Ideals of Ring of Integers Modulo m

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 $\struct {\Z_m, +, \times}$.

Recall from Ring of Integers Modulo m is Ring that $\struct {\Z_m, +}$ is the additive group of integers modulo $m$.

From Integers Modulo $m$ under Addition form Cyclic Group, $\struct {\Z_m, +}$ is a cyclic group.

By definition of ideal, $\struct {J, +}$ is a subgroup of $\struct {\Z_m, +}$.

Let $\struct {G, +}$ be a subgroup of $\struct {\Z_m, +}$.

From Subgroup of Cyclic Group is Cyclic $\struct {G, +}$ is a cyclic group.

Hence by definition of cyclic group, $\struct {G, +}$ can be generated by $d$ and denoted $\gen d$, where $d \divides m$.

By Lagrange's Theorem, for a finite cyclic group of order $k$, the order of every subgroup is a divisor of $k$.

By Bijection from Divisors to Subgroups of Cyclic Group there is exactly one 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:

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

$\set {0, 3, 6, 9}$

$\set {0, 4, 8}$

$\set {0, 6}$