Subgroup of Integers is Ideal

Theorem
Let $\struct {\Z, +}$ be the additive group of integers.

Every subgroup of $\struct {\Z, +}$ is an ideal of the ring $\struct {\Z, +, \times}$.

Proof
Let $H$ be a subgroup of $\struct {\Z, +}$.

Let $n \in \Z, h \in H$.

Then from the definition of cyclic group and Negative Index Law for Monoids:


 * $n h = n \cdot h \in \gen h \subseteq H$

The result follows.