3
$\begingroup$

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?

$\endgroup$
  • 1
    $\begingroup$ 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 Dec 22 '13 at 18:46
  • $\begingroup$ A please would be nice $\endgroup$ – Tim Ratigan Dec 22 '13 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 Dec 27 '13 at 16:40
0
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.