In abstract algebra, an integral domain is a commutative ring with 1 ≠ 0 that has no zero divisors. Integral domains are generalizations of the integers and provide a natural setting for studying divisibility. An integral domain is a commutative domain with identity.
Viewing the underlying commutative ring as a preadditive category, the above criterion on zero divisors is equivalent to the condition that every nonzero morphism is a monomorphism (hence also an epimorphism, by making use of the bilinear structure on the set of morphisms).
The condition 1 ≠ 0 only serves to exclude the trivial ring {0}.
The above is how "integral domain" is almost universally defined, but there is some variation. In particular, noncommutative integral domains are sometimes admitted, and very rarely the condition 1 ≠ 0 is omitted. However, we follow the much more usual convention of reserving the term integral domain for the commutative case and use domain for the noncommutative case. Some sources, notably Lang, use the term entire ring for integral domain
