Let R be a unitary commutative ring: let $a,b,d\in R$. I'm trying to prove that
$\langle d \rangle = \langle a,b \rangle \Leftrightarrow d = \gcd(a,b)$
I can prove the $\Rightarrow$ part. The $\Leftarrow$ part is easy to prove in a ring in which every ideal is principal, but does it holds in a generic unitary commutative ring?
[Edit] For $\gcd(a,b)$ i mean that let be $d$ an element of $R$ so $d\in \gcd(a,b)$ iff $d|a$ and $d|b$ and $\forall d' \in R: d'|a \land d'|b \Rightarrow d'|d$ . The equal sign I used before is a light abuse of notation since uniqueness is not granted using this definition.