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:

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

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 $0$.