[1][2] Integral domains are generalizations of the ring of integers and provide a natural setting for studying divisibility.
In an integral domain, every nonzero element a has the cancellation property, that is, if a ≠ 0, an equality ab = ac implies b = c. "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.
[5] 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.
Equivalently: The following rings are not integral domains.
The converse is not true in general: for example, in the quadratic integer ring
The field of fractions K of an integral domain R is the set of fractions a/b with a and b in R and b ≠ 0 modulo an appropriate equivalence relation, equipped with the usual addition and multiplication operations.
Integral domains are characterized by the condition that they are reduced (that is x2 = 0 implies x = 0) and irreducible (that is there is only one minimal prime ideal).
It follows that the unique minimal prime ideal of a reduced and irreducible ring is the zero ideal, so such rings are integral domains.
The converse is clear: an integral domain has no nonzero nilpotent elements, and the zero ideal is the unique minimal prime ideal.
The characteristic of an integral domain is either 0 or a prime number.
If R is an integral domain of prime characteristic p, then the Frobenius endomorphism x ↦ xp is injective.