Universal Property for Field of Quotients

Theorem
Let $\left({D, +, \circ}\right)$ be an integral domain.

Let $\left({F, \oplus, \cdot}\right)$ be a quotient field of $D$.

Then $F$ satisfies the following universal property:

There exists a (ring) homomorphism $\iota : D \to F$ such that:
 * for every field $\tilde F$ and
 * and:
 * for every (ring) homomorphism $\phi : D \to \tilde F$
 * there exists a unique field homomorphism $\psi : F \to \tilde F$ satisfying:
 * $\psi \iota = \phi$

That is, the following diagram commutes:


 * FieldFracComDiag.jpg

Namely we may take:
 * $\psi : a / b \mapsto \phi \left({a}\right) \phi \left({b}\right)^{-1}$