We were given this defintion in our Algebra $2$ course:
Definition $4.1$: Let $R$ be a commutative ring and $a, b\in R$. An element $d \in R$ is called a greatest common divisor, $\operatorname{gcd}(a, b)$, of $a$ and $b$ if:
• $d$ divides both $a$ and $b$ $($i.e., $\exists x$, $y \in R$ such that $a = dx$, $b = dy$$)$,
• If $e \in R$ divides both $a$ and $b$, then $e$ divides $d$.
I don't really understand how the second point could work with say the integers, for example $\operatorname{gcd}(10,12)$, if you pick $2$ to be the first $\operatorname{gcd}$, then for $e=1$ you have that it does divide both $10$ and $12$, and it does divide $2$. But then $2$ doesn't divide $1$ so $2$ is no longer a gcd according to this surely?