07-05. Rings of fractions

• $R$,
  • commutative ring

Theorem 15.

• $R$,
  • commutative ring
• $D \neq \emptyset \subseteq R$,
  • $0 \notin D$,
  • $a \notin D$, for all $a$ is a zero divisor
  • $a, b \in D$, $a b \in D$,

Then there is a commutative ring $Q$ with 1 such that

\[R \subseteq Q\]

Moreover, $Q$ has the following properties:

• (1) $\forall a \in Q$, $\exists r \in R$, $\exists d \in D$ such that $a = r d^{-1}$.
  • In particular, if \(D := R \setminues \{0\}\), then $Q$ is a field
• (2) The ring $Q$ is the smallest ring containing $R$

proof

$\Box$