The Field of Quotients Over an Integral Domain

Let I be a non degenerated non empty multiplicative loop with zero structure and let u be an element of Q(I). Then u1 is an element of I. Then u2 is an element of I. Let I be a non degenerated integral domain-like non empty double loop structure and let u, v be elements of Q(I). The functor u+ v yields an element of Q(I) and is defined by: (Def. 2) u+ v = 〈u1 · v2 + v1 ·u2, u2 · v2〉. Let I be a non degenerated integral domain-like non empty double loop structure and let u, v be elements of Q(I). The functor u · v yields an element of Q(I) and is defined as follows: (Def. 3) u · v = 〈u1 · v1, u2 · v2〉.