Greatest Common Divisor (in a Ring)

definitioncommutative-algebra

Definition

Take a commutative ring $R$, and elements $a,b\in R$ with $b\ne 0$, then we say that $a$ is a multiple of $b$ if there exists some $x\in R$ such that $a=bx$, or alternatively that $b$ divides $a$, denoted $b\mid a$. Furthermore, we say a greatest common divisor, of two elements $a,b\in R$ is some element $d\in R$ such that $d\mid a$ and $d\mid b$, and if $d^{\prime }\mid a$ and $d^{\prime }\mid b$ then $d^{\prime }\mid d$.

In different language, we have that given an ideal $I$ generated by $a,b\in R$, $d$ is the greatest common divisor of $a,b$ if

1. $I\subseteq (d)$, and
2. If $(d^{\prime })$ is an ideal containing $I$, then $(d)\subseteq (d^{\prime })$.

Examples