Subgroup of Additive Group Modulo m is Ideal of Ring

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

Let $\struct {\Z_m, +_m}$ be the additive group of integers modulo $m$.

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

Proof
Let $H$ be a subgroup of $\struct {\Z_m, +_m}$

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

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

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

But:
 * $n \cdot \map {q_m} h \in \gen {\map {q_m} h}$

where $\gen {\map {q_m} h}$ is the group generated by $\map {q_m} h$.

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

The result follows.