every PID is a UFD - alternative proof
Proof. Recall, that due to Kaplansky Theorem (see this article (http://planetmath.org/EquivalentDefinitionsForUFD) for details) it is enough to show that every nonzero prime ideal in contains a prime element.
On the other hand, recall that an element is prime if and only if an ideal generated by is nonzero and prime.
Thus, if is a nonzero prime ideal in , then (since is a PID) there exists such that . This completes the proof.
|Title||every PID is a UFD - alternative proof|
|Date of creation||2013-03-22 19:04:26|
|Last modified on||2013-03-22 19:04:26|
|Last modified by||joking (16130)|