Let $R$ be a commutative ring and let $a,b\in R$. An element $d\in R$ is a greatest common divisor of $a$ and $b$ if $d$ divides $a$ and $d$ divides $b$ and for every $d’\in R$ such that $d’$ divides $a$ and $d’$divides $b$, we have that $d’$ divides $d$.
Equivalently, $d$ is a greatest common divisor of $a$ and $b$ if the ideal generated by $d$ is the smallest principal ideal that contains both $a$ and $b$. This ideal is unique.
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