Integral domain

Not to be confused with domain of integration.

In mathematics, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. In an integral domain, every nonzero element a has the cancellation property, that is, if a ≠ 0, ab = ac implies b = c. Integral domains are generalizations of the ring of integers and provide a setting that is useful for studying divisibility.

"Integral domain" is defined almost universally as above, but there is some variation. This article follows the convention that rings have a multiplicative identity, generally denoted 1, but some authors do not follow this, by not requiring integral domains to have a multiplicative identity. Noncommutative integral domains are sometimes admitted. This article, however, follows the much more usual convention of reserving the term "integral domain" for the commutative case and using "domain" for the general case including noncommutative rings.

Some sources, notably Lang, use the term entire ring for integral domain.

Some specific kinds of integral domains are given with the following chain of class inclusions:

rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields
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