The Stacks project

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$.

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 09FJ. Beware of the difference between the letter 'O' and the digit '0'.