It will be now shown that this when R is a factorial ring, and $a_{1},…,a_{k}$ in R are not all 0. Then
$a_{1},…,a_{k}$ not relatively prime $\Leftrightarrow$ there exists a prime $p$ so that $p|a_{1},…,p|a_{k}$
$(\Rightarrow )$ Assume $a_{1},…,a_{k}$ are not relatively prime, then there is a number p which divides $a_{1},…,a_{k}$. Now it is to show that p must also be a prime. Since R is a factorial ring, that means … stuck.
$(\Leftarrow) $ Assume $p|a_{1},…,p|a_{k}$, then $a_{1},…,a_{k}$ are trivially not relatively prime.
Does anybody see a way to use that R is a factorial ring to show that p must also be a prime?