In Z[x], the ideal <2, x> is not principal. I am that the factorization of a nontrivial ideal into prime ideals is unique 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.
Would you please provide an example of a unique factorization domain in which the factorization of a nonzero ideal into prime ideals is not possible?