Universal Property of Field of Rational Fractions

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Theorem

Let $R$ be an integral domain.

Let $\struct {\map R x, \iota, x}$ be the field of rational fractions over $R$.

Let $\struct {K, f, a}$ be an ordered triple, where:

$K$ is a field
$f : R \to K$ is a unital ring homomorphism
$a$ is a transcendental element of $K$.


Then there exists a unique unital ring homomorphism $\bar f : \map R x \to K$ such that $\bar f \circ \iota = f$ and $\map {\bar f} x = a$:

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


Proof

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