Please give me an example principal ideal domain that is not a Jacobson ring. Its better this ring be commutative.


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  • $\begingroup$ You have to look for a principial ideal domain with vanishing jacobson radical. $\endgroup$ – user302982 Jan 9 '16 at 9:16
  • $\begingroup$ I meant non vanishing. Look here en.wikipedia.org/wiki/Jacobson_ring $\endgroup$ – user302982 Jan 9 '16 at 9:22
  • $\begingroup$ It hasn't example for my question. i saw it. $\endgroup$ – ruholla kh Jan 9 '16 at 11:52

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.


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