PID and UFD are equivalent in a Dedekind domain
This article shows that if is a Dedekind domain, then is a UFD if and only if it is a PID. Note that this result implies the more specific result given in the article unique factorization and ideals in ring of integers.
Since any PID is a UFD, we need only prove the other direction. So assume is a UFD, let be a nonzero (proper) prime ideal, and choose . Note that is a nonunit since is a proper ideal. Since is a UFD, we may write uniquely (up to units) as where the are distinct irreducibles in , the are positive integers, and since is not a unit. Since is prime and , it follows that some , say , is in . Then . But is prime since clearly in a UFD any ideal generated by an irreducible is prime. Since is Dedekind and thus has Krull dimension 1, it must be that and thus is principal.
|Title||PID and UFD are equivalent in a Dedekind domain|
|Date of creation||2013-03-22 17:53:45|
|Last modified on||2013-03-22 17:53:45|
|Last modified by||rm50 (10146)|