Definition:Integral Domain

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This page is about the concept of integral domain in ring theory. For other uses, see Definition:Domain.

Definition 1

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

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

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

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: $D \setminus \left\{{0_D}\right\}$.

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.