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.

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