Example 9.3.4 (Quotient fields). Recall that, given a domain $A$, there is an imbedding $A \to F$ into a field $F$ constructed from $A$ in exactly the same manner that $\mathbf{Q}$ is constructed from $\mathbf{Z}$. Formally the elements of $F$ are (equivalence classes of) fractions $a/b$, $a, b \in A$, $b \not= 0$. As usual $a/b = a'/b'$ if and only if $ab' = ba'$. The field $F$ is called the quotient field, or field of fractions, or fraction field of $A$. The quotient field has the following universal property: given an injective ring map $\varphi : A \to K$ to a field $K$, there is a unique map $\psi : F \to K$ making

$\xymatrix{ F \ar[r]_\psi & K \\ A \ar[u] \ar[ru]_\varphi }$

commute. Indeed, it is clear how to define such a map: we set $\psi (a/b) = \varphi (a)\varphi (b)^{-1}$ where injectivity of $\varphi$ assures that $\varphi (b) \not= 0$ if $b \not= 0$.

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