Field of quotients

English

Template:Lb A field all of whose elements can be represented as ordered pairs each of whose components belong to a given integral domain, such that the second component is non-zero, and so that the additive operator is defined like so: $(a, b) + (a^{'}, b^{'}) = (a b^{'} + a^{'} b, b b^{'})$ , the multiplicative operator is defined coordinate-wise, the zero is $(0, 1)$ , the unity is $(1, 1)$ , the additive inverse of $(a, b)$ is $(- a, b)$ , equivalence is defined like so: $(a, b) \equiv (a^{'}, b^{'})$ if and only if $a b^{'} = a^{'} b$ , and multiplicative inverse of a non-zero–equivalent element $(a, b)$ is $(b, a)$ .