For questions regarding integral domains, their structures, and properties. This tag should probably be accompanied by the Ring Theory tag. This tag is not for use for questions regarding integrals in analysis and calculus.

An integral domain is a commutative ring with identity and no zero divisors. That is, a commutative ring $R$ with $1$ is an integral domain if and only if for all $a, b \in R$,

$$ab = 0 \implies a = 0 \text{ or } b = 0$$

Alternatively, a commutative ring with $1$ in which the ideal $\{0\}$ is prime is an integral domain. Note that some authors do not require an integral domain to have a unit $1$.

The prototypical example of an integral domain is the ring of integers, $\Bbb{Z}$, and all fields are integral domains.

Source: Integral domain on Wikipedia.