Integral domains and fraction fields

An integral domain is a nonzero commutative ring with identity and no nonzero zero divisors. Thus a product is zero only if at least one factor is zero. Subtracting equal products proves that every nonzero factor is cancellable. Fields are domains, and the integers are a domain without being a field. A polynomial ring over a domain is also a domain, because the leading coefficients of two nonzero polynomials multiply to a nonzero leading coefficient.

The nonzero condition excludes the zero ring, where the identity equals zero. Commutativity matters for the simple fraction construction used here. Our fractions will move denominators freely through products; more general noncommutative constructions require additional hypotheses. Residue rings modulo a composite positive integer are not domains, since nontrivial factors of the modulus produce nonzero classes with zero product. These distinctions determine which algebraic procedures are available before any calculation begins.

Start with pairs consisting of a numerator a and a nonzero denominator b. Declare the pair for a over b equivalent to the pair for c over d when ad equals bc. This is the cross-multiplication rule familiar from rational numbers. Reflexivity and symmetry are immediate; transitivity uses cancellation of a nonzero denominator. That is one place where the absence of zero divisors enters the construction rather than merely simplifying a later theorem.

Define addition by a common denominator and define multiplication by multiplying numerators and denominators. Denominator products remain nonzero in a domain, and algebra shows that equivalent input pairs give equivalent results. The class of a over one embeds the original ring injectively. A nonzero fraction has nonzero numerator, so exchanging numerator and denominator gives its inverse. These properties make the collection a field, called the fraction field of the domain.

Taking the fraction field of polynomials over a field gives rational functions. A rational function is an algebraic fraction class, not merely a formula evaluated at a chosen real input. For example, the fractions with numerator x squared minus one and denominator x minus one, and with numerator x plus one and denominator one, represent the same rational function. The polynomial identity after cross multiplication proves the equality without selecting a numerical x.

However, the two displayed expressions have different immediate evaluation domains as real formulas: the first cannot be substituted at one without cancellation. Keep the rational-function identity separate from a claim that an unevaluated denominator can be set to zero. Finally, a ring with a nonzero zero divisor cannot embed in any field while preserving its products and distinct elements. Its nonzero annihilating factors would become nonzero field elements with product zero, contradicting the field's domain property.