Least common multiple - Imported Knowlpedia Content

Let $R$ be an integral domain and let $a,b\in R$ . A least common multiple of $a$ and $b$ is an element $m\in R$ such that:

$a\mid m$ and $b\mid m$ , and
if $a\mid n$ and $b\mid n$ , then $m\mid n$ .

Remarks

An lcm is unique up to associates. In settings where gcds exist, one often has the relation $m\cdot d$ is associate to $ab$ , where $d$ is a gcd of $a$ and $b$ .

Examples

In $\mathbb{Z}$ , $\mathrm{lcm}(12,18)=36$ .
In $k[x]$ , $\mathrm{lcm}(x,x^2)=x^2$ (up to multiplication by a nonzero scalar).
For any $a\in R$ , a least common multiple of $a$ and $0$ is $0$ .