Cancellation in an integral domain

Statement

Let $R$ be an integral domain 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 $a$.