Non-unique factorization of an ideal in UFD

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?

• 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. – Michael Hardy Dec 22 '13 at 18:46
• A please would be nice – Tim Ratigan Dec 22 '13 at 18:49
• 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. – hansraj Dec 27 '13 at 16:40