Let R be an integral domain and F a field such that R is a s

Let R be an integral domain and F a field such that R is a subring of F, and every element of F is equal to the quotient of two elements of R. Show that F is isomorphic to Q(R) (the field of quotients of R).

Solution

If x belongs to F, then x is of the form p/q where p and q are elements of R.

Thus let p/q be of the form simplest or p and q have no common factor between them.

Then each x can be expressed uniquely as p/q form.

Thus there exists a bijective mapping from F to Q(R)

Hence F is isomorphic to Q(R)