1. ## A simple problem about prime and irreducible

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?

2. Originally Posted by ynj
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?
of course. just choose $\displaystyle p$ to be any prime element in the prime factorization of $\displaystyle \gcd(a,b)$.

3. 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"?

4. oh,sorry, i have just searched the internet, and find a proof which says that irreducible and prime are equivalent in UFD