Definition:Integral Domain

An integral domain $$\left({R, +, \circ,}\right)$$ is a:


 * commutative ring;
 * with a unity
 * in which there are no divisors of zero, that is:

$$\forall x, y \in R: x \circ y = 0_R \Longrightarrow x = 0_R \lor y = 0_R$$

... or alternatively, in which all non-zero elements are cancellable (which is an equivalent statement from Zero Divisor Not Cancellable).

Alternative Definition
An integral domain $$\left({R, +, \circ,}\right)$$ is a commutative ring such that $$\left({R^*, \circ}\right)$$ is a monoid all of whose elements are cancellable.

This follows from the fact that an integral domain is a non-null ring with unity with no zero divisors. The result follows from Ring Less Zero Semigroup.