Universal Property of Field of Rational Fractions

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.

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.