Please give me an example principal ideal domain that is not a Jacobson ring. Its better this ring be commutative.
2026-03-30 12:22:52.1774873372
I want an example of principal ideal domain that is not a Jacobson ring (PID but non-Jacobson ring)
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Simply $\Bbb Z_{(2)}$ works: the localization of the integers at the complement of the prime ideal $2\Bbb Z$.
A localization of a PID is still a PID, and now the ring is local with a very large Jacobson radical, so zero is not an intersection of maximal ideals.