A nonzero commutative ring with unity that has no nonzero zero divisors; equivalently, a commutative ring R with 1 ≠ 0 such that ab = 0 implies a = 0 or b = 0.

Principle

The core rule is the cancellation of nonzero factors: products vanish only when one factor is zero, enabling a form of multiplicative cancellation and the embedding into a field of fractions.

Demonstration

The integers Z are an integral domain: if ab = 0 in Z then a = 0 or b = 0. Polynomial rings over a field K[x] are integral domains, so nonzero polynomials multiply to nonzero polynomials.

Misapplication

Assuming every integral domain is a unique factorization domain or that integrality implies integrally closed; for instance, not every integrally closed property or factorization property holds without extra hypotheses.

Consequence

From the absence of zero divisors one can construct the field of fractions, apply cancellation in multiplicative arguments, and develop divisibility theory similar to arithmetic in Z.

Reversal

A ring with zero divisors (e.g., Z/6Z) breaks cancellation: nonzero elements multiply to zero and many arguments using division or localization fail.

Boundary

Requires commutativity and unity and excludes the zero ring; noncommutative analogues or rings with zero divisors are outside this notion.

Semantic Tension

Tension arises between 'domain' as integral domain and other uses of 'domain' (function domain, domain of definition); also between integrality conditions and stronger factorization properties.

Synthesis

An integral domain is a commutative ring with identity that forbids nonzero zero divisors, enabling cancellation, a canonical field of fractions, and a foundation for studying divisibility and factorization under further hypotheses.