Intersection of Integer Ideals is Lowest Common Multiple

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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