Integral domain

An integral domain is a commutative ring with $1\neq 0$ such that for all $a,b\in R$, $ab=0$ implies $a=0$ or $b=0$ (equivalently, $R$ has no zero divisors ).

Integral domains are the natural setting for divisibility and factorization, and every field is an integral domain. Many constructions (e.g. forming the field of fractions ) require the domain hypothesis.

Examples:

• $\mathbb{Z}$ is an integral domain.
• If $k$ is a field, then $k[x]$ is an integral domain.
• $\mathbb{Z}/6\mathbb{Z}$ is not an integral domain since $2\cdot 3=0$ in the quotient.