Universal Property of Field of Rational Fractions

Theorem
Let $R$ be an integral domain.

Let $(R(x), \iota, x)$ be the field of rational fractions over $R$.

Let $(K, f, a)$ be an ordered triple, where:
 * $K$ is a field
 * $f : R \to K$ is a unital ring homomorphism
 * $a$ is an element of $K$.

Then there exists a unique unital ring homomorphism $\bar f : R(x) \to K$ such that $\bar f\circ\iota = f$ and $\bar f(x) = a$.


 * $\xymatrix{

R \ar[d]^\iota \ar[r]^{\forall f} & K\\ R(x) \ar[ru]_{\exists ! \bar f} }$

Proof
Use Universal Property of Polynomial Ring and Universal Poperty of Field of Fractions.