Field Of Quotients Definition

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).