Subgroup of Additive Group Modulo m is Ideal of Ring
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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:
\(\ds \paren {n + \ideal m} \times \paren {h + \ideal m}\) | \(=\) | \(\ds \map {q_m} n \times \map {q_m} h\) | where $q_m$ is the quotient mapping | |||||||||||
\(\ds \) | \(=\) | \(\ds \map {q_m} {n \times h}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map {q_m} {n \cdot h}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds n \cdot \map {q_m} h\) |
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.
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $25$. Cyclic Groups and Lagrange's Theorem: Theorem $25.4$