Subgroup of Additive Group Modulo m is Ideal of Ring

Theorem
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:
 * $$h + \left({m}\right) \in H$$, where $$\left({m}\right)$$ is a principal ideal of $$\left({\Z_m, +_m, \times_m}\right)$$, and
 * $$n \in \N^*$$.

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 a Cyclic Group, $$n \cdot q_m \left({h}\right) \in H$$.

The result follows.