Example 10.34.8. 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.
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Your email address will not be published.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.