# Intersection of Integer Ideals is Lowest Common Multiple

Jump to navigation
Jump to search

## Theorem

Let $\ideal m$ and $\ideal n$ be ideals of the integers $\Z$.

Let $\ideal k$ be the intersection of $\ideal m$ and $\ideal n$.

Then $k = \lcm \set {m, n}$.

## Proof

By Intersection of Ring Ideals is Ideal we have that $\ideal k = \ideal m \cap \ideal n$ is an ideal of $\Z$.

By Ring of Integers is Principal Ideal Domain we have that $\ideal m$, $\ideal n$ and $\ideal k$ are all necessarily principal ideals.

By Subrings of Integers are Sets of Integer Multiples we have that:

- $\ideal m = m \Z, \ideal n = n \Z$

Thus:

- $\ideal k = \set {x \in \Z: n \divides x \land m \divides x}$

The result follows by LCM iff Divides All Common Multiples.

$\blacksquare$

## Sources

- 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $5$: Rings: $\S 21$. Ideals: Example $37$