Ideals of Ring of Integers Modulo m
Contents
Theorem
Let $m \in \Z_{>0}$ be a (strictly) positive integer.
Let $\struct {\Z_m, +, \times}$ denote the ring of integers modulo $m$.
The ideals of $\struct {\Z_m, +, \times}$ are of the form:
- $d \Z / m \Z$
where $d$ is a divisor of $m$.
Proof
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Let $J$ be an ideal of $\struct {\Z_m, +, \times}$.
$\struct {J, +}$ is a subgroup of $\struct {\Z_m, +}$.
Let $\struct {G, +}$ be a subgroup of $\struct {\Z_m, +}$.
Then $\struct {G, +}$ is a cyclic subgroup generated by $\gen d$, where $d \divides m$.
We know that for a finite cyclic group of order $k$, the order of every subgroup is a divisor of $k$.
Also there is exactly one [Definition:Subgroup|subgroup]] for each divisor.
It follows that all ideals of $\struct {\Z_m, +, \times}$ are of form $\gen d$, where $d$ is a positive divisor of $m$.
$\blacksquare$
Examples
Order $12$
Let $\struct {\Z_{12}, +, \times}$ denote the ring of integers modulo $12$.
The underlying sets of the ideals of $\struct {\Z_{12}, +, \times}$ are:
- $\set {0, 2, 4, 6, 8, 10}$
- $\set {0, 3, 6, 9}$
- $\set {0, 4, 8}$
- $\set {0, 6}$
