n. 1. (Number theory) a non-zero commutative ring with a (multiplicative) identity in which the (additive) zero has no zero divisors; for example the integers, but not the integers mod m unless m is prime. A ring is an integral domain if and only if ax = ay implies x = y, and if an integral domain is finite it is a field. 2. (Algebra) a non-zero commutative ring with no zero divisors, whether or not it possesses a multiplicative identity. Compare division ring, Euclidean domain. See cancellation law.