Definition:Integral Domain

Definition 1
An integral domain $\left({D, +, \circ}\right)$ is a:


 * commutative ring which is non-null
 * with a unity
 * in which there are no (proper) zero divisors, that is:


 * $\forall x, y \in D: x \circ y = 0_D \implies x = 0_D \text{ or } y = 0_D$

... or alternatively (from the Cancellation Law of Multiplication) in which all non-zero elements are cancellable.

Definition 2
An integral domain $\left({D, +, \circ}\right)$ is a commutative ring such that $\left({D^*, \circ}\right)$ is a monoid, all of whose elements are cancellable.

In this context, $D^*$ denotes the ring $D$ without zero: $D \setminus \left\{{0_D}\right\}$.

The two definitions are equivalent.

Also known as
Some authors refer to this concept as simply a domain.

However, this conflicts with the concept of domain in set theory, in the context of mappings and relations.

Therefore, it is always best to refer to an integral domain, as to avoid possible confusion.