In a UFD in which each maximal ideal is principal, ideal generated by two relatively prime elements is whole ring

Question

Suppose that $U$ is a UFD with the property that every maximal ideal in $U$ is principal. Then I'd like to prove that if $u_1, u_2 \in U$ with $1 = \gcd(u_1, u_2)$, then $( u_1 ) + ( u_2 ) = ( u_1,u_2 ) = U$.

I'm wondering if I can show that since $(u_1, u_2)$ is not principal, then it is not maximal by assumption. Hence, if I can find a maximal ideal $M$ such that $M \subset (u_1, u_2)$, then $(u_1, u_2) = U$.

Answer 1

Consider the maximal ideal which contains $u_1$ and $u_2)$ it is principal generated by $d$ which thus divides $u_1$ and $u_2$ since the $gcd(u_1,u_2)=1$, we deduce $d=1$.