Subgroup of Integers is Ideal

Theorem
Let $$\left({\mathbb{Z}, +}\right)$$ be the Additive Group of Integers.

Every subgroup of $$\left({\mathbb{Z}, +}\right)$$ is an ideal of the ring $$\left({\mathbb{Z}, +, \times}\right)$$.

Proof
Let $$H$$ be a subgroup of $$\left({\mathbb{Z}, +}\right)$$.

Let $$n \in \mathbb{Z}, h \in H$$. Then from the definition of cyclic group and Negative Index Law for Monoids:

$$n h = n \cdot h \in \left \langle {h} \right \rangle \subseteq H$$

The result follows.