Field of fractions

Given an integral domain $𝐷$ , the field of fractions $F r a c 𝐷$ is the smallest field into which it can be embedded. #m/def/ring Let $𝐷 * = 𝐷 ∖ {0}$ . Then for any $𝑛, 𝑚 \in 𝐷$ and $𝑑, 𝑏 \in 𝐷 *$ , then $𝑛 𝑑, 𝑚 𝑏 \in F r a c 𝐷$ with

$𝑛 𝑑 = 𝑚 𝑏 ⟺ 𝑛 𝑏 = 𝑚 𝑑$
$𝑛 𝑑 + 𝑚 𝑏 = 𝑛 𝑏 + 𝑚 𝑑 𝑑 𝑏$
$𝑛 𝑑 \cdot 𝑚 𝑏 = 𝑛 𝑚 𝑑 𝑏$

which may be constructed as a quotient of the set $𝐷 \times 𝐷 *$ . We have the embedding

𝜄 𝐷 : 𝐷 ↪ F r a c 𝐷 𝑛 \mapsto 𝑛 𝑠 𝑠

for any $𝑠 \in 𝐷$ .

Proof of universal property

#missing/proof

Universal property

The field of fractions of $𝐷$ is a pair consisting of a field $F r a c 𝐷$ and injective ring homomorphism $𝜄 : 𝐷 ↪ F r a c 𝐷$ such that given any field $𝐾$ and injective ring homomorphism $𝑓 : 𝐷 \to 𝐾$ there exists a unique ring homomorphism $¯ 𝑓 : F r a c 𝐷 \to 𝐾$ so that the following diagram commutes

#state/tidy | #lang/en | #SemBr