---
tag: 0GJM24EXP9
kind: statement
role: proposition
name: Cancellation in an integral domain
msc: [13G05]
uses: [09S3PFBEPM]
sources:
  - url: https://en.wikipedia.org/wiki/Integral_domain
    locator: Properties
---

## Statement

Let $R$ be an [[integral domain|09S3PFBEPM]] and let $a, b, c \in R$ with $a \neq 0$.
If $a b = a c$, then $b = c$.

## Remarks

The proof is short: $a b = a c$ gives $a (b - c) = 0$, and $a \neq 0$ gives $b - c = 0$.
The converse also holds: a nonzero commutative ring with this cancellation property is an integral domain.
Cancellation fails in $\mathbb{Z}/6\mathbb{Z}$: $2 \cdot 1 = 2 \cdot 4$, but $1 \neq 4$.

## Explanations

### L1 human

In the integers, you can cancel a nonzero factor from both sides of an equation.
From $3 b = 3 c$ you get $b = c$.
This works in every integral domain, even when you cannot divide by $a$.