Set of Integer Multiples is Integral Ideal

Theorem
Let $m \in \Z$ be an integer.

Let $m \Z$ denote the set of integer multiples of $m$.

Then $m \Z$ is an integral ideal.

Proof
First note that $m \times 0 \in m \Z$ whatever $m$ may be.

Thus $m \Z \ne \O$.

Let $a, b \in m \Z$.

Then:

and:

Let $r \in \Z$.

Then:

Thus the conditions for $m \Z$ to be an integral ideal are fulfilled.

Hence the result.