Using Kaplansky condition, the proof of this statement is quite easy. Before knowing this condition, I tried to prove this statement by actually showing the definition of UFD holds on $S^{-1}A$ ($A$ factorial). On the course of this proof, (actually at the point where I need to prove the uniqueness of factorization after proving the existence of it--I've proved the existence) I needed to prove this for my proof steps to actually work. But this turned out false. (One of the textbooks I use actually use this statement without proving it so I also tired to employ it in my proof.)
So I'm quite stuck now. All my approaches I tried needed this at some point on the proof.
But I guess there should be an elegant proof of it, even without using the Kaplansky condition? Any suggestion or reference?
