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
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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