Lemma 10.35.14. Let $R$ be a Jacobson ring. Let $f \in R$. The ring $R_ f$ is Jacobson and maximal ideals of $R_ f$ correspond to maximal ideals of $R$ not containing $f$.

**Proof.**
By Topology, Lemma 5.18.5 we see that $D(f) = \mathop{\mathrm{Spec}}(R_ f)$ is Jacobson and that closed points of $D(f)$ correspond to closed points in $\mathop{\mathrm{Spec}}(R)$ which happen to lie in $D(f)$. Thus $R_ f$ is Jacobson by Lemma 10.35.4.
$\square$

## 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Your email address will not be published.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).

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.

## Comments (1)

Comment #1070 by Matthieu Romagny on

There are also: