Intersection of Integer Ideals is Lowest Common Multiple

April 8, 2023:  It has been suggested that this page or section be merged into Intersection of Sets of Integer Multiples. In particular: It seems to be just a ring-theoretic statement of the theorem. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Mergeto}} from the code.

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 $37$