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 idealMathworldPlanetmathPlanetmathPlanetmath in R contains a prime elementMathworldPlanetmath.

On the other hand, recall that an element pR is prime if and only if an ideal (p) generated by p is nonzero and prime.

Thus, if P is a nonzero prime ideal in R, then (since R is a PID) there exists pR such that P=(p). This completes the proof.

Title every PID is a UFD - alternative proof
Canonical name EveryPIDIsAUFDAlternativeProof
Date of creation 2013-03-22 19:04:26
Last modified on 2013-03-22 19:04:26
Owner joking (16130)
Last modified by joking (16130)
Numerical id 5
Author joking (16130)
Entry type Theorem
Classification msc 13F07
Classification msc 16D25
Classification msc 13G05
Classification msc 11N80
Classification msc 13A15