Q:

Prove a noetherian domain satisfying a special property is a PID

Let $R$ be a Noetherian domain, and assume that for all nonzero $a,b\in R$,
the greatest common divisors of $a$ and $b$ are linear combinations of $a$ and $b$. Prove that $R$ is a PID. 
Firstly I find I could prove it if I have the condition that gcd exists in the ringbut if we take the condition away I got stuck. 
Could you give me a hint？ 

A:

Let $I$ be an ideal of $R$, since $R$ is Noetherian, $I$ is generated by $(a_1,...,a_n)$ suppose that $n$ is minimal and $n>1$, let $d_1$ be the $\gcd(a_1,a_2)$, $d_1$ is a linear combination of $a_1$ and $a_2$ implies that $d_1\in I$, we deduce that $I=(d_1,a_3,..,a_n)$ contradiction since we have assumed that $I$ has at leat $n$ generators, so $n=1$ and $R$ is principal.