Least common multiple

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

Remarks

An lcm is unique up to associates. In a principal ideal domain or a unique factorization domain, if $d$ is a gcd of $a,b$ and $m$ is an lcm, then $dm$ is associate to $ab$ .

Examples

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