Integral Domain

An integral domain is a commutative ring with $1 \neq 0$ and no zero divisors: if $a \cdot b = 0$, then $a = 0$ or $b = 0$.

The absence of zero divisors means cancellation works: if $a \cdot b = a \cdot c$ and $a \neq 0$, then $b = c$. The integers $\mathbb{Z}$ and polynomial rings over fields are integral domains. Every field is an integral domain, but not every integral domain is a field.

An integral domain can always be embedded into a field — its field of fractions — just as $\mathbb{Z}$ embeds into $\mathbb{Q}$.