Ring of Integers is Principal Ideal Domain

Theorem
The integers $$\mathbb{Z}$$ form a principal ideal domain.

Proof
Let $$J$$ be an ideal of $$\mathbb{Z}$$.

Then $$J$$ is a subring of $$\mathbb{Z}$$, and so $$\left({J, +}\right)$$ is a subgroup of $$\left({\mathbb{Z}, +}\right)$$.

But by Integers Infinite Cyclic Group, the group $$\left({\mathbb{Z}, +}\right)$$ is cyclic, generated by $$1$$.

Thus by Subgroup of a Cyclic Group is Cyclic, $$\left({J, +}\right)$$ is cyclic, generated by some $$m \in \mathbb{Z}$$.

Therefore from the definition of Principal Ideal, $$J = \left\{{k m: m \in \mathbb{Z}}\right\} = \left({m}\right)$$, and is thus a principal ideal.