Ideals of Ring of Integers Modulo m

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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

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:

$\set {0, 2, 4, 6, 8, 10}$
$\set {0, 3, 6, 9}$
$\set {0, 4, 8}$
$\set {0, 6}$