set of a quotient field
1. Proposition
Let \(R\) be an Integral domain, then the quotient field is the set of equivalence classes \([(a,b)]\) for \(a,b \in R\) with
\begin{align*} [(a,b)] = [(a',b')] \Leftrightarrow a \cdot b' = a' \cdot b \end{align*}with:
1.1. addition
\begin{align*}
[(a,b)] + [(a',b')] =& [(ab' + a'b, bb')]
\end{align*}
1.2. multiplication
\begin{align*}
[(a,b)] \cdot [(a',b')] =& [(a \cdot a',b \cdot b')]
\end{align*}
1.3. inverse
\begin{align*}
[(a,b)]^{-1} =& [(b,a)]
\end{align*}