Subgroup of Additive Group Modulo m is Ideal of Ring

Theorem
Let $m \in \Z: m > 1$.

Let $\left({\Z_m, +_m}\right)$ be the additive group of integers modulo $m$.

Then every subgroup of $\left({\Z_m, +_m}\right)$ is an ideal of the ring of integers modulo $m$ $\left({\Z_m, +_m, \times_m}\right)$.

Proof
Let $H$ be a subgroup of $\left({\Z_m, +_m}\right)$

Suppose:
 * $(1): \quad h + \left({m}\right) \in H$, where $\left({m}\right)$ is a principal ideal of $\left({\Z_m, +_m, \times_m}\right)$

and
 * $(2): \quad n \in \N_{>0}$.

Then by definition of multiplication on integers and Homomorphism of Powers as applied to integers:

But:
 * $n \cdot q_m \left({h}\right) \in \left \langle {q_m \left({h}\right)}\right \rangle$

where $\left \langle {q_m \left({h}\right)}\right \rangle$ is the group generated by $q_m \left({h}\right)$.

Hence by Epimorphism from Integers to Cyclic Group, $n \cdot q_m \left({h}\right) \in H$.

The result follows.