idempotent of $R$.
When every idempotent of $R/I$ lifts, decompositions of $R/I$ into pieces can be pulled back to decompositions of $R$.
Idempotents always lift modulo a [[nil ideal|NZ1G9WRWFB]], and they lift modulo the Jacobson radical exactly in [[semiperfect rings|ZRC2S3ECFK]] among the semilocal rings.
The technique is central in the theory of finite-dimensional algebras, complete local rings and modular representation theory.

## Explanations

### L1 human

An idempotent is an element $e$ with $e^2 = e$, like a projection.
In a quotient ring, an element may be idempotent only "up to an error" in the ideal.
Lifting means correcting such an element to a true idempotent of the ring.
In $\Z/4\Z$, which is $\Z/8\Z$ modulo the nil ideal generated by $4$, every idempotent lifts.
In $\Z/6\Z$, which is $\Z$ modulo $6\Z$, the idempotent $3$ does not lift, since the only idempotents of $\Z$ are $0$ and $1$.