Commutative Ring Theory/Greatest common divisors

Definition (divisor):

Let $R$ be a ring, and let $a\in R$. A divisor of $a$ is an element $b\in R$ such that there exists $c\in R$ such that $a=bc$. The notation $b|a$ indicates that $b$ is a divisor of $a$

Definition (greatest common divisor):

Let $R$ be a commutative ring, and let $a_{1},\ldots ,a_{n}\in R$. A greatest common divisor is an element $d\in R$ such that $d|a_{j}$ for all $j\in \{1,\ldots ,n\}$, and such that for any other element $c\in R$ such that $c|a_{j}$ for all $j\in \{1,\ldots ,n\}$, we have $c|d$.

Definition (coprime):

Let $R$ be a commutative ring, and let $a_{1},\ldots ,a_{n}\in R$. These elements $a_{1},\ldots ,a_{n}$ are said to be coprime if and only if whenever $d\in R$ is such that $d|a_{j}$ for all $j\in \{1,\ldots ,n\}$, then $d\in R^{\times }$.

Proposition (a set of elements of a commutative ring divided by their greatest common divisor is coprime):

Let ${\displaystyle }$