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 $R=\mathbb Z.$

| | | | | created: 2019-06-27 21:08:24 | modified: 2019-06-27 21:50:10 | by: bookofproofs | references: [8250]