Greatest common divisor

Statement

For a finite sequence

Let $R$ be a commutative unital ring and $a_1,a_2,\dots ,a_n\in R$. An element $d\in R$ is termed a greatest common divisor or gcd of $a_1,a_2,\dots ,a_n$ if it satisfies the following equivalent conditions:

• $d|a_i$ for all $1\le i\le n$ and if $c|a_i$ for all $1\le i\le n$, then $c|d$.
• $c|a_i$ for all $1\le i\le n$ if and only if $c|d$.
• The ideal $(d)$ is the intersection of all the principal ideals of $R$ containing $(a_1,a_2,\dots ,a_n)$.

The greatest common divisor of a finite set of elements is not unique; if two elements are both greatest common divisors of $a_1,a_2,\dots ,a_n$, then they are associate elements.

For any set

Let $R$ be a commutative unital ring and $S$ be a subset of $R$. An element $d\in R$ is termed a greatest common divisor of $S$ if it satisfies the following equivalent conditions:

• $d|a$ for all $a\in S$, and if $c|a$ for all $a\in S$, then $c|d$.
• $c|a$ for all $a\in S$ if and only if $c|d$.
• The ideal $(d)$ is the intersection of all the principal ideals of $R$ containing $S$.

Facts

• Greatest common divisors of the same subset form an associate class