# Commutative Ring Theory/Greatest common divisors

Definition (divisor):

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

Definition (greatest common divisor):

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

Definition (coprime):

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

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

Let ${\displaystyle }$