Integrity, Domain of

a concept of modern algebra. The term “domain of integrity” was originally applied to the set K of algebraic integers belonging to some field P of algebraic numbers. Every number in P can be represented as the ratio of two numbers in K.

In present-day usage a domain of integrity, or integral domain, is a commutative ring where the product of nonzero elements is never zero; that is, an integral domain is a commutative ring without proper divisors of zero. Examples of integral domains are rings whose elements are numbers and the ring of polynomials with coefficients from a given field.