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:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \left({n + \left({m}\right)}\right) \times \left({h +\left({m}\right)}\right)\)	\(=\)	\(\displaystyle q_m \left({n}\right) \times q_m \left({h}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	where $q_m$ is the quotient mapping
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle q_m \left({n \times h}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle q_m \left({n \cdot h}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle n \cdot q_m \left({h}\right)\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

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.

$\blacksquare$

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 25$: Theorem $25.4$