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Definition: Generalization of the Least Common Multiple

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

• $m\mid a\quad\forall m\in M$, i.e. all elements $m\in M$ are divisors of $a$, i.e. $a$ is a common multiple of $M$, and
• $m\mid a'\quad\forall m\in M\Rightarrow a\mid a'$, i.e. $a$ divides any other common multiple $a'$ of $M.$

We express these two conditions being fulfilled simultaneously for $a$ by writing $a=\operatorname{lcm}(M).$

Notes

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

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References

Bibliography

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