The embedding is given by for any nonzero . Examples The field of fractions of the ring of integers is the field of rationals: . Let be the ring of Gaussian integers. Then , the field of Gaussian rationals. The field of fractions of a field is canonically isomorphic to the field itself. Given a field , the field of fractions of the polynomial ring in one indeterminate (which is an integral domain), is called the , field of rational fractions, or field of rational expressions and is denoted . Generalizations Localization For any commutative ring and any multiplicative set in , the localization is the commutative ring consisting of fractions with and , where now is equivalent to if and only if there exists such that . Two special cases of this are notable: If is the complement of a prime ideal , then is also denoted .When is an integral domain and is the zero ideal, is the field of fractions of . If is the set of non-zero-divisors in , then is called the total quotient ring.The total quotient ring of an integral domain is its field of fractions, but the total quotient ring is defined for any commutative ring. Note that it is permitted for to contain 0, but in that case will be the trivial ring. Semifield of fractions The semifield of fractions of a commutative semiring with no zero divisors is the smallest semifield in which it can be embedded. The elements of the semifield of fractions of the commutative semiring are equivalence classes written as with and in . See also Ore condition; condition related to constructing fractions in the noncommutative case. Projective line over a ring; alternative structure not limited to integral domains. Total ring of fractions References Field (mathematics) Commutative algebra