Example 10.35.8. A domain $R$ with finitely many maximal ideals $\mathfrak m_ i$, $i = 1, \ldots , n$ is not a Jacobson ring, except when it is a field. Namely, in this case $(0)$ is not the intersection of the maximal ideals $(0) \not= \mathfrak m_1 \cap \mathfrak m_2 \cap \ldots \cap \mathfrak m_ n \supset \mathfrak m_1 \cdot \mathfrak m_2 \cdot \ldots \cdot \mathfrak m_ n \not= 0$. In particular a discrete valuation ring, or any local ring with at least two prime ideals is not a Jacobson ring.

## Comments (0)

There are also:

• 7 comment(s) on Section 10.35: Jacobson rings

## 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Your email address will not be published.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 00G5. Beware of the difference between the letter 'O' and the digit '0'.