TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Every injective ring map from a domain into a field factors uniquely through its field of fractions

Statement

Let $D$ be an integral domain, $K$ a field, and $f : D \to K$ an injective unital ring homomorphism. There is a unique unital ring homomorphism $\tilde{f} : Frac (D) ⟶ K$ such that $\tilde{f} (d / 1) = f (d)$ for all $d \in D$ . It is injective and satisfies $\tilde{f} (a / b) = f (a) f (b)^{- 1}$ .

Facts & Assumptions

Given: An injective unital ring homomorphism $f : D \to K$ from an integral domain to a field.

[F1]

Every nonzero element of a field is a unit (Field).

[F2]

A map that sends every localisation denominator to a unit factors uniquely through the localisation, by the displayed fraction formula (Universal property of localisation: maps that invert $S$ factor uniquely through $S^{- 1} R$ ).

[F3]

The canonical map embeds $D$ in $Frac (D)$ ( $Frac (D)$ is a field and $d \mapsto d / 1$ embeds the integral domain $D$ ).

Proof

technique · direct

1.1

If $b \neq 0$ in $D$ , injectivity gives $f (b) \neq 0$ , so [F1] makes $f (b)$ a unit. Since the denominators defining $Frac (D)$ are exactly the nonzero elements, [F2] gives the unique extension and its formula.

F1F2

2.1

If $\tilde{f} (a / b) = 0$ , multiply $f (a) f (b)^{- 1} = 0$ by the unit $f (b)$ to obtain $f (a) = 0$ . Injectivity of $f$ gives $a = 0$ , hence $a / b = 0$ . Thus $\tilde{f}$ is injective.

step 1.1F3algebra∎

Depends on

Used by

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

The CRing Project, Chapter 13: Fields and Extensions (standard reference, not scraped)