Lemma 15.104.6. A product of fields is an absolutely flat ring.
Proof. Let $K_ i$ be a family of fields. If $f = (f_ i) \in \prod K_ i$, then the ideal generated by $f$ is the same as the ideal generated by the idempotent $e = (e_ i)$ with $e_ i = 0, 1$ according to whether $f_ i$ is $0$ or not. Thus $D(f) = D(e)$ is open and closed and we conclude by Lemma 15.104.5 and Algebra, Lemma 10.26.5. $\square$
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Your email address will not be published.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).
All contributions are licensed under the GNU Free Documentation License.