Definition: Generalization of the Greatest Common Divisor 

Let $(R,\cdot,+)$ be an integral domain with the multiplicative neutral element $1,$ and let $M\subseteq R$ be its subset. The element $a$ is called the greatest common divisor of $M,$ if and only if:

• $a\mid m\quad\forall m\in M$, i.e. $a$ is a divisor of all elements of $M.$
• $a’\mid m\quad\forall m\in M\Rightarrow a’\mid a$, i.e. if any other element $a’$ is dividing all elements of $M,$ then it is also dividing $a.$

We express these two conditions being fulfilled simultaneously for $a$ by writing $a=\gcd(M).$

Notes

• This definition corresponds to the special case of a greatest common divisor, if $R=\mathbb Z.$

| | | | | created: 2019-06-27 18:36:41 | modified: 2019-06-27 21:51:10 | by: bookofproofs | references: [8250]