Let D be a UFD. $\displaystyle a,b\in D$, if $\displaystyle gcd(a,b)$is not a unit, will there always exists a prime $\displaystyle p$that is the common divisor of $\displaystyle a,b$?why?
prime factoralization? but the definition of UFD says that any nonunit a can be factorlized in to product of irreducible elements, say $\displaystyle a=p_1...p_n$, where $\displaystyle p_1...p_n$. Do you mean that $\displaystyle p_i$is prime,or there is other theorem named"prime factoralization"?