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 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.

The two definitions are equivalent.

Note on Name
Some authors refer to this concept as simply a domain.

However, this conflicts with the concept of domain in mapping and relation theory.

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