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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 $R=\mathbb Z.$

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References

Bibliography

1. Koch, H.; Pieper, H.: "Zahlentheorie - Ausgewählte Methoden und Ergebnisse", Studienbücherei, 1976