Universal Property for Field of Quotients

Theorem
Let $\struct {D, +, \circ}$ be an integral domain.

Let $\struct {F, \oplus, \cdot}$ be a field of quotients 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:
 * $\psi: a / b \mapsto \map \phi a \map \phi b^{-1}$