An integral domain is a nonzero commutative ring without zero divisors. Equivalently, a product of two nonzero elements is always nonzero. The zero ring is excluded, and fails because . Every field is an integral domain, but the integers show that the converse fails: has no multiplicative inverse in . See MIT's Basic Properties of Rings, definitions 15.9–15.10 and example 15.11.
The practical benefit is multiplicative cancellation. In the integers, has at most one solution and indeed works. Yet has no integer solution. Thus multiplicative cancellation guarantees uniqueness when a solution exists; it does not guarantee division by every nonzero element. Passing to the fraction field supplies the missing solution .
The polynomial ring is also an integral domain. For example, the product of and has leading coefficient and cannot vanish. Generally the leading coefficient of a product is the product of the two leading coefficients; an integral domain prevents that coefficient from vanishing. This connects integer arithmetic to polynomial arithmetic through one reusable principle.
Historically, this connection grew far beyond elementary examples. Keith Conrad identifies the ring of integers of a number field and a ring of polynomial functions on a smooth plane curve as early important examples of Dedekind domains. This shared structure connects number theory with algebraic geometry; see Conrad's Ideal Factorization, section 6.
An **integral domain** is a specific type of algebraic structure in the field of abstract algebra. It is defined as a non-zero commutative ring with certain properties.
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