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I know that the factorization of a nontrivial ideal into prime ideals is unique in a Dedekind domain. Not all UFDs are Dedekind domains, so there must be a UFD in which there exists a nonzero ideal with non-unique factorization into prime ideals.

In non-Dedekind UFD $\mathbb{Z}[x]$, the ideal $(2, x)$ is not principal, but it has unique factorization. So this attempt fails.

Would you please provide an example of a UFD in which the unique factorization of a nonzero ideal into prime ideals is not possible?

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    $\begingroup$ Phrasing a question in this way, in a manner appropriate for a homework assignment, tends not to be welcomed here. It makes it look as if you're passing on to us a question written by someone other than yourself without including any of your own thoughts about it. $\endgroup$
    – Michael Hardy
    Commented Dec 22, 2013 at 18:46
  • $\begingroup$ A please would be nice $\endgroup$
    – Tim Ratigan
    Commented Dec 22, 2013 at 18:49
  • $\begingroup$ In Z[x], the ideal <2, x> is not principal. I am aware of the result that the factorization of a nontrivial ideal into prime ideals uniquely is possible in a Dedekind domain. Not all UFD are Dedekind domain, so there must be a UFD in which there exist a nonzero ideal with non-unique factorization into prime ideals. But I am unable to find an example. $\endgroup$
    – hansraj
    Commented Dec 27, 2013 at 16:40

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If you knew that in a Dedekind domain UFD and PID are equivalent, would you be able to come up with an example yourself?

Unique factorization domain that is not a Principal ideal domain

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