---
tag: CYCTQ0CR3K
kind: concept
name: Clean ring
msc: [16U60, 16U40]
sources:
  - url: https://en.wikipedia.org/wiki/Clean_ring
same_as:
  wikidata: Q24896331
  wikipedia: Clean ring
---

## Summary

A clean ring is a ring in which every element is the sum of an [[idempotent|0N5NJZKKN3]] and a [[unit|0XVEZDE9J8]].
The notion was introduced by W. K. Nicholson in his study of exchange rings, and every clean ring is an exchange ring.
Local rings, matrix rings over local rings and [[unit-regular rings|H4K920SVYT]] are clean.
Cleanness gives a supply of idempotents and units that helps in decomposing modules.

## Explanations

### L1 human

In a field every element is either $0$ or invertible, and $0 = 1 + (-1)$ is an idempotent plus a unit.
A clean ring generalizes this: every element can be split as "a projection plus something invertible".
In the integers this fails: the idempotents are $0$ and $1$ and the units are $\pm 1$, so the only sums are $-1, 0, 1, 2$, and $3$ is not among them.