Intersection of Integer Ideals is Lowest Common Multiple

Theorem

Let $\left({m}\right)$ and $\left({n}\right)$ be ideals of the integers $\Z$.

Let $\left({k}\right)$ be the intersection of $\left({m}\right)$ and $\left({n}\right)$.

Then $k = \operatorname{lcm} \left\{{m, n}\right\}$.

Proof

By Intersection of Ideals we have that $\left({k}\right) = \left({m}\right) \cap \left({n}\right)$ is an ideal of $\Z$.

By Ring of Integers is Principal Ideal Domain we have that $\left({m}\right)$, $\left({n}\right)$ and $\left({k}\right)$ are all necessarily principal ideals.

By Subrings of the Integers we have that:

$\left({m}\right) = m \Z, \left({n}\right) = n \Z$

Thus:

$\left({k}\right) = \left\{{x \in \Z: n \mathop \backslash x \land m \mathop \backslash x}\right\}$

The result follows by LCM iff Divides All Common Multiples.

$\blacksquare$

Sources

• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 5.21$: Example $37$