Given an integral domain, the field of fractions is the smallest field into which it can be embedded. #m/def/ring
Let .
Then for any and ,
then with
which may be constructed as a quotient of the set .
We have the embedding
for any .
Proof of universal property
#missing/proof
Universal property
The field of fractions of is a pair consisting of a field and injective ring homomorphism
such that given any field and injective ring homomorphism
there exists a unique ring homomorphism so that the following diagram commutes